{"text":"\\section{Introduction}\n\\noindent Wave maps $U: \\R^{1,3}\\to\\S^3$ from $(1+3)$-dimensional\nMinkowski space into the three-sphere are defined as critical points\nof the action functional\n\\begin{equation}\n \\label{eq:wmaction} \\int_{\\R^{1,3}}\\eta^{\\mu\\nu}\\partial_\\mu U^a\\partial_\\nu U^b\n g_{ab}\\circ U, \n\\end{equation}\nwhere $\\eta$ is the Minkowski metric, $g$ is the standard round metric\non $\\S^3$, and Einstein's summation convention is in force. By\nchoosing spherical coordinates $(t,r,\\theta,\\varphi)$ on Minkowski\nspace and hyperspherical coordinates on the three-sphere, one may\nrestrict oneself to so-called co-rotational maps which take the form\n$U(t,r,\\theta,\\varphi)=(\\psi(t,r),\\theta,\\varphi)$. Under this\nsymmetry reduction, the Euler-Lagrange equations associated to the\naction \\eqref{eq:wmaction} reduce to the single scalar wave equation\n\\begin{equation}\n \\label{eq:main}\n \\left (\\partial_t^2-\\partial_r^2-\\frac{2}{r}\\partial_r \\right )\\psi(t,r)+\\frac{\\sin(2\\psi(t,r))}{r^2}=0\n\\end{equation}\nfor the angle $\\psi$. Note that the singularity at the center $r=0$\nenforces the boundary condition $\\psi(t,0)=\\frac{m}{2}\\pi$ for $m\\in\n\\Z$. To begin with, we restrict ourselves to $m=0$. By\ntesting Eq.~\\eqref{eq:main} with $\\partial_t \\psi(t,r)$, we obtain the\nconserved energy\n\\begin{equation}\n \\label{eq:energy}\n \\int_0^\\infty \\left\n [\\tfrac12 |\\partial_t\\psi(t,r)|^2+ \\tfrac12 |\\partial_r\\psi(t,r)|^2+\\frac{\\sin^2(\\psi(t,r))}{r^2}\\right\n ]r^2 dr\n\\end{equation}\nand finiteness of the latter requires\n$\\lim_{r\\to\\infty}\\psi(t,r)=n\\pi$ for $n\\in \\Z$. \n\nDespite the existence of a positive definite energy,\nEq.~\\eqref{eq:main} develops singularities in finite time. This was\nfirst demonstrated by Shatah \\cite{Sha88} who constructed a\nself-similar solution $\\psi_T(t,r)=f_0(\\frac{r}{T-t})$ to\nEq.~\\eqref{eq:main} by a variational argument. Here, $T>0$ is a free\nparameter (the \\emph{blowup time}). In fact, $f_0(\\rho)=2\\arctan\\rho$,\nas was observed later \\cite{TurSpe90}. The solution $\\psi_T(t,r)$ is\nperfectly smooth for $t0$, and this leads to the more\ngeneral blowup solution\n\\[ \\psi_T^*(t,r):=4\\arctan\\left\n (\\frac{r}{T-t+\\sqrt{(T-t)^2+r^2}}\\right ).\n\\]\nThe skeptical reader may check by a direct computation that $\\psi_T^*$\nis indeed a solution to Eq.~\\eqref{eq:main} for all $t\\in \\R$ and\n$r>0$. The point is that $\\psi_T^*$ is smooth everywhere away from\nthe center and thus, $\\psi_T^*$ extends $\\psi_T$ smoothly beyond the\nblowup time $t=T$. Furthermore,\n\\[ \\lim_{r\\to 0+}\\psi(t,r)=2\\pi \\]\nif $t>T$, whereas $\\psi(t,0)=0$ for $t0$ we have $\\lim_{t\\to\\infty}\\psi_T^*(t,r)=2\\pi$.\n\n\n\\subsection{Statement of the main result}\nIn view of the boundary condition $\\psi(t,0)=0$ it is\nnatural to switch to the new variable\n\\[ \\widehat u(t,r):=\\frac{\\psi(t,r)}{r}. \\]\nIn terms of $\\widehat u$, Eq.~\\eqref{eq:main} reads\n\\begin{equation}\n \\label{eq:mainuhat}\n \\left (\\partial_t^2-\\partial_r^2-\\frac{4}{r}\\partial_r \\right\n )\\widehat u(t,r)+\\frac{\\sin(2r\\widehat u(t,r))-2r\\widehat u(t,r)}{r^3}=0,\n\\end{equation}\nwhich is a radial, semilinear wave equation in 5 space dimensions. For\nnotational purposes it is convenient to rewrite\nEq.~\\eqref{eq:mainuhat} as\n\\begin{equation}\n \\label{eq:mainu}\n \\left (\\partial_t^2-\\Delta_x\\right )u(t,x)=\\frac{2|x|u(t,x)-\\sin(2|x|u(t,x))}{|x|^3}\n\\end{equation}\nfor $u: \\R\\times \\R^5\\to \\R$ given by $u(t,x)=\\widehat u(t,|x|)$. By\nthe above, Eq.~\\eqref{eq:mainu} has the explicit one-parameter family\n$\\{u_T^*: T\\in \\R\\}$ of blowup solutions given by\n\\[ u_T^*(t,x):=\\frac{4}{|x|}\\arctan\\left\n (\\frac{|x|}{T-t+\\sqrt{(T-t)^2+|x|^2}}\\right ). \\]\nWe introduce the following spacetime region, depicted in\nFig.~\\ref{fig:Omega}.\n\\begin{definition}\n For $T,b\\in \\R$ we set\n \\[ \\Omega_{T,b}:=\\{(t,x)\\in \\R\\times \\R^5: 0\\leq t\\frac{d}{2}$,\nwe have $H^k(\\R^d)\\hookrightarrow\nC(\\R^d)$ and we denote by\n$H^k_{\\mathrm{rad}}(\\R^d)$\nand $H^k_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^d)$\nthe subsets of $H^k(\\R^d)$\nand $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^d)$, respectively, that consist of radial functions.\n\nAs usual, $A\\lesssim B$ means that there exists a constant $C>0$ such\nthat $A\\leq CB$. Possible dependencies of the implicit constant $C$ on\nadditional parameters follow from the context. \nWe also write $A\\simeq B$ if $A\\lesssim B$\nand $B\\lesssim A$. In general, the letter $C$ is used to denote a\nconstant that may change its value at each occurrence. For the sake of\nclarity we sometimes indicate dependencies on additional parameters by subscripts.\n\nWe follow the tradition in relativity and number the slots of functions\ndefined on Minkowski space $\\R^{1,d}$ starting at $0$, i.e.,\n$\\partial_0 u(t,x)=\\partial_t u(t,x)$. In general, Greek indices run\nfrom $0$ to $d$ whereas Latin indices run from $1$ to $d$ and\nEinstein's summation convention is in force. For the signature of the\nMinkowski metric we use the convention that spacelike vectors have\npositive lengths. \n\nFor a linear operator $\\mathbf} \\newcommand{\\mc}{\\mathcal L$ on a Banach space we denote by $\\mc\nD(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$, $\\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$, and $\\sigma_p(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ its domain,\nspectrum, and point spectrum, respectively. Furthermore, for\n$\\lambda\\in \\rho(\\mathbf} \\newcommand{\\mc}{\\mathcal L):=\\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}\\setminus \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$, we set $\\mathbf} \\newcommand{\\mc}{\\mathcal\nR_{\\mathbf} \\newcommand{\\mc}{\\mathcal L}(\\lambda):=(\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)^{-1}$.\nWe use boldface lowercase Latin letters to denote 2-component\nfunctions, e.g.~$\\mathbf f=(f_1,f_2)$ and we also use the notation\n$[\\mathbf} \\newcommand{\\mc}{\\mathcal f]_j:=f_j$ to extract the components. \n\nFinally,\n$e_1:=(1,0,0,\\dots,0)\\in \\R^d$ is the first unit vector in $\\R^d$,\nwhere the dimension $d$ follows from the context.\n\n\\section{Review of the standard Cauchy theory}\n\n\\noindent The proof of Theorem \\ref{thm:main} relies on the\nformulation of the problem in adapted hyperboloidal coordinates. In\norder to construct data on the initial hyperboloid, we employ some\nelementary results on the standard Cauchy theory which are reviewed in\nthe following. For simplicity we restrict ourselves to spatial dimensions\n$d\\geq 3$. Furthermore, we only consider wave evolution to the\nfuture starting at $t=0$. By time translation and reflection this \nis in fact already the most general situation. \n\n\\subsection{Wave propagators}\nRecall that the solution of the Cauchy problem\n\\begin{equation}\n \\label{eq:wave}\n \\left \\{ \\begin{array}{l}\n (\\partial_t^2-\\Delta_x)u(t,x)=0,\\quad (t,x)\\in \\R^{1,d} \\\\\n u(0,\\cdot)=f,\\qquad \\partial_0 u(0,\\cdot)=g\n \\end{array} \\right . \n \\end{equation}\n for $f,g\\in \\mc S(\\R^d)$, say, is given by\n \\begin{equation}\n \\label{eq:wavesol} u(t,\\cdot)=\\cos(t|\\nabla|)f+\\frac{\\sin(t|\\nabla|)}{|\\nabla|}g, \n \\end{equation}\n where $\\phi(|\\nabla|)f:=\\mc F_d^{-1}(\\phi(|\\cdot|)\\mc F_d f)$\n for $\\phi \\in C(\\R)$ and $\\mc F_d$ is the Fourier transform\n \\[ \\mc F_df(\\xi):=\\int_{\\R^d}e^{-i \\xi x}f(x)d x .\\]\n The \\emph{wave propagators} $\\cos(t|\\nabla|)$ and\n $\\frac{\\sin(t|\\nabla|)}{|\\nabla|}$ extend by continuity to\n rough data, e.g.~$(f,g)\\in \\dot H^1(\\R^d)\\times L^2(\\R^d)$.\n This yields a canonical notion of strong solutions, i.e., one\n says that $u$ solves Eq.~\\eqref{eq:wave} if\n Eq.~\\eqref{eq:wavesol} holds. Note further that for any fixed\n $t\\in \\R$, the wave propagators map $\\mc S(\\R^d)$ to itself\n since $s\\mapsto \\cos(ts)$ and $s\\mapsto \\frac{\\sin(ts)}{s}$ are\n smooth, even functions on $\\R$.\n\n \\subsection{Finite speed of propagation}\n\n The wave equation enjoys finite speed of propagation in the\n following sense.\n\n \\begin{proposition}\n \\label{prop:finite}\n Let $x_0\\in \\R^d$ and $d\\geq 3$. Then there exists a\n continuous function $\\gamma_d: [0,\\infty)\\to [1,\\infty)$ such\n that\n \\begin{align*}\n \\|\\partial_t^\\ell \\cos(t|\\nabla|)f\\|_{\\dot\n H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0))}\n &\\leq \\|f\\|_{\\dot H^{k+\\ell}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T}(x_0))} \\\\\n \\left \\|\\partial_t^\\ell \\frac{\\sin(t|\\nabla|)}{|\\nabla|}f\n \\right \\|_{\\dot H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0))}\n &\\leq \\|f\\|_{\\dot H^{k+\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T}(x_0))} \n \\end{align*}\n as well as\n \\begin{align*}\n \\|\\partial_t^\\ell \\cos(t|\\nabla|)f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0))}\n &\\leq \\gamma_d(T)\\|f\\|_{H^{1+\\ell}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T}(x_0))} \\\\\n \\left \\|\\partial_t^\\ell \\frac{\\sin(t|\\nabla|)}{|\\nabla|}f\n \\right \\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0))}\n &\\leq \\gamma_d(T) \\|f\\|_{H^\\ell(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T}(x_0))}\n \\end{align*}\n for all $f\\in \\mc S(\\R^d)$, $T>0$, $t\\in [0,T)$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and\n $\\ell\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$.\n \\end{proposition}\n\n The bounds in homogeneous Sobolev spaces $\\dot H^k$ follow\n directly from the energy identity. The $L^2$-bounds are\n slightly more involved and in order to prove them, we need the\n following result which gives us control on the $L^2$-norm in\n balls in terms of the $\\dot H^1$-norm and a boundary term.\n\n \\begin{lemma}\n \\label{lem:embed}\n Let $x_0\\in \\R^d$ and $d\\geq 3$. Then we have \n \\[ \\|f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R(x_0))}^2\\leq R^2 \\|\\nabla\n f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R(x_0))}^2+\\tfrac{d-1}{2}R \\|f\\|_{L^2(\\partial\n \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R(x_0))}^2 \\]\n for all $f\\in C^1(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R(x_0)})$ and $R>0$.\n \\end{lemma}\n\n\\begin{proof}\n By translation we may assume $x_0=0$. Introducing polar coordinates\n $r=|x|$ and $\\omega=\\frac{x}{|x|}$, we compute\n \\begin{align*}\n r^{\\frac{d-1}{2}} f(r\\omega)=\\int_0^r \\partial_s \\left [s^{\\frac{d-1}{2}}f(s\\omega)\\right ]d s\n =\\int_0^r \\left [s^{\\frac{d-1}{2}}\\partial_s f(s\\omega)+\\tfrac{d-1}{2}s^{\\frac{d-3}{2}}f(s\\omega) \\right ]d s\n \\end{align*}\n and Cauchy-Schwarz yields\n \\[ r^{d-1}|f(r\\omega)|^2 \\leq R \\int_0^R \\left |\n s^{\\frac{d-1}{2}}\\partial_s\n f(s\\omega)+\\tfrac{d-1}{2}s^{\\frac{d-3}{2}}f(s\\omega) \\right |^2\n ds \\] for all $r\\in [0,R]$. Expanding the square, we find\n \\begin{align*}\n \\tfrac{1}{R}r^{d-1}|f(r\\omega)|^2&\\leq \\int_0^R |\\partial_s f(s\\omega)|^2 s^{d-1}ds\n +(d-1)\\int_0^R \\underbrace{\\partial_s f(s\\omega)f(s\\omega)}_{\\frac12 \\partial_s [f(s\\omega)]^2}s^{d-2}ds \\\\\n &\\quad +\\left (\\frac{d-1}{2}\\right )^2 \\int_0^R f(s\\omega)^2s^{d-3}d s \\\\\n &=\\int_0^R |\\partial_s f(s\\omega)|^2 s^{d-1}ds+\\tfrac{d-1}{2}R^{d-2}f(R\\omega)^2 \\\\\n &\\quad -\\frac{(d-1)(d-3)}{4}\\int_0^R f(s\\omega)^2 s^{d-3}ds \\\\\n &\\leq \\int_0^R |\\omega^j \\partial_j f(s\\omega)|^2 s^{d-1}ds+\\tfrac{d-1}{2}R^{d-2}f(R\\omega)^2. \n \\end{align*}\n Integrating this inequality yields\n \\begin{align*} \\int_0^R \\int_{\\S^{d-1}}f(r\\omega)^2 d\\sigma(\\omega)r^{d-1}dr&\\leq R^2 \\int_0^R \\int_{\\S^{d-1}} |\\nabla f(r\\omega)|^2d\\sigma(\\omega)r^{d-1}dr \\\\\n &\\quad +\\tfrac{d-1}{2}R^d \\int_{\\S^{d-1}}f(R\\omega)^2d\\sigma(\\omega) \\\\\n &=R^2\n \\|\\nabla\n f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R)}^2+\\tfrac{d-1}{2}R\\|f\\|_{L^2(\\partial\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_R)}^2,\n \\end{align*}\n which is the claim.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:finite}]\n Let $u(t,\\cdot)=\\cos(t|\\nabla|)f$. Then\n $u(t,\\cdot), \\partial_t u(t,\\cdot)\\in \\mc S(\\R^d)$ for all\n $t\\in \\R$, $u\\in C^\\infty(\\R^{1,d})$, $u(0,\\cdot)=f$,\n $\\partial_0 u(0,\\cdot)=0$, and $(\\partial_t^2 -\\Delta_x)u(t,x)=0$.\n Since $\\partial^\\alpha u(t,\\cdot)$ for any multi-index\n $\\alpha\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0^d$ satisfies the same equation, it is sufficient to\n consider the case $k=1$. Furthermore, by translation invariance we\n may assume $x_0=0$. We start with the case $\\ell=0$. A\n straightforward computation yields\n \\[ \\frac{d}{dt}\\left [\\int_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}}\\left (|\\nabla_x\n u(t,x)|^2+|\\partial_t u(t,x)|^2 \\right )dx \\right ] \\leq 0, \\]\ncf.~the proof of Lemma \\ref{lem:apxenid},\n and thus,\n \\[ \\|u(t,\\cdot)\\|_{\\dot H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}^2=\\|\\nabla\n u(t,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}^2\\leq \\|\\nabla\n u(0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2+\\|\\partial_0 u(0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2\n =\\|f\\|_{\\dot H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2 \\] since $\\partial_0 u(0,\\cdot)=0$.\n\n For the $L^2$-bound we appeal to Lemma \\ref{lem:embed} and note that\n the energy may be augmented by a boundary term that does not destroy\n the monotonicity. Indeed, we have\n \\[ \\frac{d}{dt}\\left [\\int_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}}\\left (|\\nabla_x\n u(t,x)|^2+|\\partial_t u(t,x)|^2 \\right )dx +\n \\frac{1}{T-t}\\int_{\\partial \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d}u(t,\\omega)^2 d\\sigma(\\omega)\n \\right ] \\leq 0, \\] \nsee Lemma \\ref{lem:apxenid}.\nConsequently, Lemma \\ref{lem:embed} implies\n \\begin{align*}\n \\|u(t,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}^2\n &\\leq (T-t)^2\\left [ \\|\\nabla u(t,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}^2 \n +\\tfrac{d-1}{2}(T-t)^{-1}\\|u(t,\\cdot)\\|_{L^2(\\partial\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}^2\\right ] \\\\\n &\\leq \\tfrac{d-1}{2}(T-t)^2 \\left [\\|\\nabla\n u(0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2\n +\\|\\partial_0 u(0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2+T^{-1}\\|u(0,\\cdot)\\|_{L^2(\\partial\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2 \\right ] \\\\\n &\\leq \\widetilde \\gamma_d(T) \\|u(0,\\cdot)\\|_{H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}^2\n \\end{align*}\nfor a continuous function $\\widetilde\\gamma_d: [0,\\infty)\\to [1,\\infty)$,\n where the last step follows from the trace theorem. For\n $\\ell\\geq 1$ we repeat the above arguments with $u$ replaced by\n $\\partial_0^\\ell u$ and use the equation to transform temporal\n derivatives into spatial ones. The proof for the sine propagator is\n identical.\n\\end{proof}\n\n\\begin{remark}\n By approximation, finite speed of propagation holds for rough data\n as well.\n\\end{remark}\n\nIn view of Proposition \\ref{prop:finite} it is natural to extend the\ndefinition of the wave propagators to functions defined on balls only.\nThis is most conveniently realized by means of Sobolev extensions.\n\n\\begin{lemma}\n \\label{lem:extension}\nLet $d\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ and $x_0\\in \\R^d$.\nFor any $r>0$ there exists a linear map $\\mc E_{r,x_0,d}: L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))\\to\nL^2(\\R^d)$ such that $\\mc E_{r,x_0,d} f|_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0)}=f$ a.e.~and $f\\in\nH^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))$ for $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ implies $\\mc E_{r,x_0,d} f\\in\nH^k(\\R^d)$. Furthermore, there exists a constant $C_{k,d}>0$ such that\n\\[ \\|\\mc E_{r,x_0,d} f\\|_{H^k(\\R^d)}\\leq C_{k,d} \\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))} \\]\nfor all $r>0$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$, $x_0\\in \\R^d$, and $f\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))$.\n\\end{lemma}\n\n\\begin{proof}\n From e.g.~\\cite{AdaFou03} we infer the existence of an\n extension $\\mc E_d: L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d)\\to L^2(\\R^d)$ such that $\\mc E_d\n f|_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d}=f$ a.e.~and $\\|\\mc E_d f\\|_{H^k(\\R^d)}\\leq C_{k,d}\n \\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d)}$ for all $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$ and all $f\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d)$.\nNote further that $f\\in H^d(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d)$ implies $\\mc E_d f\\in\nH^d(\\R^d)\\hookrightarrow C(\\R^d)$ by Sobolev embedding and thus, $f$\nand $\\mc E_d f$ may be identified with continuous functions such that\n$\\mc E_d f|_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d}=f$.\nFor $f\\in H^d(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))$ we now set \n$\\mc E_{r,x_0,d}f(x):=\\mc E_d (f_{1\/r}(\\cdot+x_0\/r))_r(x-x_0)$, where\n$f_\\lambda(x):=f(\\frac{x}{\\lambda})$ for any $\\lambda>0$.\nBy density, $\\mc E_{r,x_0,d}$ extends to all of $L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))$ and\nit is straightforward to verify that $\\mc E_{r,x_0,d}$ has the desired properties.\n\\end{proof}\n\n\\begin{definition}\n\\label{def:wploc}\nLet $T>0$, $t\\in [0,T)$, and $d\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $d\\geq 3$. Then we define\n\\[ \\cos(t|\\nabla|), \\frac{\\sin(t|\\nabla|)}{|\\nabla|}:\nL^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d(x_0))\\to L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0)) \\] by\n\\begin{align*}\n \\cos(t|\\nabla|)f&:=\\left . \\left (\\cos(t|\\nabla|)\\mc E_{T,x_0,d}f\n \\right ) \\right |_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0)} \\\\\n \\frac{\\sin(t|\\nabla|)}{|\\nabla|}f&:=\\left (\\left .\n\\frac{\\sin(t|\\nabla|)}{|\\nabla|}\\mc E_{T,x_0,d}f\\right ) \\right |_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0)},\n\\end{align*}\nwhere $\\mc E_{T,x_0,d}$ is a Sobolev extension as in Lemma \\ref{lem:extension}.\n\\end{definition}\n\n\\begin{remark}\n Proposition \\ref{prop:finite} implies that Definition\n \\ref{def:wploc} is independent of the extension chosen and\n that the wave propagators are bounded linear maps from $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d(x_0))$ to\n $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0))$ for all $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$, $T>0$, $t\\in [0,T)$, and $x_0\\in \\R^d$.\n\\end{remark}\n\n\\subsection{Local well-posedness of semilinear wave\n equations}\\label{Sec:LWP}\n\nNext, we turn to the local Cauchy problem for nonlinear wave equations\nof the form\n\\begin{equation}\n \\label{Eq:NLW5d}\n \\left \\{\n \\begin{array}{l}\n \\partial_t^2 u(t,\\cdot)-\\Delta u(t,\\cdot) = \\mc N(u(t,\\cdot)) \\\\\n u(0,\\cdot) = f, \\quad \\partial_0 u(0,\\cdot) = g\n \\end{array} \\right . ,\n\\end{equation}\nwhere $\\mc N$ is some nonlinear operator. In fact, we are going to\nrestrict ourselves to the following class of \\emph{admissible}\nnonlinearities.\n\n\\begin{definition}\n \\label{def:adm}\n Let $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ and $x_0\\in \\R^d$, $d\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$. \nA map $\\mc N: H^k(\\R^d)\\to H^{k-1}_\\mathrm{loc}(\\R^d)$ is called \\emph{$(k,x_0)$-admissible}\n iff\n$\\mc N(0)=0$ and for any $R\\geq 1$ there exists a constant\n $C_{R,k,x_0,d}>0$ such that\n \\[ \\|\\mc N(f)-\\mc N(g)\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))}\\leq\n C_{R,k,x_0,d}\n \\|f-g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))} \\]\n for all $r\\in [0,R]$ and all $f,g\\in H^k(\\R^d)$ satisfying\n\\[ \\|f\\|_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0)}+\\|g\\|_{B_r^d(x_0)}\\leq R. \\]\n\\end{definition}\n\n\\begin{remark}\n For any $r>0$, a $(k,x_0)$-admissible nonlinearity $\\mc N$\n naturally restricts to a map $\\mc N_r: H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_r(x_0))\\to\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_r(x_0))$ by \n \\[ \\mc N_{r}(f):=\\mc N(\\mc E_{r,x_0,d} f)|_{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_r(x_0)}, \\]\n where $\\mc E_{r,x_0,d}$ is a Sobolev extension as in Lemma\n \\ref{lem:extension}. The Lipschitz bound in Definition\n \\ref{def:adm} ensures that $\\mc N_r$ is independent of the extension\n chosen. For notational convenience we will identify $\\mc N$ with\n $\\mc N_r$.\n\\end{remark}\n\n\\begin{definition}\n Let $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$, $T>0$, $x_0\\in \\R^d$, $d\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $T'\\in (0,T)$. The Banach\n space $X^k_{T,x_0}(T')$ consists of functions\n \\[ u: \\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0) \\to \\R \\]\n such that $u(t,\\cdot)\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0))$ for each\n $t\\in [0,T']$ and the map\n $t\\mapsto \\|u(t,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0))}$ is continuous on\n $[0,T']$. Furthermore, we set\n \\[ \\|u\\|_{X^k_{T,x_0}(T')}:=\\max_{t\\in\n [0,T']}\\|u(t,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0))}. \\]\nFor brevity we write $X_T^k(T'):=X_{T,0}^k(T')$.\n\\end{definition}\n\nAppealing to Duhamel's principle, we consider the following notion of\nsolutions.\n\\begin{definition}\n Let $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $T>0$, $T'\\in (0, T)$, and $x_0\\in \\R^d$, $d\\geq 3$.\n Furthermore, assume that $\\mc N$ is $(k,x_0)$-admissible. We say\n that a function\n \\[ u: \\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)\\to \\R \\]\n is a \\emph{strong $H^k$ solution} of Eq.~\\eqref{Eq:NLW5d} in the\n truncated lightcone\n $\\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)\\subset \\R^{1,d}$\n iff $u\\in X^k_{T,x_0}(T')$ and\n \\[\n u(t,\\cdot)=\\cos(t|\\nabla|)u(0,\\cdot)+\\frac{\\sin(t|\\nabla|)}{|\\nabla|}\\partial_0 u(0,\\cdot)\n +\\int_0^t \\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\mc N(u(s,\\cdot))ds \\]\n for all $t\\in [0,T']$. \n\\end{definition}\n\n\n\n\n\\begin{theorem}[Local existence in lightcones]\n \\label{thm:LWP}\n Let $k \\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $M_0, T > 0$, and $x_0\\in \\R^d$, $d\\geq 3$.\n Furthermore, assume that $\\mc N$ is $(k,x_0)$-admissible. Then\n there exists a $T'\\in (0, T)$ such that for all\n $ (f,g) \\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T(x_0)) \\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d(x_0))$ satisfying\n \\[ \\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T(x_0))}+\\|g\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d(x_0))} \\leq M_0, \\]\n the initial value problem Eq.~\\eqref{Eq:NLW5d} has a strong $H^k$\n solution $u_{f,g}$ in the truncated lightcone\n $\\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)$. Furthermore,\n $\\partial_0 u_{f,g}\\in X^{k-1}_{T,x_0}(T')$ and the solution map\n \\[ (f,g)\\mapsto (u_{f,g},\\partial_0 u_{f,g}) \\]\n is Lipschitz as a function from (a ball in)\n $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T(x_0))\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T(x_0))$ to\n $X^k_{T,x_0}(T') \\times X^{k-1}_{T,x_0}(T')$.\n\\end{theorem}\n\n\\begin{proof}\n Without loss of generality we may assume $x_0=0$. \nWe set\n $M:=2M_0 \\gamma$, where\n $\\gamma:=\\max_{s\\in [0,\\frac{T}{2}]}\\gamma_d(T-s)$ and $\\gamma_d$\n is the continuous function from Proposition \\ref{prop:finite}.\n Furthermore, for $T'\\in [0,\\frac{T}{2}]$ we set\n \\[ Y(T'):=\\{u\\in X_T^k(T'): \\|u\\|_{X^k_T(T')}\\leq\n M\\} \\]\n and define a map $\\mc K_{f,g}$ on\n $Y(T')$ by\n \\[ \\mc\n K_{f,g}(u)(t):=\\cos(t|\\nabla|)f+\\frac{\\sin(t|\\nabla|)}{|\\nabla|}g\n +\\int_0^t \\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\mc N(u(s,\\cdot))\n ds,\\quad t\\in [0,T']. \\]\n Let $u\\in Y(T')$. From Proposition \\ref{prop:finite} and\n Definition \\ref{def:adm} we\n infer the existence of a constant $\\alpha>0$\n such that\n \\begin{align*}\n \\|\\mc K_{f,g}(u)(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}\n &\\leq \\gamma\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}+\\gamma\\|g\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)} \n + \\gamma\\int_0^t \\|\\mc N(u(s,\\cdot))\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-s})}ds \\\\\n &\\leq \\frac{M}{2}+\\alpha\\gamma\\int_0^t \\|u(s,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-s})} ds \\\\\n &\\leq \\frac{M}{2}+\\alpha\\gamma T'\\|u\\|_{X^k_T(T')} \\\\\n&\\leq \\frac{M}{2}+\\alpha\\gamma T'M\n \\end{align*}\n for all $t\\in [0,T']$. Consequently, by choosing $T'>0$ small\n enough, we obtain\n \\[ \\|\\mc K_{f,g}(u)\\|_{X^k_T(T')}\\leq M, \\]\n which means that $\\mc K_{f,g}(u)\\in Y(T')$ whenever\n $u\\in Y(T')$. Similarly, for $u,v\\in Y(T')$, we\n infer\n \\begin{align*}\n \\|\\mc K_{f,g}(u)(t)-\\mc K_{f,g}(v)(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}\n &\\leq \\gamma\\int_0^t \\|\\mc N(u(s,\\cdot))-\\mc N(v(s,\\cdot))\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-s})}ds \\\\\n &\\leq \\alpha\\gamma\\, T' \\|u-v\\|_{X^k_T(T')}\n \\end{align*}\n for all $t\\in [0,T']$, which yields\n \\[ \\|\\mc K_{f,g}(u)-\\mc K_{f,g}(v)\\|_{X^k_T(T')}\\leq \\tfrac12\n \\|u-v\\|_{X^k_T(T')} \\]\n upon choosing $T'>0$ sufficiently small. Thus, since $Y(T')$ is a\n closed subset of the Banach space $X_T^k(T')$, the contraction\n mapping principle implies the existence of a fixed point\n $u_{f,g}\\in Y(T')$ of $\\mc K_{f,g}$. Furthermore, we have\n \\begin{equation}\n \\label{eq:lwpdtu}\n \\partial_t u_{f,g}(t,\\cdot)=\\partial_t \\cos(t|\\nabla|)f+\\partial_t\n \\frac{\\sin(t|\\nabla|)}{|\\nabla|}g+\\int_0^t \\partial_t\n \\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\mc N(u_{f,g}(s,\\cdot))ds \n \\end{equation}\nand Proposition \\ref{prop:finite} yields\n \\begin{align*}\n \\|\\partial_t u_{f,g}(t,\\cdot)\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}\n &\\lesssim \n \\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}+\\|g\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}+\\int_0^t \\|\\mc N(u_{f,g}(s,\\cdot))\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-s})}ds \\\\\n &\\lesssim M_0+\\|u_{f,g}\\|_{X^k_T(T')}\n \\end{align*}\n for all $t\\in [0,T']$, which shows\n $\\partial_0 u_{f,g}\\in X^{k-1}_T(T')$.\n\n It remains to prove the Lipschitz continuity of the solution map\n $(f,g)\\mapsto u_{f,g}$. We have\n \\begin{align*}\n \\|u_{f,g}(t)-u_{\\widetilde f, \\widetilde g}(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}\n &=\\|\\mc\n K_{f,g}(u_{f,g})(t)-\\mc K_{\\widetilde f,\\widetilde\n g}(u_{\\widetilde f, \\widetilde g})(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})} \\\\\n &\\leq \\|\\mc K_{f,g}(u_{f,g})(t)-\\mc\n K_{f,g}(u_{\\widetilde f, \\widetilde g})(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})} \\\\\n &\\quad +\\|\\mc K_{f,g}(u_{\\widetilde f, \\widetilde g})(t)-\\mc K_{\\widetilde\n f,\\widetilde g}(u_{\\widetilde f, \\widetilde g})(t)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})} \\\\\n &\\leq \\tfrac12 \\|u_{f,g}- u_{\\widetilde f,\\widetilde\n g}\\|_{X^k_T(T')} \\\\\n &\\quad +\\gamma_d(T)\\|f-\\widetilde\n f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}+\\gamma_d(T)\\|g-\\widetilde\n g\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}\n \\end{align*}\n for all $t\\in [0,T']$ and thus,\n \\[ \\|u_{f,g}-u_{\\widetilde f,\\widetilde g}\\|_{X^k_T(T')}\\lesssim\n \\|(f,g)-(\\widetilde f,\\widetilde g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)\\times\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T)}. \\] \nFinally, from Eq.~\\eqref{eq:lwpdtu} we infer\n \\begin{align*}\n \\|\\partial_t u_{f,g}(t,\\cdot)-\\partial_t u_{\\widetilde\n f,\\widetilde g}(t,\\cdot)\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d)}\n&\\lesssim \\|(f,g)-(\\widetilde f,\\widetilde g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)\\times\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)} \\\\\n&\\quad +\\int_0^t \\|\\mc N(u_{f,g}(s,\\cdot))-\\mc\n N(u_{\\widetilde f,\\widetilde g}(s,\\cdot))\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-s})}ds \\\\\n&\\lesssim \\|(f,g)-(\\widetilde f,\\widetilde g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)\\times\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)} \\\\\n&\\quad +\\|u_{f,g}-u_{\\widetilde f,\\widetilde g}\\|_{X^k_T(T')} \\\\\n&\\lesssim \\|(f,g)-(\\widetilde f,\\widetilde g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)\\times\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)}\n \\end{align*}\nfor all $t\\in [0,T']$, which finishes the proof.\n\\end{proof}\n\nFinite speed of propagation is valid for nonlinear equations as\nwell. This is expressed by the following uniqueness result.\n\n\\begin{theorem}[Uniqueness in lightcones]\n \\label{thm:uniq}\n Let $k \\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $T > 0$, $T'\\in [0,T)$, and $x_0\\in \\R^d$,\n $d\\geq 3$. Furthermore, assume that $\\mc N$ is\n $(k,x_0)$-admissible. Suppose $u$ and $v$ are both strong $H^k$\n solutions of Eq.~\\eqref{Eq:NLW5d} in the truncated lightcone\n $\\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)$ with the same\n initial data, i.e., $u(0,\\cdot)=v(0,\\cdot)$ and\n $\\partial_0 u(0,\\cdot)=\\partial_0 v(0,\\cdot)$. Then $u=v$.\n\\end{theorem}\n\n\\begin{proof}\n We have\n\\[ u(t,\\cdot)-v(t,\\cdot)=\\int_0^t\n\\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\left [\\mc N(u(s,\\cdot))-\\mc\n N(v(s,\\cdot))\\right ]ds \\]\nand thus,\n\\begin{align*}\n\\| u(t,\\cdot)-v(t,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})}\n&\\lesssim \\int_0^t \\|\\mc N(u(s,\\cdot))-\\mc\n N(v(s,\\cdot))\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-s}^d)}ds \\\\\n&\\lesssim \\int_0^t \\|u(s,\\cdot)-v(s,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-s}^d)}ds\n\\end{align*}\nfor all $t\\in [0,T']$. Consequently, Gronwall's inequality yields\n$\\|u(t,\\cdot)-v(t,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d)}=0$ for all $t\\in [0,T']$.\n\\end{proof}\n\n\\subsection{Upgrade of regularity}\nNow we take a different viewpoint and \\emph{assume} that we already\nhave a strong $H^k$ solution. We would then like to conclude that the\nsolution is in fact smooth, provided the data are smooth. To this\nend, we need to strengthen the assumptions on the nonlinearity. We\nstart with an auxiliary result which will also be useful later in a\ndifferent context.\n\n\\begin{lemma}\n \\label{lem:Moser}\nLet $d\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $x_0\\in \\R^d$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k>\\frac{d}{2}$. Furthermore,\nlet $F\\in C^\\infty(\\R\\times \\R^d)$, $F(0,x)=\\partial_1 F(0,x)=0$ for all $x\\in \\R^d$,\nand for $f: \\R^d\\to\\R$ set\n\\[ \\mc N(f)(x):=F(f(x),x). \\]\nThen $\\mc N$ maps $H^k(\\R^d)$ to $H^{k}_\\mathrm{loc}(\\R^d)$ and for\nany $R\\geq 1$\nthere exists a constant $C_{R,k,x_0,d}>0$ such that\n\\begin{align*}\n \\|\\mc N(f)-\\mc N(g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))}\\leq \n C_{R,k,x_0,d}\\left (\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))}+\\|g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))}\\right )\\|f-g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d(x_0))} \n\\end{align*}\nfor all $r\\in [0,R]$ and all $f,g\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)$ satisfying\n$\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}+\\|g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\\leq R$.\nIn particular, $\\mc N$ is $(k,x_0)$-admissible. \n\\end{lemma}\n\n\\begin{proof}\n We assume without loss of generality that $x_0=0$ and\n note that the assumption $k>\\frac{d}{2}$ implies\n $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)\\hookrightarrow C(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d})$ for any\n $r>0$. Thus, elements of $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)$ can be identified with\n continuous functions. Furthermore, $H^k(\\R^d)$ is a Banach algebra\n and thus,\n \\begin{equation}\n \\begin{split}\n \\label{eq:algBR}\n \\|fg\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)} &=\\|\\mc E_r f\\mc E_r g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)} \\leq\n \\|\\mc E_r f\\mc E_r g\\|_{H^k(\\R^d)}\\lesssim \\|\\mc E_r\n f\\|_{H^k(\\R^d)}\\|\\mc E_r g\\|_{H^k(\\R^d)} \\\\\n &\\lesssim \\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\\|g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\n \\end{split}\n \\end{equation}\nfor all $r>0$ and $f,g\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)$, where $\\mc E_r:=\\mc E_{r,0,d}$\nis an extension as in Lemma \\ref{lem:extension}.\n\nNow we use the\nfundamental theorem of calculus to obtain the identity\n \\begin{align*}\n \\mc N(f)(x)-\\mc N(g)(x)\n &=F(f(x),x)-F(g(x),x)=\\int_0^1 \\partial_s F\\big\n (sf(x)+(1-s)g(x),x\\big )ds \\\\\n &=[f(x)-g(x)]\\int_0^1 \\partial_1 F\\big\n (sf(x)+(1-s)g(x),x\\big )ds \\\\\n&=[f(x)-g(x)]\\int_0^1 \\mc N'\\big\n (sf+(1-s)g\\big )(x)ds \n \\end{align*}\nfor all $x\\in \\R^d$, where $\\mc N'(f)(x):=\\partial_1 F(f(x),x)$.\nWe claim that $\\mc N'$ maps $H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)$ to itself for any $r>0$ and\nthat for any $R\\geq 1$, there exists a continuous function $\\gamma_R:\n[0,\\infty)\\to [0,\\infty)$ such that\n\\begin{equation}\n \\label{eq:MoserN'}\n \\|\\mc N'(f)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\\leq \\gamma_R(\\|\\mc E_r f\\|_{H^k(\\R^d)})\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\n\\end{equation}\nfor all $r\\in (0,R]$ and $f\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)$.\nAssume for the moment that this is true. \nThen Eq.~\\eqref{eq:algBR} and the triangle inequality yield\n\\begin{align*}\n \\|\\mc N(f)-\\mc N(g)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_r)}\n&\\lesssim \\|f-g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\\int_0^1 \\left \\|\n \\mc N'\\big(sf+(1-s)g\\big)\\right \\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}ds.\n\\end{align*}\nFurthermore,\n\\begin{align*}\n\\int_0^1 &\\left \\|\n \\mc N'\\big(sf+(1-s)g\\big)\\right \\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}ds \\\\\n&\\leq \\int_0^1 \\gamma_R\\left (\\|s\\mc E_r f+(1-s)\\mc E_r g\\|_{H^k(\\R^d)}\\right)\\|sf+(1-s)g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}ds \\\\\n&\\leq \\left (\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}+\\|g\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\\right\n )\\int_0^1 \\gamma_R\\left (\\|s\\mc E_r f+(1-s)\\mc E_r g\\|_{H^k(\\R^d)}\\right)ds\n\\end{align*}\nfor all $r\\in (0,R]$. This yields the stated bound and finishes the proof.\nConsequently, it remains to prove Eq.~\\eqref{eq:MoserN'}.\n\nTo this end, we employ a smooth cut-off $\\chi_R:\\R^d\\to [0,1]$\nsatisfying $\\chi_R(x)=1$ for $|x|\\leq R$ and $\\chi_R(x)=0$ for\n$|x|\\geq 2R$. We set $F_R(u,x):=\\chi_R(x)\\partial_1 F(u,x)$. Then $F_R\\in\nC^\\infty(\\R\\times \\R^d)$ and for any compact $K\\subset \\R$ and any\nmulti-index $\\alpha\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0^{1+d}$, we have $\\partial^\\alpha F_R\\in\nL^\\infty(K\\times \\R^d)$.\nFurthermore, by assumption, $F_R(0,x)=0$ for all $x\\in \\R^d$.\nThus, by Moser's inequality, see e.g.~\\cite{Rau12}, Theorem\n6.4.1, $x\\mapsto F_R(f(x),x)$ belongs to $H^k(\\R^d)$ for any $f\\in\nH^k(\\R^d)$ and there exists a continuous function $\\widetilde\\gamma_R:\n[0,\\infty)\\to [0,\\infty)$ such that\n\\begin{align*}\n \\|\\mc N'(f)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\n &=\\|\n F_R(\\mc E_r f(\\cdot),\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)}\n \\leq \\|F_R(\\mc E_r f(\\cdot),\\cdot)\\|_{H^k(\\R^d)} \\\\\n&\\leq \\widetilde\\gamma_R\\left (\\|\\mc E_r f\\|_{H^k(\\R^d)}\\right )\\|\\mc\n E_r f\\|_{H^k(\\R^d)} \\\\\n&\\lesssim \\widetilde\\gamma_R\\left (\\|\\mc E_r f\\|_{H^k(\\R^d)}\\right )\\|f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_r^d)},\n\\end{align*}\nwhich proves Eq.~\\eqref{eq:MoserN'}.\n\\end{proof}\n\n\\begin{theorem}[Upgrade of regularity]\n \\label{thm:reg}\n Let $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k>\\frac{d}{2}$, $T>0$, $T'\\in [0,T)$, and\n $x_0\\in \\R^d$, $d\\geq 3$. Furthermore, assume that the nonlinear\n operator is given by $\\mc N(f)(x)=F(f(x),x)$ for a function\n $F\\in C^\\infty(\\R\\times\\R^d)$ satisfying $F(0,x)=\\partial_1 F(0,x)=0$ for all\n $x\\in\\R^d$. Suppose that $u$ is a strong $H^k$ solution of\n Eq.~\\eqref{Eq:NLW5d} in the truncated lightcone\n $\\bigcup_{t\\in [0,T']}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)$. If\n $u(0,\\cdot), \\partial_0 u(0,\\cdot) \\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_T})$\n then $u\\in C^\\infty (\\overline{\\bigcup_{t\\in [0,T']}\\{t\\}\\times\n \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)})$ and $u$ is a classical solution, i.e.,\n\\[ (\\partial_t^2 -\\Delta_x)u(t,x)=F(u(t,x),x) \\]\nfor all $t\\in [0,T']$ and $x\\in \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0)}$.\n\\end{theorem}\n\n\\begin{proof}\nWithout loss of generality we assume $x_0=0$. \nBy assumption, we have\n\\begin{equation}\n\\label{eq:usmooth}\nu(t,\\cdot)=\\cos(t|\\nabla|)u(0,\\cdot)+\\frac{\\sin(t|\\nabla|)}{|\\nabla|}\\partial_0\nu(0,\\cdot)+\\int_0^t\n\\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\mc N(u(s,\\cdot))ds \n\\end{equation}\nfor all $t\\in [0,T']$. Furthermore, Lemma \\ref{lem:Moser} yields $\\mc\nN(u(t,\\cdot))\\in H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d)$ for all $t\\in [0,T']$ and\nfrom Eq.~\\eqref{eq:usmooth} we infer\n\\begin{align*}\n \\|u(t,\\cdot)\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d)}\n&\\lesssim\n \\|u(0,\\cdot)\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)}+\\|\\partial_0 u(0,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_T^d)}+\\int_0^t \\|\\mc\n N(u(s,\\cdot))\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-s}^d)}ds \\\\\n&\\lesssim \\int_0^t \\|u(s,\\cdot)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-s}^d)}ds \\\\\n&\\lesssim \\|u\\|_{X^k_T(T')},\n\\end{align*}\nwhich implies $u(t,\\cdot)\\in H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d)$ for all\n$t\\in [0,T']$. Inductively, we find $u(t,\\cdot)\\in H^\\ell(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t})$\nfor all $t\\in [0,T']$ and any $\\ell\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$. By Sobolev embedding we\ntherefore obtain $u(t,\\cdot)\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d})$. The\nsame type of argument yields\n$\\partial_t u(t,\\cdot)\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d})$.\nFurthermore, with $\\mc E_T:=\\mc E_{T,0,d}$ the extension from Lemma\n\\ref{lem:extension}, we infer\n\\begin{align*}\n\\partial_t^2 u(t,\\cdot)\n&=\\partial_t^2 \\cos(t|\\nabla|)\\mc E_T u(0,\\cdot)+\\partial_t^2\n\\frac{\\sin(t|\\nabla|)}{|\\nabla|}\\mc E_T \\partial_0\nu(0,\\cdot) \\\\\n&\\quad +\\int_0^t\n\\partial_t^2 \\frac{\\sin((t-s)|\\nabla|)}{|\\nabla|}\\mc E_{T-s}\\mc\n N(u(s,\\cdot))ds \n+\\mc E_{T-t} \\mc N(u(t,\\cdot)) \\\\\n&=\\Delta u(t,\\cdot)+\\mc E_{T-t}\\mc N(u(t,\\cdot))\n\\end{align*}\nand thus, $\\partial_t^2 u(t,x)-\\Delta_x u(t,x)=F(u(t,x),x)$ for all\n$t\\in [0,T']$ and $x\\in \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{T-t}^d(x_0)}$. Inductively, it\nfollows that $u\\in C^\\infty (\\overline{\\bigcup_{t\\in [0,T']}\\{t\\}\\times\n \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^d_{T-t}(x_0)})$.\n\\end{proof}\n\n\\subsection{Application to the wave maps equation}\nTo conclude this section, we show that the above theory applies to the\nwave maps equation. To this end it suffices to prove that the\nnonlinearity in Eq.~\\eqref{eq:mainu} satisfies the hypotheses of Lemma\n\\ref{lem:Moser}.\n\n\\begin{lemma}\n \\label{lem:wmsmooth}\nLet $F: \\R\\times \\R^5\\to \\R$ be given by\n\\[ F(u,x):=\\frac{2|x|u-\\sin(2|x|u)}{|x|^3}. \\]\nThen $F(0,x)=\\partial_1 F(0,x)=0$ for all $x\\in \\R^5$ and $F\\in\nC^\\infty(\\R\\times \\R^5)$.\n\\end{lemma}\n\n\\begin{proof}\n From Taylor's theorem with integral remainder,\n \\[ f(t_0+t)=\\sum_{n=0}^N\n \\frac{f^{(n)}(t_0)}{n!}t^n+\\frac{t^{N+1}}{N!}\\int_0^1\n f^{(N+1)}(t_0+st)(1-s)^N ds, \\] we infer\n\\[ \\sin(2|x|u)=2|x|u-4|x|^3 u^3\\int_0^1 \\cos(2s|x|u)(1-s)^2\nds \\]\nand thus,\n\\[ F(u,x)=4u^3\\int_0^1 \\cos(2s|x|u)(1-s)^2 ds. \\]\nSince cosine is an even function, it follows that $F\\in C^\\infty(\\R\\times \\R^5)$\nand $F(0,x)=\\partial_1 F(0,x)=0$ for all $x\\in \\R^5$ is obvious.\n\\end{proof}\n\n\n\n\\section{The wave equation in hyperboloidal similarity coordinates}\n\n\\noindent In this section we study the \\emph{free} wave equation on\n$\\R^{1,d}$ in hyperboloidal similarity coordinates. In fact, we will\nfocus on the dimensions $d=1$ and $d=5$ and restrict ourselves to the\nradial case.\n\n\n\\subsection{Coordinate systems}\nThroughout this paper we use three different coordinate systems on\n(portions of) $\\R^{1,d}$, which we consistently denote by\n\\begin{align*}\n (t,x)&=(t,x^1,\\dots, x^d)=(x^0, x^1, \\dots, x^d) \\in \\R^{1+d} \\\\\n (\\tau,\\xi)&=(\\tau,\\xi^1,\\dots,\\xi^d)= (\\xi^0,\\xi^1,\\dots,\\xi^d) \\in \\R^{1+d} \\\\\n (s,y)&=(s,y^1,\\dots,y^d)=(y^0,y^1,\\dots ,y^d) \\in \\R^{1+d}.\n\\end{align*}\nNaturally, $(t,x)$ are the standard Cartesian coordinates where the\nMinkowski metric takes the form $\\eta=\\mathrm{diag}(-1,1,\\dots,1)$.\nThe \\emph{standard similarity coordinates} $(\\tau,\\xi)$ are defined by\n\\[ (t,x)=(T-e^{-\\tau}, e^{-\\tau}\\xi), \\]\nwhere $T\\in \\R$ is a free parameter. Strictly speaking, the\ncoordinates $(\\tau,\\xi)$ depend on $T$ but we suppress this in the\nnotation. We have\n\\begin{align*}\n \\partial_\\tau u(T-e^{-\\tau},e^{-\\tau}\\xi)&=e^{-\\tau}\\partial_0 u(T-e^{-\\tau},e^{-\\tau}\\xi)-e^{-\\tau}\\xi^j\\partial_j u(T-e^{-\\tau},e^{-\\tau}\\xi) \\\\\n \\partial_{\\xi^j}u(T-e^{-\\tau},e^{-\\tau}\\xi)&=e^{-\\tau}\\partial_j u(T-e^{-\\tau},e^{-\\tau}\\xi)\n\\end{align*}\nand as a consequence, the wave operator is given by\n\\begin{align*}\n -\\partial^\\mu\\partial_\\mu &u(T-e^{-\\tau}, e^{-\\tau}\\xi) \\\\\n &=e^{2\\tau}\n \\left [ \\partial_\\tau^2 + 2\\xi^j \\partial_{\\xi^j}\\partial_\\tau -(\\delta^{jk}-\\xi^j\\xi^k)\\partial_{\\xi^j}\\partial_{\\xi^k}\n +\\partial_\\tau+2\\xi^j\\partial_{\\xi^j} \\right ]u(T-e^{-\\tau}, e^{-\\tau}\\xi).\n\\end{align*}\nThe coordinates $(s,y)$ are defined by\n\\[ (t,x)=(T+e^{-s}h(y), e^{-s}y), \\]\nwhere again $T\\in \\R$ is a free parameter and\n\\[ h(y):=\\sqrt{2+|y|^2}-2 \\]\nis called the \\emph{height function}. Note that the choice $h(y)=-1$\nyields the standard similarity coordinates $(\\tau,\\xi)$ from\nabove. By the chain rule, we infer\n\\begin{align*}\n \\partial_s u(T+e^{-s}h(y),e^{-s}y)&=-e^{-s}h(y)\\partial_0\n u(T+e^{-s}h(y),e^{-s}y)-e^{-s}y^j\\partial_ju(T+e^{-s}h(y),e^{-s}y) \\\\\n \\partial_{y^j} u(T+e^{-s}h(y),e^{-s}y)&=e^{-s}\\partial_j\n h(y)\\partial_0\n u(T+e^{-s}h(y),e^{-s}y)+e^{-s}\\partial_j u(T+e^{-s}h(y),e^{-s}y).\n\\end{align*}\nFor brevity we introduce the following notation for the partial\nderivatives expressed in the new coordinates.\n\\begin{definition}\n \\label{def:mcD}\n We define\n \\begin{align*}\n (\\Dh_0 v)(s,y)&:=\\frac{e^s}{y^k\\partial_k h(y)-h(y)}\\left [\n \\partial_s+y^k\\partial_{y^k} \\right ]v(s,y) \\\\\n (\\Dh_j v)(s,y)&:=e^s\\partial_{y^j} v(s,y)-\\partial_j h(y)(\\Dh_0 v)(s,y)\n \\end{align*}\n\\end{definition}\nThen we have $\\mc D_\\mu v(s,y)=\\partial_\\mu u(T+e^{-s}h(y),e^{-s}y)$\nand thus,\n\\[ \\partial^\\mu \\partial_\\mu u(T+e^{-s}h(y),e^{-s}y)=\\mc D^\\mu\\mc\nD_\\mu v(s,y), \\]\nwhere $v(s,y)=u(T+e^{-s}h(y),e^{-s}y)$. Note that by construction,\nthe differential operators $\\mc D_\\mu$ and $\\mc D_\\nu$ commute. In\nthe case $d=1$ we have\n\\begin{align*}\n \\Dh_0 v(s,y)&=\\frac{e^s}{y h'(y)-h(y)}(\\partial_s+y\\partial_y)v(s,y) \\\\\n \\Dh_1 v(s,y)&=-\\frac{e^s}{yh'(y)-h(y)}[h'(y)\\partial_s+h(y)\\partial_y]v(s,y).\n\\end{align*}\n\nFinally, we note that there is a convenient direct relation between\nthe coordinates $(\\tau,\\xi)$ and $(s,y)$ given by\n\\begin{equation}\n \\label{eq:sytauxi}\n (\\tau,\\xi)=\\left (s-\\log(-h(y)), -\\frac{y}{h(y)} \\right ).\n\\end{equation}\nIn particular, this implies the identity\n\\begin{equation}\n \\label{eq:sytauxiwave}\n -\\mc D^\\mu\\mc D_\\mu v(s,y)=e^{2\\tau}\n \\left [ \\partial_\\tau^2 + 2\\xi^j \\partial_{\\xi^j}\\partial_\\tau -(\\delta^{jk}-\\xi^j\\xi^k)\\partial_{\\xi^j}\\partial_{\\xi^k}\n +\\partial_\\tau+2\\xi^j\\partial_{\\xi^j} \\right ]w(\\tau,\\xi),\n\\end{equation}\nwhere $v(s,y)=w(s-\\log(-h(y)), -y\/h(y))$.\n\n \\subsection{Control of the wave evolution}\n Let $u\\in C^2(\\R^{1,1})$ satisfy the wave equation\n \\[ \\partial_t^2 u(t,x)-\\partial_x^2 u(t,x)=0. \\]\n Furthermore, assume that $u(t,\\cdot)$ is odd for all $t\\in \\R$. In\n HSC we obtain\n \\begin{equation}\n \\begin{split}\n \\label{eq:wavev} 0&=\\Dh_0^2 v-\\Dh_1^2\n v=(\\Dh_0-\\Dh_1)(\\Dh_0+\\Dh_1)v =(\\Dh_0+\\Dh_1)(\\Dh_0-\\Dh_1)v\n \\end{split}\n \\end{equation}\n where $v(s,y)=u(T+e^{-s}h(y),e^{-s}y)$. If we set\n $v_\\pm:=\\mc D_0 v\\pm \\mc D_1 v$, Eq.~\\eqref{eq:wavev} implies\n \\begin{equation}\n \\label{eq:wavevpm}\n \\begin{split}\n [1-h'(y)]\\partial_s v_-(s,y)&=-[y-h(y)]\\partial_y v_-(s,y) \\\\\n [1+h'(y)]\\partial_s v_+(s,y)&=-[y+h(y)]\\partial_y v_+(s,y).\n \\end{split}\n \\end{equation}\n Note that $y-h(y)$ has a unique zero at $y=-\\frac12$ and $y-h(y)<0$\n for $y<-\\frac12$. Geometrically, $y=\\pm\\frac12$ is the boundary of\n the backward lightcone with tip $(T,0)$. By testing the first\n equation with $v_-$ and integrating over $[-R, R]$, we find\n \\begin{align}\n \\label{eq:energyv-}\n \\frac{d}{ds}\\int_{-R}^{R}v_-(s,y)^2 [1-h'(y)]dy &=-\\int_{-R}^{R} \\partial_y [v_-(s,y)^2] [y-h(y)]dy \\nonumber \\\\\n &= -v_-(s,y)^2 [y-h(y)]\\Big |_{-R}^{R}\n +\\int_{-R}^{R} v_-(s,y)^2 [1-h'(y)]dy \\nonumber \\\\\n &\\leq \\int_{-R}^{R} v_-(s,y)^2 [1-h'(y)]dy,\n \\end{align}\n provided $R\\geq \\frac12$. Integration with respect to $s$ and\n $1-h'(y)\\simeq 1$ for $y\\in [-R,R]$ yield the bound\n \\[ \\|v_-(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim\n e^{s\/2}\\|v_-(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\]\n for all $s\\geq s_0$ and any fixed $s_0$. Analogously, we infer\n \\begin{equation}\n\\label{eq:aprioriv+}\n \\|v_+(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim\n e^{s\/2}\\|v_+(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}. \n\\end{equation}\n Consequently, from\n \\begin{equation}\n \\label{eq:vvpm}\n \\begin{split}\n \\partial_s v(s,y)&=\\tfrac12 e^{-s}[y-h(y)]v_-(s,y)-\\tfrac12 e^{-s}[y+h(y)]v_+(s,y) \\\\\n \\partial_y v(s,y)&=-\\tfrac12 e^{-s}[1-h'(y)] v_-(s,y)+\\tfrac12\n e^{-s}[1+h'(y)]v_+(s,y)\n \\end{split}\n \\end{equation}\n we obtain the bound\n \\begin{align*}\n \\|v(s,\\cdot)\\|_{\\dot H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|\\partial_s v(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}&\\lesssim e^{-s}\\left (\\|v_-(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|v_+(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\right ) \\\\\n &\\lesssim e^{-s\/2}\\left (\\|v_-(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|v_+(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\right ) \\\\\n &\\lesssim e^{-s\/2}\\left (\\|v(s_0,\\cdot)\\|_{\\dot H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|\\partial_0 v(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\right ),\n \\end{align*}\n where we have used the fact that $yh'(y)-h(y)\\geq \\frac12$ for all\n $y\\in \\R$. Since $u(t,\\cdot)$ is assumed to be odd, we have the\n boundary condition $v(s,0)=u(T+e^{-s}h(0),0)=0$ for all $s\\in \\R$ and\n thus,\n \\[ v(s,y)=\\int_0^y \\partial_{y'} v(s,y')dy' \\]\n which, by Cauchy-Schwarz, yields the final \\emph{energy estimate}\n \\[ \\|v(s,\\cdot)\\|_{H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|\\partial_s\n v(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim e^{-s\/2}\\left\n (\\|v(s_0,\\cdot)\\|_{H^1(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|\\partial_0\n v(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\right ). \\]\n We generalize to higher derivatives.\n\n \\begin{lemma}\n \\label{lem:vHk}\n Fix $s_0\\in \\R$, $R\\geq \\frac12$, $T\\in \\R$, and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 2$. Furthermore, assume\n that $u\\in C^k(\\R^{1,1})$ satisfies\n \\[ \\partial_t^2 u(t,x)-\\partial_x^2 u(t,x)=0 \\]\n and suppose $u(t,\\cdot)$ is odd for all $t\\in \\R$. Let\n $v(s,y):=u(T+e^{-s}h(y),e^{-s}y)$. Then we have the bounds\n \\[ \\| v(s,\\cdot)\\|_{H^\\ell(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+ \\|\\partial_s\n v(s,\\cdot)\\|_{H^{\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim \\,e^{-s\/2}\\left (\\|\n v(s_0,\\cdot)\\|_{H^\\ell(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|\\partial_0\n v(s_0,\\cdot)\\|_{H^{\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\right ) \\]\n for all $s\\geq s_0$ and all $\\ell\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ satisfying $\\ell\\leq k$.\n \\end{lemma}\n \n \n\\begin{proof}\n Define the differential operators\n \\[ (L_\\pm f)(y):=-\\frac{y\\pm h(y)}{1\\pm h'(y)}f'(y),\\qquad (D_\\pm\n f)(y):=\\frac{1}{1\\pm h'(y)}f'(y). \\]\n Then Eq.~\\eqref{eq:wavevpm} can be written as\n \\begin{equation}\n \\label{eq:Lv} \n \\partial_s v_\\pm(s,\\cdot)=L_\\pm v_\\pm(s,\\cdot). \n \\end{equation}\n We have the commutator relation $[D_\\pm,L_\\pm]=-D_\\pm$ and thus,\n applying $D_\\pm^j$ to Eq.~\\eqref{eq:Lv}, for $0\\leq j\\leq k-1$, yields\n \\[ \\partial_s D_\\pm^jv_\\pm(s,\\cdot)=D_\\pm^j L_\\pm\n v_\\pm(s,\\cdot)=L_\\pm D_\\pm^j\n v_\\pm(s,\\cdot)-jD_{\\pm}^jv_\\pm(s,\\cdot). \\]\n Consequently, by testing with $[1\\pm h'(y)]D_\\pm^j v_\\pm(s,\\cdot)$,\n we infer\n \\begin{align}\n \\label{eq:kbound}\n \\|D_\\pm^jv_\\pm(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}^2\\lesssim e^{(1-2j) s} \\|D_\\pm^j v_\\pm(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}^2\n \\end{align}\n for any $0\\leq j\\leq k-1$. Now we claim that, for any $\\ell\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$,\n \\begin{equation}\n \\label{eq:Hkequiv}\n \\|f\\|_{H^\\ell(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\simeq \\sum_{j=0}^\\ell \\|D_\\pm^j f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} .\n \\end{equation}\n Suppose for the moment that Eq.~\\eqref{eq:Hkequiv} is true. Then\n Eq.~\\eqref{eq:kbound} implies\n \\begin{align*}\n \\|v_\\pm(s,\\cdot)\\|_{H^{\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\n&\\simeq \\sum_{j=0}^{\\ell-1}\\|D_\\pm^j v_\\pm(s,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\n\\lesssim e^{s\/2} \\sum_{j=0}^{\\ell-1}\\|D_\\pm^j v_\\pm(s_0,\\cdot)\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\\\\n&\\lesssim e^{s\/2}\\|v_\\pm(s_0,\\cdot)\\|_{H^{\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\n \\end{align*}\n for any $0\\leq\\ell\\leq k$\n and the claim follows from Eq.~\\eqref{eq:vvpm} and the boundary\n condition $v(s,0)=0$.\n\n It remains to prove Eq.~\\eqref{eq:Hkequiv}. Note that\n $1\\pm h'(y)\\gtrsim 1$ for all $y\\in \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R$. Consequently, the bound\n $\\|D_\\pm^j f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim \\|f\\|_{H^j(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}$ is trivial.\n Conversely,\n \\[ D_\\pm^{\\ell}f(y)=\\sum_{j=0}^{\\ell-1}\n a_{\\pm,j}(y)f^{(j)}(y)+\\frac{1}{1\\pm h'(y)}f^{(\\ell)}(y) \\]\n for functions $a_{\\pm,j}\\in C^\\infty(\\R)$ and thus,\n \\[ \\|f^{(\\ell)}\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim\n \\|D_\\pm^{\\ell}f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\sum_{j=0}^{\\ell-1}\\|f^{(j)}\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\n \\lesssim \\|D_\\pm^{\\ell}f\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}+\\|f\\|_{H^{\\ell-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}. \\]\n Consequently, the claim follows inductively.\n\\end{proof}\n\n\\begin{remark}\n Lemma \\ref{lem:vHk} shows that the full range of energy bounds is\n available in the HSC. Even better, the evolution decays\n exponentially in these coordinates. This is a scaling effect.\n\\end{remark}\n\n\\subsection{Radial wave evolution in 5 space dimensions}\nLet $u\\in C^\\infty(\\R^{1,d})$ satisfy\n$(\\partial_t^2-\\Delta)u(t,\\cdot)=0$ and suppose $u(t,\\cdot)$ is\nradial. Then there exists a function\n$\\widehat u \\in C^\\infty(\\R^{1,1})$ such that\n$u(t,x)=\\widehat u(t,|x|)$. In addition, $\\widehat u(t,\\cdot)$ is even\nand satisfies\n\\[ (\\partial_t^2-\\partial_r^2-\\tfrac{d-1}{r}\\partial_r)\\widehat\nu(t,r)=0. \\]\nIt is well-known that radial wave evolution in five space dimensions\ncan be reduced to the case $d=1$. \nThis is a consequence of the intertwining identity\\footnote{Similar\n formulas exist for all odd dimensions.}\n\\begin{equation}\n\\label{eq:ident51}\n\\partial_r^2 (r^2\\partial_r+3r)=(r^2\\partial_r+3r)(\\partial_r^2+\\tfrac{4}{r}\\partial_r). \n\\end{equation}\nMore precisely, let $\\Omega\\subset \\R^{1,1}$ be a domain and suppose\n$\\widehat u\\in C^3(\\Omega)$.\nThen it follows directly from Eq.~\\eqref{eq:ident51} that\n$(\\partial_t^2-\\partial_r^2-\\frac{4}{r}\\partial_r)\\widehat u(t,r)=0$ implies\n$(\\partial_t^2-\\partial_r^2)\\widetilde u(t,r)=0$, where $\\widetilde\nu(t,r):=(r^2\\partial_r+3r)\\widehat u(t,r)$.\nThe converse is slightly more subtle. To begin with,\n$(\\partial_t^2-\\partial_r^2)\\widetilde u(t,r)=0$ implies that\n$(\\partial_t^2-\\partial_r^2-\\frac{4}{r}\\partial_r)\\widehat u(t,r)$\nbelongs to the kernel of $r^2\\partial_r+3r$. The equation\n$(r^2\\partial_r+3r)U(t,r)=0$ has the general solution $U(t,r)=\\frac{f(t)}{r^3}$ for a\nfree function $f$. Consequently, we obtain\n$(\\partial_t^2-\\partial_r^2-\\frac{4}{r}\\partial_r)\\widehat u(t,r)=0$ \\emph{but\n only for those $t$ where $(t,0)\\in \\Omega$}.\nThis appears to cause problems for the evolution in HSC, cf.~Fig.~\\ref{fig:HSC}.\n\nIn order to deal with this issue, we first\nrecall that\n\\[ \\mc D_0 \\widehat v(s,\\eta)=e^s\nh_1(\\eta)(\\partial_s+\\eta\\partial_\\eta)\\widehat v(s,\\eta),\\qquad\nh_1(\\eta):=\\frac{1}{\\eta h'(\\eta)-h(\\eta)} \\]\nand thus,\n\\begin{align*}\n\\mc D_0^2 \\widehat v(s,\\eta)\n&=\ne^{2s}h_1(\\eta)^2\\left [\\partial_s^2+2\\eta\\partial_s\\partial_\\eta\n +\\eta^2\\partial_\\eta^2\n+\\left (\\eta\\tfrac{h_1'(\\eta)}{h_1(\\eta)}+1\\right )\\partial_s+\\left \n(\\eta^2\\tfrac{h_1'(\\eta)}{h_1(\\eta)}+2\\eta\\right )\\partial_\\eta\n\\right ]\\widehat v(s,\\eta).\n\\end{align*}\nSimilarly,\n\\[ \\mc D_1 \\widehat v(s,\\eta)=-e^s\nh_1(\\eta)[h'(\\eta)\\partial_s+h(\\eta)\\partial_\\eta]\\widehat\nv(s,\\eta) \\]\nand therefore,\n\\begin{align*}\n \\mc D_1^2 \\widehat v(s,\\eta)\n=\ne^{2s}h_1(\\eta)^2 \\Big [\n&h'(\\eta)^2\\partial_s^2+2h'(\\eta)h(\\eta)\\partial_s\\partial_y+h(\\eta)^2\\partial_\\eta^2 \\\\\n&+\\Big ( h''(\\eta)h(\\eta)+h'(\\eta)^2+h'(\\eta)h(\\eta)\\tfrac{h_1'(\\eta)}{h_1(\\eta)}\\Big)\\partial_s \\\\\n&+\\Big ( \\tfrac{h_1'(\\eta)}{h_1(\\eta)}h(\\eta)^2+2h'(\\eta)h(\\eta)\\Big\n )\\partial_\\eta\n\\Big ]\\widehat v(s,\\eta) .\n\\end{align*}\nConsequently, the radial, $d$-dimensional wave equation in HSC,\n\\[ \\mc D_0^2 \\widehat v(s,\\eta)-\\mc D_1^2 \\widehat\nv(s,\\eta)-\\frac{(d-1)e^s}{\\eta}\\mc D_1 \\widehat v(s,\\eta)=0, \\]\ncan be written as the system\n\\begin{equation}\n\\label{eq:wavedHSCsys}\n \\partial_s \\begin{pmatrix}\\widehat v(s,\\cdot) \\\\ \\partial_s\n \\widehat v(s,\\cdot) \\end{pmatrix}=\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal L}_d \\begin{pmatrix}\n \\widehat v(s,\\cdot) \\\\ \\partial_s \\widehat v(s,\\cdot) \\end{pmatrix}, \n\\end{equation}\nwith the spatial differential operator\n\\[ \\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal L}_d \\begin{pmatrix} \\widehat f_1 \\\\ \\widehat\n f_2 \\end{pmatrix}:=\\begin{pmatrix} \\widehat f_2 \\\\\nc_{12}\\widehat f_1''\n+c^d_{11}\\widehat f_1'+c_{21}\\widehat f_2'\n+c^d_{20}\\widehat f_2 \\end{pmatrix} \\]\nand the coefficients\n\\begin{align*}\nc_{12}(\\eta)&=\\frac{h(\\eta)^2-\\eta^2}{1-h'(\\eta)^2} \\\\\nc^d_{11}(\\eta)&=-\\frac{1}{1-h'(\\eta)^2}\\left [\n\\frac{(d-1)h(\\eta)}{\\eta\n h_1(\\eta)}+[\\eta^2-h(\\eta)^2]\\frac{h_1'(\\eta)}{h_1(\\eta)}\n+2[\\eta-h'(\\eta)h(\\eta)] \\right ] \\\\\nc_{21}(\\eta)&=2\\frac{h'(\\eta)h(\\eta)-\\eta}{1-h'(\\eta)^2} \\\\\nc^d_{20}(\\eta)&=-\\frac{1}{1-h'(\\eta)^2}\\left [ \\frac{(d-1)h'(\\eta)}{\\eta\n h_1(\\eta)}+[\\eta-h'(\\eta)h(\\eta)]\\frac{h_1'(\\eta)}{h_1(\\eta)}\n-h''(\\eta)h(\\eta)-h'(\\eta)^2+1 \\right ].\n\\end{align*}\nThe equation $\\widetilde u(t,r)=(r^2\\partial_r+3r)\\widehat u(t,r)$ in HSC reads\n\\begin{equation}\n\\begin{split}\n\\label{eq:intertwHSC}\n \\widetilde v(s,\\eta)&=e^{-2s}\\eta^2 \\mc D_1 \\widehat\n v(s,\\eta)+3e^{-s}\\eta \\widehat v(s,\\eta) \\\\\n&=-e^{-s}\\eta^2 h_1(\\eta)h'(\\eta)\\partial_s\\widehat\n v(s,\\eta)-e^{-s}\\eta^2 h_1(\\eta)h(\\eta)\\partial_\\eta\\widehat\n v(s,\\eta)\n+3e^{-s}\\eta\\widehat v(s,\\eta) \\\\\n&=:e^{-s}\\left [\na_{11}(\\eta)\\partial_\\eta+a_{10}(\\eta)+a_{20}(\\eta)\\partial_s\\right ]\\widehat v(s,\\eta).\n\\end{split}\n\\end{equation}\nand differentiation with respect to $s$ yields\n\\begin{align*}\n \\partial_s \\widetilde v(s,\\eta)\n=&e^{-s}\\left\n [-a_{11}(\\eta)\\partial_\\eta-a_{10}(\\eta)+[a_{10}(\\eta)-a_{20}(\\eta)]\\partial_s\n+a_{11}(\\eta)\\partial_\\eta\\partial_s + a_{20}(\\eta)\\partial_s^2\\right\n ]\\widehat v(s,\\eta).\n\\end{align*}\nIf we assume for the moment that $\\widehat v$ solves\nEq.~\\eqref{eq:wavedHSCsys} with $d=5$, we may replace $\\partial_s^2 \\widehat\nv(s,\\eta)$ by lower-order derivatives in $s$. Explicitly, this yields\n\\begin{equation}\n\\begin{split}\n\\label{eq:intertwHSC_s}\n \\partial_s \\widetilde v(s,\\eta)\n=e^{-s}\\Big [ &a_{20}(\\eta)c_{12}(\\eta)\\partial_\\eta^2\n+[a_{20}(\\eta)c^5_{11}(\\eta)-a_{11}(\\eta)]\\partial_\\eta\n-a_{10}(\\eta) \\\\\n&+[a_{20}(\\eta)c_{21}(\\eta)+a_{11}(\\eta)]\\partial_\\eta\\partial_s \\\\\n&+[a_{10}(\\eta)+a_{20}(\\eta)(c^5_{20}(\\eta)-1)]\\partial_s\n\\Big ]\\widehat v(s,\\eta).\n\\end{split}\n\\end{equation}\nWe combine Eqs.~\\eqref{eq:intertwHSC} and \\eqref{eq:intertwHSC_s} into\nthe single vector-valued equation\n\\[ \\begin{pmatrix}\\widetilde v(s,\\cdot) \\\\ \\partial_s \\widetilde\n v(s,\\cdot)\n\\end{pmatrix}=e^{-s}\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal D}_5\n\\begin{pmatrix}\\widehat v(s,\\cdot) \\\\ \\partial_s \\widehat\n v(s,\\cdot)\n\\end{pmatrix} \\]\nwith the spatial differential operator\n\\[ \\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal D}_5 \\begin{pmatrix}\n\\widehat f_1 \\\\ \\widehat f_2 \\end{pmatrix}\n=\\begin{pmatrix}\na_{11} \\widehat f_1' + a_{10}\\widehat f_1+a_{20}\\widehat f_2 \\\\\nb_{12}\\widehat f_1''\n+b_{11}\\widehat f_1'\n+b_{10}\\widehat f_1\n+b_{21}\\widehat f_2'\n+b_{20}\\widehat f_2\n\\end{pmatrix} \\]\nand the coefficients\n\\begin{align*}\n b_{12}&=a_{20}c_{12} & b_{11}&=a_{20}c^5_{11}-a_{11} & \nb_{10}&=-a_{10} \\\\\n b_{21}&=a_{20}c_{21}+a_{11} & b_{20}&=a_{10}+a_{20}(c^5_{20}-1).\n\\end{align*}\nThe intertwining relation Eq.~\\eqref{eq:ident51} now manifests itself\nas \n\\begin{equation}\n \\label{eq:D5L5}\n \\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal D}_5\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal L}_5 \\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal f}=\n\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal L}_1\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal D}_5 \\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal f}+\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal\n D}_5\\widehat{\\mathbf} \\newcommand{\\mc}{\\mathcal f},\n\\end{equation}\nwhich may be verified by a straightforward (but, admittedly,\nlengthy) computation.\n\n\n \\begin{definition}\n \\label{def:D}\n For $R>0$, $\\eta\\in [-R,R]$, and\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f \\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5})^2$ radial, we set\n $\\mathbf} \\newcommand{\\mc}{\\mathcal E_1 \\mathbf} \\newcommand{\\mc}{\\mathcal f(\\eta):=\\mathbf} \\newcommand{\\mc}{\\mathcal f(\\eta e_1)$. Furthermore,\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal D_5 \\mathbf} \\newcommand{\\mc}{\\mathcal f:=\\widehat {\\mathbf} \\newcommand{\\mc}{\\mathcal D}_5\\mathbf} \\newcommand{\\mc}{\\mathcal E_1 \\mathbf} \\newcommand{\\mc}{\\mathcal f. \\]\n \\end{definition}\n\n\\begin{definition}\n Let $I\\subset \\R$ be a symmetric interval around the origin and\n $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$. Then we set\n \\begin{align*}\n C^k_\\pm(I)&:=\\{f\\in C^k(I): f(x)=\\pm f(-x) \\mbox{ for all }x\\in I\\} \\\\\n C^\\infty_\\pm(I)&:=\\{f\\in C^\\infty(I): f(x)=\\pm f(-x) \\mbox{ for all }x\\in I\\} \\\\\n H^k_\\pm(I)&:=\\{f\\in H^k(I): f(x)=\\pm f(-x) \\mbox{ for all }x\\in I\\} \n \\end{align*}\n\\end{definition}\n\nThe following result establishes the key mapping properties of\n$\\mathbf} \\newcommand{\\mc}{\\mathcal D_5$ which, in conjunction with Lemma \\ref{lem:vHk}, yield the\ndesired energy bounds for $v$. The proof is rather lengthy and\ntherefore postponed to the appendix.\n\n\n\\begin{proposition}\n \\label{prop:D}\n Fix $R>0$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k\\geq 2$. Then the operator $\\mathbf} \\newcommand{\\mc}{\\mathcal D_5$\n extends to a bijective map\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal D_5: H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5) \\times\n H_\\mathrm{rad}^{k}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\to H^{k}_-(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}_-(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R) \\]\n and we have\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal D_5\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\simeq \\|\\mathbf} \\newcommand{\\mc}{\\mathcal\n f\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_R)\\times H^{k}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_R)} \\]\n for all\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times\n H^{k}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$.\n\\end{proposition}\n\n\\begin{proof}\n See Section \\ref{sec:proofD}.\n\\end{proof}\n\n\n\n\n\\subsection{Semigroup formulation}\n \nSo far we have proved a priori bounds, i.e., we have assumed that the\nsolution already exists. Now we turn to the proof of existence. To\nthis end, we employ the machinery of strongly continuous semigroups.\nAs before, we start with the case $d=1$. From above we know that if\n$u$ satisfies $\\partial_t^2 u(t,x)-\\partial_x^2 u(t,x)=0$ and\n$v(s,y)=u(T+e^{-s}h(y),e^{-s}y)$, then $v_\\pm:=\\mc D_0v\\pm \\mc D_1 v$\nsatisfy\n\\[ [1\\pm h'(y)]\\partial_s v_\\pm(s,y)=-[y\\pm h(y)]\\partial_y\nv_\\pm(s,y). \\] Equivalently,\n\\[ \\partial_s v_\\pm(s,\\cdot)=L_\\pm v_\\pm(s,\\cdot) \\]\nwith the spatial differential operator\n\\[ L_\\pm f(y):=-\\frac{y\\pm h(y)}{1\\pm h'(y)}f'(y). \\]\n\n\n\\begin{proposition}\n \\label{prop:gen}\n Let $R\\geq \\tfrac12$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$. Then the operator\n $L_\\pm: C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})\\subset H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\to\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$\n is closable and its closure $\\overline{L_\\pm}$ generates a strongly\n continuous one-parameter semigroup $S_\\pm$ on $H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ with\n the bound\n \\[ \\|S_\\pm(s)f\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim\n e^{s\/2}\\|f\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)} \\]\n for all $s\\geq 0$ and $f\\in H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$.\n\\end{proposition}\n\n\\begin{proof}\n We define two inner products on $L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ by\n \\[ (f|g)_\\pm:=\\int_{-R}^R f(y)\\overline{g(y)}[1\\pm h'(y)]dy \\]\n and denote the induced norms by $\\|\\cdot\\|_\\pm$. A straightforward\n integration by parts using $R\\geq\\frac12$ yields the bound\n \\[ \\Re (L_\\pm f|f)_\\pm\\leq \\tfrac12 \\|f\\|_\\pm^2 \\]\n for all $f\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$,\n cf.~Eq.~\\eqref{eq:energyv-}. Furthermore, we set\n \\[ D_\\pm f(y)=\\frac{1}{1\\pm h'(y)}f'(y) \\]\n and define an inner product\n \\[ (f|g)_{\\pm,k-1}:=\\sum_{j=0}^{k-1} (D_\\pm^j f|D_\\pm^j g)_\\pm \\]\n with induced norm $\\|\\cdot\\|_{\\pm,k-1}$. Recall from the proof of\n Lemma \\ref{lem:vHk} that\n $\\|\\cdot\\|_{\\pm,k-1}\\simeq \\|\\cdot\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}$. We have the\n commutator relation $[D_\\pm,L_\\pm]=-D_\\pm$ and thus,\n \\begin{align*}\n \\Re(L_\\pm f|f)_{\\pm,k-1}&=\\Re \\sum_{j=0}^{k-1} (D_\\pm^j L_\\pm f|D_\\pm^j f)_\\pm=\\sum_{j=0}^{k-1} \\left [\\Re (L_\\pm D_\\pm^j f|D_\\pm^j f)_\\pm-j(D_\\pm^j f|D_\\pm^j f)_\\pm \\right ] \\\\\n &\\leq \\tfrac12 \\sum_{j=0}^{k-1} \\|D_\\pm^j f\\|_\\pm^2 \\\\\n &=\\tfrac12 \\|f\\|_{\\pm,k-1}^2\n \\end{align*}\n for all $f\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$. Thus, by the\n Lumer-Phillips Theorem \\cite{EngNag00} it suffices to prove that the\n range of $1-L_\\pm$ is dense in $H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$. In other words, we\n have to show that for each given $F\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$,\n there exists an $f\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ such that\n $(1-L_\\pm)f=F$. The equation $(1-L_+)f=F$ reads\n \\[ \\frac{y+h(y)}{1+h'(y)}f'(y)+f(y)=F(y). \\]\n An explicit solution is given by\n \\begin{align*}\n f(y)&=\\frac{1}{y+h(y)}\\int_{1\/2}^y [1+h'(t)]F(t)dt \\\\\n &=\n \\frac{y-\\frac12}{y+h(y)}\\int_0^1 [1+h'(\\tfrac12+t(y-\\tfrac12))]F(\\tfrac12+t(y-\\tfrac12))dt. \n \\end{align*}\n Since $y=\\frac12$ is the only zero of $y+h(y)$ and\n $1+h'(\\frac12)\\not=0$, it is evident that\n $f\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$. Analogously, one proves the\n density of the range of $1-L_-$ and we are done.\n\\end{proof}\n\nThe next lemma shows that the closure $\\overline{L_\\pm}$ acts as a\nclassical differential operator, provided the underlying Sobolev space\ncontains $C^1(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$.\n\n\\begin{lemma}\n \\label{lem:closure}\n Let $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 3$, and consider the closure\n $\\overline{L_\\pm}$ of the operator\n $L_\\pm: C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R} )\\subset H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\to\n H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$.\n Then $\\mc D(\\overline{L_\\pm})\\subset C^{k-2}(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ and\n we have\n \\[ \\overline{L_\\pm} f(y)=-\\frac{y\\pm h(y)}{1\\pm h'(y)}f'(y) \\]\n for all $f\\in \\mc D(\\overline{L_\\pm})$.\n\\end{lemma}\n\n\\begin{proof}\n Let $f\\in \\mc D(\\overline{L_\\pm})$. By definition, there exists a\n sequence $(f_n)_{n\\in\\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}}\\subset C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ such that\n $f_n\\to f$ and $L_\\pm f_n\\to \\overline{L_\\pm}f$ in\n $H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$. By Sobolev embedding we see that\n $f \\in C^{k-2}(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ and\n \\begin{align*}\n \\left | L_\\pm f_n(y)+\\frac{y\\pm h(y)}{1\\pm h'(y)}f'(y)\\right |&\\lesssim \n \\|f_n'-f'\\|_{L^\\infty(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\lesssim \\|f_n-f\\|_{H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\\to 0\n \\end{align*}\n as $n\\to\\infty$.\n\\end{proof}\n\nAs a corollary, we obtain classical solutions for the half-wave\nequations.\n\n\\begin{corollary}\n \\label{cor:class}\n Let $R\\geq \\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k\\geq 3$. Furthermore, let\n $f_\\pm \\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ and set\n \\[ v_\\pm(s,y):=S_\\pm(s)f_\\pm(y), \\]\n where $S_\\pm$ is the semigroup on $H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ from Proposition\n \\ref{prop:gen}. Then\n $v_\\pm\\in C^1([0,\\infty)\\times \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ and\n \\[ (\\mc D_0\\mp \\mc D_1)v_\\pm(s,y)=0. \\]\n\\end{corollary}\n\n\\begin{proof}\n Since\n $f_\\pm \\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})\\subset \\mc\n D(\\overline{L_\\pm})$,\n semigroup theory implies\n $\\partial_s v_\\pm(s,\\cdot)=\\overline{L_\\pm}v_\\pm(s,\\cdot)$.\n Consequently, Lemma \\ref{lem:closure} finishes the proof.\n\\end{proof}\n\nNow we can easily construct a semigroup that produces a solution to\nthe one-dimensional wave equation in HSC.\n\n\\begin{definition}\n Let $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k\\geq 3$. For\n $(f_1,f_2)\\in H^{k}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ and\n $(f_-,f_+)\\in H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ we set\n \\begin{align*} \\mathbf} \\newcommand{\\mc}{\\mathcal A\\left (\\begin{array}{c} f_1 \\\\ f_2 \\end{array}\n \\right )(y):= \\frac{1}{yh'(y)-h(y)}\\left (\n \\begin{array}{c}\n (y+h(y))f_1'(y)+(1+h'(y))f_2(y) \\\\\n (y-h(y))f_1'(y)+(1-h'(y))f_2(y)\n \\end{array} \n \\right )\n \\end{align*}\n and\n \\[\n \\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{c}f_- \\\\ f_+\\end{array} \\right )(y):=\n \\frac12 \\left (\\begin{array}{c}\n -\\int_0^y (1-h'(t))f_-(t)dt+\\int_0^y (1+h'(t))f_+(t)dt \\\\\n (y-h(y))f_-(y)-(y+h(y))f_+(y)\n \\end{array} \\right ).\n \\]\n Furthermore, for $s\\geq 0$, we define\n $\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s): H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R) \\to\n H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ by\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s):=e^{-s}\\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{cc}S_-(s) & 0 \\\\\n 0 &\n S_+(s) \\end{array}\n \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal\n A, \\]\n where $S_\\pm$\n are the\n semigroups on\n $H^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$\n constructed in\n Proposition\n \\ref{prop:gen}.\n \\end{definition}\n\n As the following\n result shows,\n $\\mathbf} \\newcommand{\\mc}{\\mathcal S_1$ is the\n solution operator\n for the\n one-dimensional\n wave equation in\n HSC with a\n Dirichlet\n condition at the center.\n\n \\begin{proposition}\n \\label{prop:S1}\n Let $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in C_-^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})^2$ and set\n $v(s,y):=[\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_1(y)$. Then\n $v\\in C^2([0,\\infty)\\times \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$, $v(s,\\cdot)$ is\n odd for all $s\\geq 0$, and we have\n $\\partial_s v(s,y)=[\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_2(y)$ as well as\n \\[ \\mc D_0^2 v(s,y)-\\mc D_1^2 v(s,y)=0 \\]\nfor all $(s,y)\\in [0,\\infty)\\times \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R}$.\n Furthermore, the family $\\{\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s): s\\geq 0\\}$ forms a\n strongly continuous semigroup of bounded operators on\n $H_-^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)\\times H_-^{k-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)$ with generator\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal L_1 =\\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{cc} \\overline{L_-} & 0\n \\\\ 0 & \\overline{L_+} \\end{array} \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal A-\\mathbf} \\newcommand{\\mc}{\\mathcal I. \\]\n \\end{proposition}\n\n\n\\begin{proof}\nWe define $v_\\pm$ by\n\\[ \\begin{pmatrix}v_-(s,\\cdot) \\\\ v_+(s,\\cdot)\\end{pmatrix}:=\n\\begin{pmatrix}S_-(s) & 0 \\\\ 0 & S_+(s)\\end{pmatrix}\\mathbf} \\newcommand{\\mc}{\\mathcal A\\mathbf} \\newcommand{\\mc}{\\mathcal f. \\]\nFrom Corollary \\ref{cor:class} we have $v_\\pm \\in C^1([0,\\infty)\\times\n\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R})$ and $(\\mc D_0\\mp \\mc D_1)\nv_\\pm=0$. Furthermore,\n\\begin{align*}\n v_-(0,-y)&=\\frac{1}{-yh'(-y)-h(-y)}\\left\n [(-y+h(-y))f_1'(-y)+(1+h'(-y))f_2(-y)\\right] \\\\\n&=\\frac{1}{yh'(y)-h(y)}\\left [(-y+h(y))f_1'(y)-(1-h'(y))f_2(y)\\right]\n \\\\\n&=-v_+(0,y).\n\\end{align*}\nSince $v_-(s,-y)$ satisfies the same equation as $v_+(s,y)$, it\nfollows from the a priori bound Eq.~\\eqref{eq:aprioriv+} that\n\\[ \\|v_+(s,\\cdot)+v_-(s,-(\\cdot))\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}\n\\lesssim e^{s\/2}\\|v_+(0,\\cdot)+v_-(0,-(\\cdot))\\|_{L^2(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R)}=0 \\]\nand thus, \n$v_-(s,-y)=-v_+(s,y)$ for all $(s,y)\\in [0,\\infty)\\times \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R}$.\nConsequently,\nthe function\n $y\\mapsto -(1-h'(y))v_-(s,y)+(1+h'(y))v_+(s,y)$ is even, whereas\n $y\\mapsto (y-h(y))v_-(s,y)-(y+h(y))v_+(s,y)$ is odd.\nThis shows that $\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)$ maps odd functions to odd functions.\n The first statement now follows from Eq.~\\eqref{eq:vvpm} (or a straightforward computation). To prove\n the semigroup property, we first note that $\\mathbf} \\newcommand{\\mc}{\\mathcal A\\mathbf} \\newcommand{\\mc}{\\mathcal B=\\mathbf} \\newcommand{\\mc}{\\mathcal I$ and\n thus,\n \\begin{align*}\n \\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s+t)&=e^{-s-t}\\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{cc}S_-(s+t) & 0 \\\\\n 0 & S_+(s+t) \\end{array} \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal A \\\\\n &=e^{-s-t}\\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{cc}S_-(s) & 0 \\\\\n 0 & S_+(s) \\end{array} \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal A\\mathbf} \\newcommand{\\mc}{\\mathcal B\n \\left (\\begin{array}{cc}S_-(t) & 0 \\\\\n 0 & S_+(t) \\end{array} \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal A \\\\\n &=\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(t)\n \\end{align*}\n for all $s,t\\geq 0$. Furthermore, it is obvious that\n $s\\mapsto \\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)$ is strongly continuous. \nFinally, $\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(0)\\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal B\\mathbf} \\newcommand{\\mc}{\\mathcal A \\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal f$ since $\\mathbf} \\newcommand{\\mc}{\\mathcal f$ is\nodd.\nThe statement about the generator is obvious.\n\\end{proof}\n\nBy conjugating with $\\mathbf} \\newcommand{\\mc}{\\mathcal D_5$, we obtain the solution operator for the\n5-dimensional wave equation. This leads to the main result of this\nsection.\n\n\\begin{definition}\n \\label{def:S5}\n For $s\\geq 0$ we define\n $\\mathbf} \\newcommand{\\mc}{\\mathcal S_5(s): H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times\n H^k_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5) \\to H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times\n H^k_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$ by\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal S_5(s):=e^s\\mathbf} \\newcommand{\\mc}{\\mathcal D_5^{-1}\\mathbf} \\newcommand{\\mc}{\\mathcal S_1(s)\\mathbf} \\newcommand{\\mc}{\\mathcal D_5. \\]\n\\end{definition}\n\n\\begin{theorem}\n \\label{thm:S5}\n The family $\\{\\mathbf} \\newcommand{\\mc}{\\mathcal S_5(s): s\\geq 0\\}$ forms a strongly continuous\n semigroup of bounded operators on\n $H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times H^k_{\\mathrm{rad}}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$ and\n we have\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal S_5(s)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}\\lesssim\n e^{s\/2} \\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)} \\]\n for all $s\\geq 0$ and\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times\n H^k_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$.\n The generator $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5$ of $\\mathbf} \\newcommand{\\mc}{\\mathcal S_5$ is given by\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal L_5=\\mathbf} \\newcommand{\\mc}{\\mathcal D_5^{-1}\\mathbf} \\newcommand{\\mc}{\\mathcal B\\left (\\begin{array}{cc}\\overline{L_-} & 0 \\\\\n 0 & \\overline{L_+}\\end{array}\n \\right )\\mathbf} \\newcommand{\\mc}{\\mathcal A\\mathbf} \\newcommand{\\mc}{\\mathcal D_5. \\]\n Furthermore, the function\n $v(s,\\cdot)=[\\mathbf} \\newcommand{\\mc}{\\mathcal S_5(s) \\mathbf} \\newcommand{\\mc}{\\mathcal\n f]_1$\n belongs to\n $C^2([0,\\infty)\\times\n \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5})$\n and satisfies\n \\[ \\mc D_0^2 v-\\mc D^j\\mc\n D_j v=0. \\]\n Finally,\n $\\partial_s v(s,\\cdot)=[\\mathbf} \\newcommand{\\mc}{\\mathcal\n S_5(s)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_2$.\n \\end{theorem}\n\n Finally, we obtain the\n explicit form of $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5$.\n To keep equations within\n margins, we define the\n following auxiliary\n quantities.\n\n \\begin{definition}\n \\label{def:H}\n We set\n \\begin{align*}\n H_0{}^0(s,y)&:=\\frac{e^s}{y^\\ell\\partial_\\ell h(y)-h(y)} & H_0{}^j(s,y)&:=\\frac{e^s y^j}{y^\\ell\\partial_\\ell h(y)-h(y)} \\\\\n H_j{}^0(s,y)&:=-\\frac{e^s\\partial_j h(y)}{y^\\ell\\partial_\\ell h(y)-h(y)} &\n H_j{}^k(s,y)&:=e^s\\delta_j{}^k-\\frac{e^s\\partial_j h(y)}{y^\\ell\\partial_\\ell h(y)-h(y)}y^k\n \\end{align*}\n \\end{definition}\n Then we have\n $\\mc\n D_\\mu=H_\\mu{}^\\nu \\partial_\\nu$,\n see Definition \\ref{def:mcD},\n and thus,\n \\begin{equation}\n \\begin{split}\n \\label{eq:DmuDmu}\n \\mc D^\\mu\\mc\n D_\\mu&=H^{\\mu\\nu}\\partial_\\nu(H_\\mu{}^\\lambda\\partial_\\lambda)\n =H^{\\mu\\nu}H_\\mu{}^\\lambda\\partial_\\nu\\partial_\\lambda+H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^\\lambda\\partial_\\lambda \\\\\n &=H^{\\mu\n 0}H_\\mu{}^0\\partial_0^2+2\n H^{\\mu\n j}H_\\mu{}^0\\partial_j\\partial_0\n +H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^0\\partial_0\n +H^{\\mu\n j}H_\\mu{}^k \\partial_j\\partial_k+H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^j \\partial_j.\n \\end{split}\n \\end{equation}\n It follows that\n $\\mc D^\\mu\\mc D_\\mu v=0$ is\n equivalent to\n \\[ \\partial_0 \\left\n (\\begin{array}{c}v\n \\\\ \\partial_0\n v \\end{array} \\right )\n =\\left (\\begin{array}{c}\\partial_0 v \\\\\n -\\frac{H^{\\mu\n j}H_\\mu{}^k}{H^{\\mu\n 0}H_\\mu{}^0}\\partial_j\\partial_k\n v-\\frac{H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^j}{H^{\\mu\n 0}H_\\mu{}^0}\\partial_j\n v -2\\frac{H^{\\mu\n j}H_\\mu{}^0}{H^{\\mu\n 0}H_\\mu{}^0}\\partial_j \\partial_0\n v-\\frac{H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^0}{H^{\\mu\n 0}H_\\mu{}^0} \\partial_0\n v \\end{array}\n \\right )\\] and thus,\n \\begin{equation}\n \\label{eq:L5}\n \\mathbf} \\newcommand{\\mc}{\\mathcal L_5 \\left (\\begin{array}{c} f_1 \\\\ f_2 \\end{array} \\right )=\n \\left (\\begin{array}{c}\n f_2 \\\\ \n -\\frac{H^{\\mu j}H_\\mu{}^k}{H^{\\mu 0}H_\\mu{}^0}\\partial_j\\partial_k f_1-\\frac{H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^j}{H^{\\mu 0}H_\\mu{}^0}\\partial_j f_1\n -2\\frac{H^{\\mu j}H_\\mu{}^0}{H^{\\mu 0}H_\\mu{}^0}\\partial_j f_2-\\frac{H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^0}{H^{\\mu 0}H_\\mu{}^0} f_2 \\end{array} \\right ).\n \\end{equation}\n \n\n \n\n \\section{Wave maps in hyperboloidal similarity coordinates}\n\n \\noindent Now we return to the wave maps equation\n \\begin{equation}\n \\label{eq:mainuu}\n \\left (\\partial_t^2-\\Delta_x\\right )u(t,x)=\\frac{2|x|u(t,x)-\\sin(2|x|u(t,x))}{|x|^3}\n \\end{equation}\n with the one-parameter family $\\{u_T^*: T\\in \\R\\}$ of blowup\n solutions given by\n \\[ u_T^*(t,x)=\\frac{4}{|x|}\\arctan\\left\n (\\frac{|x|}{T-t+\\sqrt{(T-t)^2+|x|^2}}\\right ). \\]\n\n\n \\subsection{Perturbations of the blowup solution}\n \\label{sec:pert}\n We would like to study the stability of $u^*_{T}$ and thus, we\n insert the ansatz $u(t,x)=u^*_{T}(t,x)+\\widetilde u(t,x)$ into\n Eq.~\\eqref{eq:mainuu} which yields\n \\begin{equation}\n \\label{eq:mainutilde}\n \\left [\\partial_t^2-\\Delta_x+V_T(t,x)\\right ]\\widetilde\n u(t,x)=F_T(\\widetilde u(t,x),t,x),\n \\end{equation}\n where\n \\begin{align*}\n V_T(t,x)&=\\frac{2\\cos(2|x|u_T^*(t,x))-2}{|x|^2} \n \\end{align*}\n and\n \\begin{align*}\n F_T(\\widetilde u(t,x),t,x)=-|x|^{-3}\\Big [ \n &\\sin\\big (2|x|u_T^*(t,x)+2|x|\\widetilde u(t,x)\\big ) \\\\\n &-\\sin(2|x|u_T^*(t,x))-2|x|\\cos(2|x|u_T^*(t,x))\\widetilde u(t,x)\n \\Big ].\n \\end{align*}\n\n In hyperboloidal similarity coordinates,\n Eq.~\\eqref{eq:mainutilde} reads\n \\begin{equation}\n \\label{eq:mainv}\n -\\mc D^\\mu\\mc D_\\mu v(s,y)+V_T(\\eta_T(s,y))v(s,y)=F_T(v(s,y),\\eta_T(s,y)),\n \\end{equation}\n where, as always, $\\eta_T(s,y)=(T+e^{-s}h(y), e^{-s}y)$ and\n $v(s,y)=\\widetilde u(\\eta_T(s,y))$. By definition of\n $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5$, Eq.~\\eqref{eq:mainv} is equivalent to\n \\begin{equation}\n \\begin{split}\n \\label{eq:vsys}\n \\partial_s \\begin{pmatrix}\n v(s,y) \\\\\n \\partial_s v(s,y)\n \\end{pmatrix}\n =& \\mathbf} \\newcommand{\\mc}{\\mathcal L_5\\begin{pmatrix}\n \\ v(s,\\cdot) \\\\\n \\partial_s v(s,\\cdot)\n \\end{pmatrix}(y) +\\begin{pmatrix} 0 \\\\ (H^{\\mu\n 0}(s,y)H_\\mu{}^0(s,y))^{-1}V_T(\\eta_T(s,y)) v(s,y)\n \\end{pmatrix} \\\\\n &-\\begin{pmatrix}\n 0 \\\\\n (H^{\\mu 0}(s,y)H_\\mu{}^0(s,y))^{-1} F_T(\n v(s,y),\\eta_T(s,y))\n \\end{pmatrix}.\n \\end{split}\n \\end{equation}\n Note that\n \\[ H^{\\mu 0}(s,y)H_\\mu{}^0\n (s,y)=-e^{2s}\\frac{1-\\partial^jh(y)\\partial_j\n h(y)}{[y^\\ell \\partial_\\ell h(y)-h(y)]^2}=:e^{2s}H(y)^{-1} \\]\n and\n \\[ u_T^*(\\eta_T(s,y))=\\frac{4e^s}{|y|}\\arctan\\left\n (\\frac{|y|}{\\sqrt{|y|^2+h(y)^2}-h(y)}\\right\n )=:e^s\\alpha_0(y). \\]\n Observe that $\\alpha_0\\in C^\\infty(\\R^5)$. Consequently,\n \\[ V(y):=(H^{\\mu 0}(s,y)H_\\mu{}^0(s,y))^{-1}V_T(\\eta_T(s,y))=\n H(y)\\frac{2\\cos(2|y|\\alpha_0(y))-2}{|y|^2} \\]\n is independent of $s$. Furthermore,\n \\begin{align*}\n F_T(v(s,y),\\eta_T(s,y))\n =-e^{3s}|y|^{-3}\\Big [ \n &\\sin\\big (2e^{-s}|y|u_T^*(\\eta_T(s,y))+2e^{-s}|y|v(s,y)\\big ) \\\\\n &-\\sin\\big(2e^{-s}|y|u_T^*(\\eta_T(s,y))\\big ) \\\\\n &-2e^{-s}|y|\\cos \\big (2e^{-s}|y|u_T^*(\\eta_T(s,y))\\big )v(s,y)\n \\Big ]\n \\end{align*}\n and we can write\n \\[(H^{\\mu 0}(s,y)H_\\mu{}^0(s,y))^{-1} F_T(\n v(s,y),\\eta_T(s,y))=e^s H(y)N(e^{-s}|y|v(s,y),y), \\] where\n \\begin{align*}\n N(p,y):=-\\frac{ \n \\sin(2|y|\\alpha_0(y)+2p)\n -\\sin(2|y|\\alpha_0(y))-2\\cos(2|y|\\alpha_0(y))p}{|y|^3}.\n \\end{align*}\n In order to obtain an autonomous equation, we rescale and write\n Eq.~\\eqref{eq:vsys} in terms of\n \\[ \\Phi(s)(y):=\\begin{pmatrix} \\phi_1(s)(y) \\\\\n \\phi_2(s)(y)\\end{pmatrix}:=e^{-s}\\begin{pmatrix} v(s,y)\n \\\\ \\partial_s v(s,y) \\end{pmatrix}. \\] This yields\n \\begin{equation}\n \\label{eq:Psi}\n \\partial_s \\Phi(s)=(\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I+\\mathbf} \\newcommand{\\mc}{\\mathcal L')\\Phi(s)+\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s)), \n \\end{equation}\n where\n \\begin{align*}\n \\mathbf} \\newcommand{\\mc}{\\mathcal L' \\begin{pmatrix}\n f_1 \\\\ f_2 \\end{pmatrix}(y)&:=\\begin{pmatrix} 0 \\\\ V(y) f_1(y) \\end{pmatrix} \\\\\n \\mathbf} \\newcommand{\\mc}{\\mathcal N \\left (\\begin{array}{c}\n f_1 \\\\ f_2 \\end{array} \\right )(y)&:=\\left (\\begin{array}{c} 0 \\\\ -H(y) N(|y|f_1(y),y) \\end{array} \\right ).\n \\end{align*}\n In the following, we write $\\mathbf} \\newcommand{\\mc}{\\mathcal L:=\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I+\\mathbf} \\newcommand{\\mc}{\\mathcal L'$.\n\n\n\n\n\\subsection{Existence of the linearized evolution}\nThe rest of this section is devoted to the analysis of\nEq.~\\eqref{eq:Psi}. The first step is to develop a sufficiently good\nunderstanding of the linearized equation that is obtained from\nEq.~\\eqref{eq:Psi} by dropping the nonlinearity. We start with a\nsimple lemma that constructs a semigroup $\\mathbf} \\newcommand{\\mc}{\\mathcal S$ which governs the\nlinearized flow. In particular, this yields the well-posedness of the\nlinearized Cauchy problem in the sense of semigroup theory.\n\n\n\\begin{definition}\n For $R>0$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$ we set\n \\[ \\mc H^{k}_R:=H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\times\n H^{k}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5) \\] and\n \\[ \\|(f_1,f_2)\\|_{\\mc\n H^{k}_R}^2:=\\|f_1\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}^2+\\|f_2\\|_{H^{k}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}^2 \\]\n\\end{definition}\n\n\\begin{lemma}\n \\label{lem:S}\n Let $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k\\geq 3$. Then the operator\n $\\mathbf} \\newcommand{\\mc}{\\mathcal L=\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I+\\mathbf} \\newcommand{\\mc}{\\mathcal L'$ is the generator of a strongly\n continuous semigroup $\\{\\mathbf} \\newcommand{\\mc}{\\mathcal S(s): s\\geq 0\\}$ on $\\mc H^k_R$.\n Furthermore, every $\\lambda\\in \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ with\n $\\Re\\lambda>-\\frac12$ is an eigenvalue with finite algebraic\n multiplicity.\n\\end{lemma}\n\n\\begin{proof}\n Since $H^{k+1}_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)\\hookrightarrow\n H^k_{\\mathrm{rad}}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$ is compact and $V\\in C^\\infty(\\R^5)$, it\n follows that $\\mathbf} \\newcommand{\\mc}{\\mathcal L': \\mc H_R^k\\to\\mc H_R^k$ is compact and the\n bounded perturbation theorem implies that $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I+\\mathbf} \\newcommand{\\mc}{\\mathcal L'$\n generates a semigroup $\\mathbf} \\newcommand{\\mc}{\\mathcal S(s)$ on $\\mc H_R^k$.\nNow suppose $\\lambda\\in \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ and\n$\\Re\\lambda>-\\frac12$. Since $\\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I)\\subset \\{z\\in \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}:\n\\Re z\\leq -\\frac12\\}$, we have the identity $\\lambda \\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L=[\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal\nR_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)](\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_5+\\mathbf} \\newcommand{\\mc}{\\mathcal\nI)$. Consequently, $1\\in \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda))$ and by\nthe compactness of $\\mathbf} \\newcommand{\\mc}{\\mathcal L'$ we see that in fact $1\\in \\sigma_p(\\mathbf} \\newcommand{\\mc}{\\mathcal\nL'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda))$.\nThis means that there exists a nonzero $\\mathbf} \\newcommand{\\mc}{\\mathcal g\\in \\mc H_R^k$ in the\nkernel of $\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)$. Thus, $\\mathbf} \\newcommand{\\mc}{\\mathcal\nf:=\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal g$ is nonzero, belongs to $\\mc\nD(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$, and satisfies\n\\[ (\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f= \n[\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal\nR_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)](\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_5+\\mathbf} \\newcommand{\\mc}{\\mathcal\nI)\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal g=[\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal\nR_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)]\\mathbf} \\newcommand{\\mc}{\\mathcal g=\\mathbf} \\newcommand{\\mc}{\\mathcal 0.\n \\]\n In other words, $\\mathbf} \\newcommand{\\mc}{\\mathcal f$ is an eigenfunction of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$ to the\n eigenvalue $\\lambda\\in \\sigma_p(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$. Finally, suppose that\n $\\lambda$ has infinite algebraic multiplicity. Then, by \\cite{Kat95},\n p.~239, Theorem 5.28, $\\lambda$ would belong to the essential\n spectrum of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$. This, however, is impossible since\n $\\lambda\\notin \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I)$ and the essential spectrum is\n stable under compact perturbations, see \\cite{Kat95}, p.~244, Theorem\n 5.35.\n\\end{proof}\n\n\n\n\\subsection{Spectral analysis of the generator}\n\nNext, we turn to the analysis of the point spectrum of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$. As a\nmatter of fact, the spectral analysis of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$ is essentially\nindependent of the particular choice of the height function $h$ and\ncan be reduced to the case $h(y)=-1$. This will allow us to utilize\nthe spectral information from \\cite{CosDonXia16, CosDonGlo17} to show\nthat the only unstable eigenvalue of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$ is $\\lambda=1$.\n\n\\begin{definition}\n \\label{def:f1}\n We set\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*(y):=\\begin{pmatrix}f_{1,1}^*(y) \\\\\n f_{1,2}^*(y)\\end{pmatrix} :=\\frac{1}{|y|^2+h(y)^2}\\begin{pmatrix}1\n \\\\ 2\\end{pmatrix}. \\]\n\\end{definition}\n\n\\begin{lemma}\n \\label{lem:sigmapL}\n Let $R\\geq \\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, and $k\\geq 4$. Furthermore, let\n $\\mathbf} \\newcommand{\\mc}{\\mathcal L: \\mc D(\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\subset \\mc H^k_R\\to\\mc H^k_R$ be the operator\n defined in Lemma \\ref{lem:S}. Then\n $\\ker(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)=\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\rangle$. Moreover, if\n $\\lambda\\in \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ and $\\Re\\lambda\\geq 0$, then $\\lambda=1$.\n\\end{lemma}\n\n\\begin{proof}\n Obviously, $\\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5})^2$ and thus,\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\in \\mc D(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$. The blowup solution $u_T^*$ satisfies\n \\[\n (\\partial_t^2-\\Delta_x)u_T^*(t,x)=\\frac{2|x|u_T^*(t,x)-\\sin(2|x|u_T^*(t,x))}{|x|^3} \\]\n and differentiating this equation with respect to $T$ yields\n \\[ (\\partial_t^2-\\Delta_x)\\partial_T\n u_T^*(t,x)=-\\frac{2\\cos(2|x|u_T^*(t,x))-2}{|x|^2}\\partial_T\n u_T^*(t,x).\n \\]\n A straightforward computation yields\n \\[ \\partial_T u_T^*(t,x)=\\frac{4}{|x|}\\partial_T \\arctan\\left\n (\\frac{|x|}{T-t+\\sqrt{(T-t)^2+|x|^2}}\\right )\n =-\\frac{2}{(T-t)^2+|x|^2} \\] and thus,\n \\[ (\\partial_T\n u_T^*)(T+e^{-s}h(y),e^{-s}y)=-\\frac{2e^{2s}}{|y|^2+h(y)^2}.\n \\]\n Consequently, $\\partial_s (e^s \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*)=\\mathbf} \\newcommand{\\mc}{\\mathcal L (e^s \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*)$,\n which is equivalent to $(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$ and thus,\n $\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\rangle \\subset \\ker(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)$. The reverse\n inclusion is a simple consequence of basic ODE theory since we\n restrict ourselves to radial functions.\n\n Suppose now that $\\lambda\\in \\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ and $\\Re\\lambda\\geq 0$.\n By Lemma \\ref{lem:S} it follows that $\\lambda\\in \\sigma_p(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$\n and thus, there exists a nontrivial\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f=(f_1,f_2)\\in \\mc D(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ such that\n $(\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$. Equivalently,\n $\\partial_s (e^{\\lambda s} \\mathbf} \\newcommand{\\mc}{\\mathcal f)=\\mathbf} \\newcommand{\\mc}{\\mathcal L(e^{\\lambda s} \\mathbf} \\newcommand{\\mc}{\\mathcal f)$ or\n \\[ \\partial_s (e^{(\\lambda+1)s}\\mathbf} \\newcommand{\\mc}{\\mathcal f)=(\\mathbf} \\newcommand{\\mc}{\\mathcal L_5+\\mathbf} \\newcommand{\\mc}{\\mathcal\n L')(e^{(\\lambda+1)s}\\mathbf} \\newcommand{\\mc}{\\mathcal f). \\]\n By Sobolev embedding, the function $v(s,y):=e^{(\\lambda+1)s}f_1(y)$\n belongs to $C^2(\\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2})$ and by definition of $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5$\n and $\\mathbf} \\newcommand{\\mc}{\\mathcal L'$, $v$ satisfies\n \\begin{equation}\n \\label{eq:sigmaPLv}\n -\\mc D^\\mu \\mc D_\\mu v(s,y)+V_T(\\eta_T(s,y))v(s,y)=0 \n \\end{equation}\n for all $(s,y)\\in \\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}$. Note that $v$ is nontrivial\n since the first component of $(\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$ reads\n $\\lambda f_1-f_2+f_1=0$. Now recall that\n \\[ V_T(\\eta_T(s,y))=e^{2s}\\frac{2\\cos(2|y|\\alpha_0(y))-2}{|y|^2} \\]\n and, since $h(y)<0$ for all $y\\in \\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1\/2}^5}$, we can\n write\n \\[ \\alpha_0(y)=\\frac{4}{|y|}\\arctan\\left\n (\\frac{|y|}{\\sqrt{|y|^2+h(y)^2}-h(y)}\\right\n )=\\frac{4}{|y|}\\arctan\\left\n (\\frac{-|y|\/h(y)}{1+\\sqrt{1+|y|^2\/h(y)^2}}\\right ). \\]\n Consequently,\n \\[\n V_T(\\eta_T(s,y))=e^{2s}h(y)^{-2}\\frac{2\\cos(2|y|\\alpha_0(y))-2}{|y|^2\/h(y)^2}=e^{2s}h(y)^{-2}V_0(y\/h(y)) \\]\n with\n \\begin{equation}\n \\label{def:V0}\n V_0(\\xi)=\\frac{2}{|\\xi|^2}\\left [\\cos\\left (8\\arctan\\left\n (\\frac{|\\xi|}{1+\\sqrt{1+|\\xi|^2}}\\right)\\right)-1\\right\n ]=-\\frac{16}{(1+|\\xi|^2)^2}. \n \\end{equation}\n Therefore, by setting $v(s,y):=w(s-\\log(-h(y)),-y\/h(y))$,\n Eq.~\\eqref{eq:sigmaPLv} transforms into\n \\[ \\left [\n \\partial_\\tau^2+2\\xi^j\\partial_{\\xi^j}\\partial_\\tau-(\\delta^{jk}-\\xi^j\\xi^k)\\partial_{\\xi^j}\\partial_{\\xi^k}\n +\\partial_\\tau+2\\xi^j\\partial_{\\xi^j}+V_0(\\xi)\\right\n ]w(\\tau,\\xi)=0 \\]\n for all $(\\tau,\\xi)\\in \\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5$, see\n Eq.~\\eqref{eq:sytauxiwave}. Explicitly, we have\n \\begin{align*}\n w(\\tau,\\xi)&=v\\left (\\tau+\\log\\left (\\frac{2}{2+\\sqrt{2(1+|\\xi|^2)}}\\right ), \\frac{2\\xi}{2+\\sqrt{2(1+|\\xi|^2)}}\\right ) \\\\\n &=e^{(\\lambda+1)\\tau} \\left (\\frac{2}{2+\\sqrt{2(1+|\\xi|^2)}}\\right )^{\\lambda+1}f_1\\left (\\frac{2\\xi}{2+\\sqrt{2(1+|\\xi|^2)}}\\right ) \\\\\n &=:e^{(\\lambda+1)\\tau}f(\\xi)\n \\end{align*}\n and thus, $f$ satisfies\n \\begin{equation}\n \\label{eq:specf}\n \\left [\n -(\\delta^{jk}-\\xi^j\\xi^k)\\partial_{\\xi^j}\\partial_{\\xi^k}+2(\\lambda+2)\\xi^j\\partial_{\\xi^j}\n +(\\lambda+1)(\\lambda+2)+V_0(\\xi)\\right ]f(\\xi)=0 \n \\end{equation}\n for all $\\xi\\in \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5$. Note that $f\\in H^5(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5)$ and thus, by\n Sobolev embedding, $f\\in C^2(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5})$. Furthermore, since\n $f$ is radial, we may write $f(\\xi)=\\widehat f(|\\xi|)\/|\\xi|$ for a\n nontrivial odd function $\\widehat f\\in C^2([0,1])$. In terms of\n $\\widehat f$, Eq.~\\eqref{eq:specf} reads\n \\[ -(1-\\rho^2)\\widehat f''(\\rho)-\\frac{2}{\\rho}\\widehat\n f'(\\rho)+2(\\lambda+1)\\rho \\widehat f'(\\rho) +\\lambda(\\lambda+1)\\widehat\n f(\\rho)+\\frac{2(1-6\\rho^2+\\rho^4)}{\\rho^2(1+\\rho^2)^2}\\widehat\n f(\\rho)=0 \\]\n for $\\rho\\in (0,1)$. Frobenius' method yields\n $\\widehat f\\in C^\\infty([0,1])$ and thus, by \\cite{CosDonXia16,\n CosDonGlo17}, we conclude that $\\lambda=1$.\n\\end{proof}\n\n\n\\begin{remark}\n The proof of Lemma \\ref{lem:sigmapL} shows that the existence of the\n eigenvalue $\\lambda=1$ is a mere consequence of the time translation\n symmetry of the wave maps equation \\eqref{eq:main}. Consequently,\n this instability of the linearized flow does \\emph{not} indicate an\n instability of the blowup profile.\n\\end{remark}\n\nBy Lemma \\ref{lem:sigmapL}, the eigenvalue $1\\in \\sigma_p(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ is\nisolated. This allows us to define the corresponding spectral\nprojection.\n\n\\begin{definition}\n Fix $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$, and let\n $\\mathbf} \\newcommand{\\mc}{\\mathcal L: \\mc D(\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\subset \\mc H^k_R\\to\\mc H^k_R$ be the operator\n from Lemma \\ref{lem:S}. Furthermore, let $\\gamma: [0,2\\pi]\\to \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}$\n be given by $\\gamma(t)=1+\\frac12 e^{i t}$. Then we set\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal P:=\\frac{1}{2\\pi i}\\int_\\gamma \\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n L}(\\lambda)d\\lambda. \\]\n\\end{definition}\n\n\\begin{proposition}\n \\label{prop:P}\n The projection $\\mathbf} \\newcommand{\\mc}{\\mathcal P$ commutes with the semigroup $\\mathbf} \\newcommand{\\mc}{\\mathcal S(s)$ and we\n have\n \\[ \\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P=\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\rangle. \\]\n\\end{proposition}\n\n\\begin{proof}\n The fact that $\\mathbf} \\newcommand{\\mc}{\\mathcal P$ commutes with $\\mathbf} \\newcommand{\\mc}{\\mathcal S(s)$ follows from the\n abstract theory, see e.g.~\\cite{Kat95, EngNag00}. To prove the\n statement about $\\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P$, we first recall from Lemma \\ref{lem:S}\n that $\\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P\\subset \\mc D(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ is finite-dimensional.\n Consequently, the part $\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P}$ of $\\mathbf} \\newcommand{\\mc}{\\mathcal L$ in $\\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P$\n is an operator acting on a finite-dimensional Hilbert space with\n spectrum $\\sigma(\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P})=\\{1\\}$. This implies that\n $\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P}$ is nilpotent. Thus, there exists an\n $\\ell\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ such that $(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P})^\\ell=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$. We\n claim that $\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P}=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$. Suppose this were not\n true, i.e., $\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P}\\not= \\mathbf} \\newcommand{\\mc}{\\mathcal 0$. Then, by Lemma\n \\ref{lem:sigmapL},\n \\[ \\mathrm{rg}\\, (\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P})\\subset \\ker(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal\n P})\\subset \\ker(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)=\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^* \\rangle \\]\n and thus, there exists an\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f=(f_1,f_2)\\in \\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P \\subset H^5_\\mathrm{rad}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_R)\\times\n H_\\mathrm{rad}^4(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_R)\\subset C^2(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}})\\times\n C^1(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}})$ such that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*=(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_{\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P})\\mathbf} \\newcommand{\\mc}{\\mathcal f=(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f=(2\\mathbf} \\newcommand{\\mc}{\\mathcal\n I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal L')\\mathbf} \\newcommand{\\mc}{\\mathcal f. \\]\n From the explicit form of $\\mathbf} \\newcommand{\\mc}{\\mathcal L_5$ in Eq.~\\eqref{eq:L5} we infer\n $f_{1,1}^*=2f_1-f_2$ and\n \\begin{align*}\n H^{\\mu 0}H_\\mu{}^0(s,y)f_{1,2}^*(y)=&H^{\\mu j}H_\\mu{}^k(s,y)\\partial_j\\partial_k f_1(y)\n +H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^j\\partial_j f_1(y) \\\\\n &+2H^{\\mu j}H_\\mu{}^0(s,y)\\partial_j f_2(y)\n +[2H^{\\mu\n 0}H_{\\mu}{}^0(s,y)+H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^0(s,y)]\n f_2(y) \\\\\n &-e^{2s}h(y)^{-2}V_0(y\/h(y))f_1(y)\n \\end{align*}\n for all $(s,y)\\in \\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}$. The potential $V_0$ is given\n in Eq.~\\eqref{def:V0}. Consequently,\n \\begin{equation}\n \\begin{split}\n \\label{eq:Pf1}\n e^{-2s}&H^{\\mu j}H_\\mu{}^k(s,y)\\partial_j\\partial_k f_1(y)\n +e^{-2s}\\left [ H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^j(s,y) + 4H^{\\mu j}H_\\mu{}^0(s,y)\\right ]\\partial_j f_1(y) \\\\\n &+2e^{-2s}[2H^{\\mu 0}H_\\mu{}^0(s,y)+H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^0(s,y)]\n f_1(y)-h(y)^{-2}V_0(y\/h(y))f_1(y)=G(y),\n \\end{split}\n \\end{equation}\n where\n \\begin{align*}\n G(y)&=2e^{-2s}H^{\\mu j}H_\\mu{}^0(s,y)\\partial_j f_{1,1}^*(y)\n +e^{-2s}[2H^{\\mu 0}H_{\\mu}{}^0(s,y)+H^{\\mu\\nu}\\partial_\\nu\n H_\\mu{}^0(s,y)]f_{1,1}^*(y) \\\\\n &\\quad +e^{-2s}H^{\\mu 0}H_\\mu{}^0(s,y)f_{1,2}^*(y) \\\\\n &=2e^{-2s}H^{\\mu j}H_\\mu{}^0(s,y)\\partial_j f_{1,1}^*(y)\n +e^{-2s}[4H^{\\mu 0}H_\\mu{}^0(s,y)+H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^0(s,y)]f_{1,1}^*(y).\n \\end{align*}\n Obviously, $G$ is radial and belongs to\n $C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}})$.\n Explicitly, we have\n \\begin{align*}\n e^{-2s}H^{\\mu j}H_\\mu{}^0(s,y)\n &=-e^{-2s}H_0{}^jH_0{}^0(s,y)+e^{-2s}H^{k j}H_k{}^0(s,y) \\\\\n &=-\\frac{y^j}{[y^\\ell\\partial_\\ell h(y)-h(y)]^2}\n -\\frac{\\partial^j h(y)}{y^\\ell\\partial_\\ell\n h(y)-h(y)}+\\frac{\\partial^k h(y)\\partial_k\n h(y)}{[y^\\ell\\partial_\\ell h(y)-h(y)]^2}y^j \\\\\n &=-\\frac{1-\\partial^k h(y)\\partial_k h(y)}{[y^\\ell\\partial_\\ell\n h(y)-h(y)]^2}y^j\n -\\frac{\\partial^j h(y)}{y^\\ell\\partial_\\ell h(y)-h(y)} \\\\\n &=-h_1(|y|)\\left [h_1(|y|)\\left [1-\\widehat h'(|y|)^2\\right ]+\\frac{\\widehat\n h'(|y|)}{|y|}\\right ]y^j,\n \\end{align*}\n where $\\widehat h(|y|):=h(y)=\\sqrt{2+|y|^2}-2$ and\n \\[ h_1(|y|):=\\frac{1}{|y|\\widehat h'(|y|)-\\widehat h(|y|)}. \\] \nNext, \n \\begin{align*}\n e^{-2s}H^{\\mu 0}H_\\mu{}^0(s,y)\n &=-e^{-2s}H_0{}^0H_0{}^0(s,y)+e^{-2s}H^{j 0}H_j{}^0(s,y) \n=-\\frac{1-\\partial^j h(y)\\partial_j h(y)}{[y^\\ell\\partial_\\ell\n h(y)-h(y)]^2} \\\\\n &=-h_1(|y|)^2\\left [1-\\widehat h'(|y|)^2\\right].\n \\end{align*}\n Furthermore, we have\n \\begin{align*} H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^0\n &=H^{\\mu 0}\\partial_0 H_\\mu{}^0+H^{\\mu k}\\partial_k H_\\mu{}^0\n =H^{\\mu 0}H_\\mu{}^0+H^{0k}\\partial_k H_0{}^0+H^{jk}\\partial_k\n H_j{}^0 \\\\\n &=H^{\\mu 0}H_\\mu{}^0-H_0{}^k\\partial_k H_0{}^0+H^{jk}\\partial_k H_j{}^0\n \\end{align*} \n and \n \\begin{align*}\n e^{-2s}H_0{}^k\\partial_k H_0{}^0(s,y)\n &=\\frac{y^k}{y^\\ell \\partial_\\ell\n h(y)-h(y)}\\partial_{y^k}\\frac{1}{y^\\ell\\partial_\\ell h(y)-h(y)}\n=|y|h_1'(|y|)h_1(|y|).\n \\end{align*}\n Finally,\n \\begin{align*}\n e^{-2s}H^{jk}\\partial_k H_j{}^0(s,y)\n &=-\\left (\\delta^{jk}-\\frac{\\partial^j h(y)y^k}{y^\\ell \\partial_\\ell\n h(y)-h(y)}\\right )\n \\partial_{y^k}\\frac{\\partial_j h(y)}{y^\\ell\\partial_\\ell\n h(y)-h(y)} \\\\\n &=-\\frac{\\partial^j\\partial_j h(y)}{y^\\ell \\partial_\\ell\n h(y)-h(y)}\n -\\partial_j h(y)\\partial_{y_j}\\frac{1}{y^\\ell\\partial_\\ell\n h(y)-h(y)} \\\\\n &\\quad +\\frac{\\partial^j h(y)y^k}{[y^\\ell\\partial_\\ell\n h(y)-h(y)]^2}\\partial_j\\partial_k h(y) \\\\\n &\\quad +\\frac{\\partial^j h(y)\\partial_j h(y)}{y^\\ell\\partial_\\ell\n h(y)-h(y)}\n y^k\\partial_{y^k}\\frac{1}{y^\\ell\\partial_\\ell h(y)-h(y)}\n \\end{align*}\n and thus, in terms of $\\widehat h$ and $h_1$,\n \\begin{align*}\n e^{-2s}H^{jk}\\partial_k H_j{}^0(s,y)\n &=-h_1(|y|)\\left [\\widehat h''(|y|)+\\frac{4\\widehat\n h'(|y|)}{|y|}\\right ]-h_1'(|y|)\\widehat h'(|y|) \\\\\n &\\quad +|y|h_1(|y|)^2\\widehat h'(|y|)\\widehat\n h''(|y|)+|y|h_1'(|y|)h_1(|y|)\\widehat h'(|y|)^2.\n \\end{align*}\n In summary,\n \\begin{align*}\n e^{-2s}H^{\\mu\\nu}\\partial_\\nu H_\\mu{}^0(s,y)\n &=-h_1(|y|)\\left [\\widehat h''(|y|)+\\frac{4\\widehat\n h'(|y|)}{|y|}\\right ]\n +|y|h_1(|y|)^2\\widehat h''(|y|)\\widehat h'(|y|) \\\\\n &\\quad -\\left [h_1(|y|)^2+|y|h_1'(|y|)h_1(|y|)\\right]\\left [1-\\widehat\n h'(|y|)^2\\right]\n -h_1'(|y|)\\widehat h'(|y|).\n \\end{align*}\n With these explicit expressions at hand it is straightforward to\n check that $G(y)<0$ for all $y\\in \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1\/2}^5$.\n In particular, $G$ has no zeros in $\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1\/2}$ and this will be the\n key property.\n\n\n Observe that $(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$ implies that $f_{1,1}^*$\n solves Eq.~\\eqref{eq:Pf1} with $G=0$. We claim that another solution\n is given by\n \\[ \\widetilde\n f_{1,1}^*(y)=\\frac{1}{|y|^2+h(y)^2}\\int_{1\/2}^{-|y|\/h(y)}\\frac{(1+\\rho^2)^2}{\\rho^4(1-\\rho^2)}d\\rho. \\]\n To see this, we start from the radial version of\n Eq.~\\eqref{eq:specf} with $\\lambda=1$,\n \\begin{equation}\n \\label{eq:Pfrad}\n -(1-\\rho^2)f''(\\rho)-\\frac{4}{\\rho}f'(\\rho)+6\\rho f'(\\rho)+6f(\\rho)-\\frac{16}{(1+\\rho^2)^2}f(\\rho)=0. \n \\end{equation}\n Eq.~\\eqref{eq:Pfrad} is of the form $f''+pf'+qf=0$ with\n $p(\\rho)=\\frac{4}{\\rho}-\\frac{2\\rho}{1-\\rho^2}$. Consequently, the\n Wronskian $W(f,g)$ of two solutions $f,g$ of Eq.~\\eqref{eq:Pfrad} is\n given by\n \\[ W(f,g)(\\rho)=W(f,g)(\\tfrac12)e^{-\\int_{1\/2}^\\rho\n p(t)dt}=\\tfrac{3}{64}W(f,g)(\\tfrac12)\\frac{1}{\\rho^4(1-\\rho^2)}. \\]\n Note that $\\rho\\mapsto \\frac{1}{1+\\rho^2}$ is a solution of\n Eq.~\\eqref{eq:Pfrad} (cf.~the proof of Lemma \\ref{lem:sigmapL}) and\n thus, by the reduction formula, another solution is given by\n \\[ \\rho\\mapsto \\frac{1}{1+\\rho^2}\\int_{1\/2}^\\rho\n \\frac{(1+r^2)^2}{r^4(1-r^2)}dr. \\]\n As a consequence, we see that the function\n \\[ w(\\tau,\\xi):=e^{2\\tau}\\frac{1}{1+|\\xi|^2}\\int_{1\/2}^{|\\xi|}\n \\frac{(1+\\rho^2)^2}{\\rho^4(1-\\rho^2)}d\\rho \\] satisfies\n \\[ e^{2\\tau}\\left [\n \\partial_\\tau^2+2\\xi^j\\partial_{\\xi^j}\\partial_\\tau-(\\delta^{jk}-\\xi^j\\xi^k)\\partial_{\\xi^j}\\partial_{\\xi^k}\n +\\partial_\\tau+2\\xi^j\\partial_{\\xi^j}+V_0(\\xi)\\right\n ]w(\\tau,\\xi)=0 \\]\n for all $(\\tau,\\xi)\\in \\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5\\setminus \\{0\\}$. This means\n that $v(s,y)=w(s-\\log(-h(y)),-y\/h(y))$ satisfies\n \\[ -\\mc D^\\mu\\mc D_\\mu v(s,y)+e^{2s}h(y)^{-2}V_0(y\/h(y))v(s,y)=0 \\]\n for all $(s,y)\\in \\R\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1\/2}^5\\setminus \\{0\\}$,\n cf.~Eq.~\\eqref{eq:sytauxiwave}.\n We have\n \\begin{align*}\n v(s,y)&=w(s-\\log(-h(y)),-y\/h(y))=e^{2s}h(y)^{-2}\\frac{1}{1+|y|^2\/h(y)^2}\n \\int_{1\/2}^{-|y|\/h(y)}\\frac{(1+\\rho^2)^2}{\\rho^4(1-\\rho^2)}d\\rho \\\\\n &=e^{2s}\\widetilde f_{1,1}^*(y)\n \\end{align*}\n and thus, $\\widetilde f_{1,1}^*$ satisfies Eq.~\\eqref{eq:Pf1} with\n $G=0$ and for all $y\\in \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1\/2}^5\\setminus \\{0\\}$, as claimed.\n\n By definition, we have\n \\begin{align*}\n e^{-2s}H^{\\mu j}H_\\mu{}^k(s,y)\n &=e^{-2s}[-H_0{}^jH_0{}^k(s,y)+H^{mj}H_m{}^k(s,y)] \\\\\n &=\\delta^{jk}-\\frac{1-\\partial^m h(y)\\partial_m h(y)}{[y^\\ell \\partial_\\ell h(y)-h(y)]^2}y^jy^k\n -\\frac{1}{y^\\ell \\partial_\\ell h(y)-h(y)}\\left [y^j \\partial^k h(y)+y^k\\partial^j h(y)\\right ]\n \\end{align*}\n and thus, if $f(y)=\\widehat f(|y|)$, we obtain\n \\begin{align*}\n e^{-2s}H^{\\mu j}H_\\mu{}^k(s,y)\\partial_j\\partial_k f(y)\n &=\\widehat f''(|y|)+\\frac{4}{|y|}\\widehat f'(|y|)\n-\\frac{1-\\partial^j h(y)\\partial_j h(y)}{[y^\\ell\\partial_\\ell h(y)-h(y)]^2}|y|^2\\widehat f''(|y|) \\\\\n &\\quad -\\frac{2y^j\\partial_j h(y)}{y^\\ell\\partial_\\ell h(y)-h(y)}\\widehat f''(|y|) \\\\\n &=[1-a(|y|)]\\widehat f''(|y|)+\\frac{4}{|y|}\\widehat f'(|y|),\n \\end{align*}\n where\n \\begin{align*}\n a(|y|)&=\\frac{2\\sqrt{2+|y|^2}-1}{2(\\sqrt{2+|y|^2}-1)^2}|y|^2.\n \\end{align*}\n Consequently, if we write $f_1(y)=\\widehat f_1(|y|)$, we see that\n Eq.~\\eqref{eq:Pf1} is of the form\n \\begin{equation}\n \\label{eq:Pf1hat}\n \\widehat f_1''(\\eta)+\\widetilde p(\\eta)\\widehat\n f_1'(\\eta)+\\widetilde q(\\eta)f_1(\\eta)\n=\\frac{\\widehat G(\\eta)}{1-a(\\eta)},\\qquad \\eta\\in (0,\\tfrac12), \n \\end{equation}\n where $G(y)=\\widehat G(|y|)$. By the above, the homogeneous version of\n Eq.~\\eqref{eq:Pf1hat} has the solutions \\begin{align*}\n \\phi(\\eta)&=f_{1,1}^*(\\eta e_1)=\\frac{1}{\\eta^2+h(\\eta e_1)^2} \\\\\n \\psi(\\eta)&=\\widetilde\n f_{1,1}^*(\\eta\n e_1)=\\phi(\\eta)\\int_{1\/2}^{-\\eta\/h(\\eta\n e_1)}\n \\frac{(1+\\rho^2)^2}{\\rho^4(1-\\rho^2)}d\\rho.\n \\end{align*}\n As for the asymptotic\n behavior, we note that\n $\\phi\\in\n C^\\infty([0,\\frac12])$\n whereas\n \\begin{align*}\n |\\psi(\\eta)|&\\simeq\n \\eta^{-3}|\\log(\\tfrac12-\\eta)|,\n & |\\psi'(\\eta)|&\\simeq \\eta^{-4}(\\tfrac12-\\eta)^{-1} \n \\end{align*}\n for all\n $\\eta\\in (0,\\frac12)$.\n Furthermore, we have\n $W(\\phi,\\psi)(\\eta)\\simeq\n \\eta^{-4}(\\frac12-\\eta)^{-1}$.\n Consequently, by the\n variation of constants\n formula, there exist\n constants $c_1, c_2\\in \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}$\n such that\n \\begin{equation}\n \\begin{split}\n \\label{eq:Pvoc}\n \\widehat f_1(\\eta)=&c_1 \\phi(\\eta)+c_2\\psi(\\eta) \\\\\n &-\\phi(\\eta)\\int_0^\\eta\n \\frac{\\psi(\\rho)}{W(\\phi,\\psi)(\\rho)}\\frac{\\widehat\n G(\\rho)}{1-a(\\rho)}d\\rho\n +\\psi(\\eta)\\int_0^\\eta\n \\frac{\\phi(\\rho)}{W(\\phi,\\psi)(\\rho)}\\frac{\\widehat\n G(\\rho)}{1-a(\\rho)}d\\rho\n \\end{split}\n \\end{equation} \n for $\\eta\\in\n (0,\\frac12)$.\n Taking the limit\n $\\eta\\to 0+$ yields $c_2=0$\n since\n $\\widehat f_1, \\widehat G\\in\n C([0,\\frac12])$.\n Note further that\n \\[ \\lim_{\\eta\\to \\frac12-}\n \\int_0^\\eta\n \\frac{\\psi(\\rho)}{W(\\phi,\\psi)(\\rho)}\\frac{\\widehat\n G(\\rho)}{1-a(\\rho)}d\\rho \\]\n exists and thus, by sending\n $\\eta\\to \\frac12-$,\n Eq.~\\eqref{eq:Pvoc} implies\n \\[ \\int_0^{1\/2}\n \\frac{\\phi(\\rho)}{W(\\phi,\\psi)(\\rho)}\\frac{\\widehat\n G(\\rho)}{1-a(\\rho)}d\\rho=0. \\]\n But this is impossible\n because the integrand has no\n zeros in $(0,\\frac12)$.\n This contradiction shows\n that\n $(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L)\\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$\n for all $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in \\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P$\n and from Lemma\n \\ref{lem:sigmapL} we\n conclude that\n $\\mathrm{rg}\\,\\mathbf} \\newcommand{\\mc}{\\mathcal P=\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal\n f_1^*\\rangle$.\n \\end{proof}\n\n\\subsection{Control of the linearized flow}\nWe arrive at the main result on the linearized flow.\n\n\\begin{theorem}\n \\label{thm:S}\n Fix $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$ and let $\\mathbf} \\newcommand{\\mc}{\\mathcal S$ be the\n semigroup on $\\mc H^k_R$ from Lemma \\ref{lem:S}. Then there exist\n $\\omega_0,M>0$ such that\n \\begin{align*}\n \\mathbf} \\newcommand{\\mc}{\\mathcal S(s)\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal f&=e^s\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal f \\\\\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal S(s)(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^{k}}&\\leq \n M e^{-\\omega_0 s}\\|(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H^k_R}\n \\end{align*}\n for all $s\\geq 0$ and $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in \\mc H^k_R$.\n\\end{theorem}\n\n\\begin{proof}\n The first statement follows directly from Lemma \\ref{lem:sigmapL}\n and Proposition \\ref{prop:P}. As for the second statement, we first\n claim that there exists an $N\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ such that\n \\begin{equation}\n \\label{eq:SR}\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^k}\\lesssim \\|\\mathbf} \\newcommand{\\mc}{\\mathcal\n f\\|_{\\mc H_R^k} \n \\end{equation}\n for all $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in \\mc H_R^k$ and all\n $\\lambda\\in \\Omega_N:=\\{z\\in \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}: \\Re z\\geq -\\frac14, |z|\\geq N\\}$.\n Indeed, from Theorem \\ref{thm:S5} we infer\n $\\Omega_N\\subset \\rho(\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I)$ and thus, for any\n $\\lambda\\in \\Omega_N$ we have the identity\n $\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L=[\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal\n I}(\\lambda)](\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_5+\\mathbf} \\newcommand{\\mc}{\\mathcal I)$\n which shows that $\\lambda\\in \\rho(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ if and only if the\n operator $\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)$ is bounded\n invertible. By a Neumann series argument we see that this is the\n case if $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\|_{\\mc\n H^k_R}<1$. Recall that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f(y)=\\begin{pmatrix} 0 \\\\\n V(y)[\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_1(y)\n \\end{pmatrix}\n \\]\n and from the first component of the identity\n $(\\lambda\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal L_5+\\mathbf} \\newcommand{\\mc}{\\mathcal I)\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f=\\mathbf} \\newcommand{\\mc}{\\mathcal f$\n we infer\n \\[ (\\lambda+1)[\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_1-[\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_2=f_1. \\]\n Consequently, by noting that $V\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5})$, we\n obtain\n \\begin{align*}\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^k}\n &\\lesssim \\|[\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f]_1\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}\n \\lesssim |\\lambda|^{-1}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H^k_R}+|\\lambda|^{-1}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal\n R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H^k_R} \\\\\n &\\lesssim |\\lambda|^{-1}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H^k_R}.\n \\end{align*}\n If $N\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ is sufficiently large, we therefore have\n $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal L'\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L_5-\\mathbf} \\newcommand{\\mc}{\\mathcal I}(\\lambda)\\|_{\\mc H_R^k}\\leq \\frac12$\n for all $\\lambda\\in \\Omega_N$ and Eq.~\\eqref{eq:SR} follows.\n\n Furthermore, from Lemma \\ref{lem:sigmapL} we infer the existence of\n an $\\omega_0>0$ such that\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L}(\\lambda)(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\|_{\\mc H^k_R}\\lesssim 1 \\]\n for all $\\lambda\\in \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}$ satisfying $\\Re\\lambda\\geq -\\omega_0$ and\n $|\\lambda|\\leq N$. Combining this with Eq.~\\eqref{eq:SR} we obtain\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal R_{\\mathbf} \\newcommand{\\mc}{\\mathcal L}(\\lambda)(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\|_{\\mc H^k_R}\\lesssim 1 \\]\n for all $\\lambda\\in \\mathbb{C}} \\newcommand{\\Z}{\\mathbb{Z}$ with $\\Re\\lambda\\geq -\\omega_0$.\n Consequently, an application of the Gearhart-Pr\\\"uss Theorem (see\n e.g.~\\cite{EngNag00}, p.~302, Theorem 1.11) finishes the proof.\n\\end{proof}\n\n\\begin{definition}\n From now on, $\\omega_0>0$ denotes the corresponding constant from\n Theorem \\ref{thm:S}.\n\\end{definition}\n\n\\subsection{Bounds on the nonlinearity}\n\nNext, we show that the nonlinearity is locally Lipschitz.\n\n\\begin{lemma}\n \\label{lem:N}\n Fix $R, M>0$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 2$. Then we have the bound\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\mathbf} \\newcommand{\\mc}{\\mathcal f)-\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\mathbf} \\newcommand{\\mc}{\\mathcal g)\\|_{\\mc H_R^k}\\lesssim \\left (\\|\\mathbf} \\newcommand{\\mc}{\\mathcal\n f\\|_{\\mc H_R^k}+\\|\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc H_R^k}\\right )\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc\n H_R^k}\n \\]\n for all $\\mathbf} \\newcommand{\\mc}{\\mathcal f,\\mathbf} \\newcommand{\\mc}{\\mathcal g\\in \\mc H_R^k$ satisfying\n $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^k}, \\|\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc H_R^k}\\leq M$.\n\\end{lemma}\n\n\\begin{proof}\n Recall that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal N \\begin{pmatrix} f_1 \\\\ f_2 \\end{pmatrix}(y)\n =\\begin{pmatrix} 0 \\\\ -H(y)N(|y|f_1(y), y) \\end{pmatrix},\n \\]\n where $H\\in C^\\infty(\\overline{\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5})$ and\n \\begin{align*}\n N(|y|f_1(y),y)&=-\\frac{\\sin(2|y|\\alpha_0(y)+2|y|f_1(y))-\\sin(2|y|\\alpha_0(y))-2|y|\\cos(2|y|\\alpha_0(y))f_1(y)}{|y|^3}.\n \\end{align*}\n From Taylor's theorem with integral remainder we infer\n \\begin{align*}\n \\sin(x_0+x)-\\sin(x_0)-\\cos(x_0)x&=-\\frac{\\sin\n x_0}{2}x^2-\\frac{x^3}{2}\\int_0^1\n \\cos(x_0+tx)(1-t)^2 dt\n \\end{align*}\n and thus,\n \\begin{align*}\n N(|y|f_1(y),y)&=\\frac{2\\sin(2|y|\\alpha_0(y))}{|y|}f_1(y)^2 \\\\\n &\\quad +4f_1(y)^3\\int_0^1 \n \\cos(2|y|\\alpha_0(y)+2t|y|f_1(y))(1-t)^2 dt \\\\\n &=\\frac{2\\sin(2|y|\\alpha_0(y))}{|y|}f_1(y)^2+f_1(y)^3\\Phi_0(f_1(y),|y|),\n \\end{align*}\n where\n \\[ \\Phi_0(u,|y|)=4\\int_0^1 \\cos \\left (2|y|\\left\n (\\alpha_0(y)+tu\\right) \\right )(1-t)^2 dt.\n \\]\n Note that $y\\mapsto \\frac{2\\sin(2|y|\\alpha_0(y))}{|y|}$ belongs to\n $C^\\infty(\\R^5)$. Furthermore, $\\Phi_0\\in C^\\infty(\\R\\times\\R)$ and\n $\\partial_u^j \\Phi_0(u,\\cdot)$ is even for any $j\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0$.\n Consequently, the map $(u,y)\\mapsto \\Phi_0(u, |y|)$ belongs to\n $C^\\infty(\\R\\times \\R^5)$. We set $\\mc N(f)(y):=-H(y)N(|y|f(y),y)$. \n Then, by Lemma \\ref{lem:Moser},\n \\begin{align*}\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\mathbf} \\newcommand{\\mc}{\\mathcal f)-\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\mathbf} \\newcommand{\\mc}{\\mathcal g)\\|_{\\mc H^k_R}\n&\\lesssim \\|\\mc N(f_1)-\\mc N(g_1)\\|_{H^k(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}\n\\leq \\|\\mc N(f_1)-\\mc N(g_1)\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)} \\\\\n&\\lesssim \\left (\\|f_1\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}+\\|g_1\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}\\right\n )\n\\|f_1-g_1\\|_{H^{k+1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)} \\\\\n&\\lesssim \\left ( \\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H^k_R}+\\|\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc H^k_R}\\right )\n\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc H^k_R}\n \\end{align*}\nsince $k+1\\geq 3>\\frac52$.\n\\end{proof}\n\n\\subsection{Existence of the nonlinear evolution}\nNow we turn to the full equation \\eqref{eq:Psi}. By Duhamel's\nprinciple,\n\\begin{equation}\n \\label{eq:Psiw}\n \\Phi(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\Phi(s_0)+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds'\n\\end{equation}\nis a weak formulation of Eq.~\\eqref{eq:Psi}. In general, this\nequation will not have a solution for all $s\\geq s_0$ due to the\none-dimensional instability of the linearized flow. Consequently, as\nan intermediate step, we modify Eq.~\\eqref{eq:Psiw} according to the\nLyapunov-Perron method from dynamical systems theory.\n\\begin{definition}\n For $R\\geq\\frac12$, $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$, $s_0\\in \\R$, and $\\omega_0>0$\n from Theorem \\ref{thm:S}, we set\n \\[ \\mc X_R^k(s_0):=\\{\\Phi\\in C([s_0,\\infty),\\mc H_R^k):\n \\|\\Phi\\|_{\\mc X_R^k(s_0)}<\\infty\\}, \\] where\n \\[ \\|\\Phi\\|_{\\mc X_R^k(s_0)}:=\\sup_{s>s_0}\\left [e^{\\omega_0\n s}\\|\\Phi(s)\\|_{\\mc H_R^k}\\right ]. \\]\n Furthermore, we define\n $\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}: \\mc X_R^k(s_0)\\times \\mc H_R^k \\to \\mathrm{rg}\\, \\mathbf} \\newcommand{\\mc}{\\mathcal P$ by\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}(\\Phi, \\mathbf} \\newcommand{\\mc}{\\mathcal f):=\\mathbf} \\newcommand{\\mc}{\\mathcal P\\left (\\mathbf} \\newcommand{\\mc}{\\mathcal f+\\int_{s_0}^\\infty\n e^{s_0-s'}\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds' \\right ). \\]\n\\end{definition}\nInstead of Eq.~\\eqref{eq:Psiw} we now consider the modified equation\n\\begin{equation}\n \\label{eq:Psim}\n \\Phi(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\left [\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}(\\Phi,\\mathbf} \\newcommand{\\mc}{\\mathcal f)\\right ]+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal\n S(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds'.\n\\end{equation}\nThe point of this modification is that it tames the instability, as\nthe following result shows.\n\n\\begin{proposition}\n \\label{prop:mod}\n Fix $R\\geq\\frac12$, $s_0\\in \\R$, and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$. Then there\n exists a $c>0$ and a $\\delta>0$ such that for any\n $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\in \\mc H_R^k$ satisfying\n $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^k}\\leq \\frac{\\delta}{c}$, there exists a unique\n solution $\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}\\in C([s_0,\\infty),\\mc H_R^k)$ to\n Eq.~\\eqref{eq:Psim} that satisfies $\\|\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(s)\\|_{\\mc H_R^k}\\leq \\delta e^{-\\omega_0 s}$ for all\n $s\\geq s_0$. Furthermore, the solution map $\\mathbf} \\newcommand{\\mc}{\\mathcal f\\mapsto \\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}$ is\n Lipschitz as a function from (a small ball in) $\\mc H_R^k$ to\n $\\mc X_R^k(s_0)$.\n\\end{proposition}\n\n\\begin{proof}\nWe set $\\mc Y_\\delta:=\\{\\Phi\\in \\mc X_R^k(s_0): \\|\\Phi\\|_{\\mc\n X_R^k(s_0)}\\leq \\delta\\}$ and\n\\[ \\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s):= \\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\left [\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}(\\Phi,\\mathbf} \\newcommand{\\mc}{\\mathcal f)\\right ]+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal\n S(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds'. \\]\nLet $\\Phi\\in \\mc Y_\\delta$. By Theorem \\ref{thm:S},\n\\begin{align*}\n\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)\n&=e^{s-s_0}[\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}(\\Phi,\\mathbf} \\newcommand{\\mc}{\\mathcal f)]+\\int_{s_0}^s\n e^{s-s'}\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds' \\\\\n&=-\\int_s^\\infty e^{s-s'}\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds'\n\\end{align*}\nand, since $\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\mathbf} \\newcommand{\\mc}{\\mathcal 0)=\\mathbf} \\newcommand{\\mc}{\\mathcal 0$, Lemma \\ref{lem:N} yields\n\\begin{align*}\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)\\|_{\\mc H_R^k}\n&\\lesssim\ne^s\\int_s^\\infty e^{-s'}\n\\|\\Phi(s')\\|_{\\mc H_R^k}^2 ds'\n\\lesssim \\|\\Phi\\|_{\\mc X_R^k(s_0)}^2 e^s\\int_s^\\infty e^{-s'-2\\omega_0\n s'}ds' \\\\\n&\\lesssim \\delta^2 e^{-2\\omega_0 s}\n\\end{align*}\nfor all $s\\geq s_0$.\nFurthermore,\n\\begin{align*}\n (\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)\n&=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal f+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s')(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds'\n\\end{align*}\nand thus, by Theorem \\ref{thm:S},\n\\begin{align*}\n \\|(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)\\|_{\\mc H_R^k}\n&\\lesssim e^{-\\omega_0 s}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f\\|_{\\mc H_R^k}+\\int_{s_0}^s\n e^{-\\omega_0(s-s')}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))\\|_{\\mc H_R^k}ds' \\\\\n&\\lesssim \\tfrac{\\delta}{c}e^{-\\omega_0 s}\n+e^{-\\omega_0 s}\\int_{s_0}^s e^{\\omega_0 s'}\\|\\Phi(s')\\|_{\\mc H_R^k}^2\n ds' \\\\\n&\\lesssim \\tfrac{\\delta}{c}e^{-\\omega_0 s} + \\|\\Phi\\|_{\\mc\n X_R^k(s_0)}^2 e^{-\\omega_0\n s}\\int_{s_0}^s e^{-\\omega_0 s'}ds' \\\\\n&\\lesssim \\tfrac{\\delta}{c}e^{-\\omega_0 s} +\\delta^2 e^{-\\omega_0 s}\n\\end{align*}\nfor all $s\\geq s_0$. \nConsequently, by choosing $c>0$ large enough and $\\delta>0$ small enough, we infer $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Phi)(s)\\|_{\\mc H_R^k}\\leq \\delta e^{-\\omega_0 s}$ for all $s\\geq\ns_0$. In other words,\n$\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)\\in \\mc Y_\\delta$ for all $\\Phi\\in\\mc Y_\\delta$.\n\nNext, we show that $\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}$ is a contraction on $\\mc Y_\\delta$.\nFor $\\Phi,\\Psi\\in \\mc Y_\\delta$ we have\n\\begin{align*}\n \\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)-\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Psi)(s)=-\\int_s^\\infty e^{s-s'}\\mathbf} \\newcommand{\\mc}{\\mathcal P\\left [\n\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))-\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Psi(s'))\\right ]ds'\n\\end{align*}\nand thus, by Lemma \\ref{lem:N},\n\\begin{align*}\n \\|\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)-\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Psi)(s)\\|_{\\mc H_R^k}\n&\\lesssim e^s \\int_s^\\infty e^{-s'}\\left [\\|\\Phi(s')\\|_{\\mc\n H_R^k}+\\|\\Psi(s')\\|_{\\mc H_R^k}\\right ]\\|\\Phi(s')-\\Psi(s')\\|_{\\mc\n H_R^k}ds' \\\\\n&\\lesssim \\delta \\|\\Phi-\\Psi\\|_{\\mc X_R^k(s_0)}e^s\\int_s^\\infty\n e^{-s'-2\\omega_0 s'}ds' \\\\\n&\\lesssim \\delta e^{-2\\omega_0 s}\\|\\Phi-\\Psi\\|_{\\mc X_R^k(s_0)}\n\\end{align*}\nfor all $s\\geq s_0$.\nSimilarly,\n\\begin{align*}\n(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)-(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Psi)(s)=\n\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s')(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\left [\n\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))-\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Psi(s'))\\right ]ds'\n\\end{align*}\nand thus, by Theorem \\ref{thm:S} and Lemma \\ref{lem:N},\n\\begin{align*}\n \\|&(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi)(s)-(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Psi)(s)\\|_{\\mc H_R^k} \\\\\n&\\lesssim \\int_{s_0}^s e^{-\\omega_0(s-s')}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))-\\mathbf} \\newcommand{\\mc}{\\mathcal\n N(\\Psi(s'))\\|_{\\mc H_R^k}ds' \\\\\n&\\lesssim \\int_{s_0}^s e^{-\\omega_0(s-s')}\n\\left [\\|\\Phi(s')\\|_{\\mc\n H_R^k}+\\|\\Psi(s')\\|_{\\mc H_R^k}\\right ]\\|\\Phi(s')-\\Psi(s')\\|_{\\mc\n H_R^k} ds' \\\\\n&\\lesssim \\delta \\|\\Phi-\\Psi\\|_{\\mc X_R^k(s_0)}e^{-\\omega_0\n s}\\int_{s_0}^s e^{-\\omega_0 s'}ds' \\\\\n&\\lesssim \\delta e^{-\\omega_0 s}\\|\\Phi-\\Psi\\|_{\\mc X_R^k(s_0)}\n\\end{align*}\nfor all $s\\geq s_0$. In summary, $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_\\mathbf} \\newcommand{\\mc}{\\mathcal f(\\Phi)-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n f}(\\Psi)\\|_{\\mc X_R^k(s_0)}\\lesssim \\delta \\|\\Phi-\\Psi\\|_{\\mc\n X_R^k(s_0)}$ for all $\\Phi,\\Psi\\in \\mc Y_\\delta$ and upon choosing\n$\\delta>0$ sufficiently small, the contraction mapping principle\nyields the existence of a unique fixed point $\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}\\in \\mc\nY_\\delta$ of $\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}$.\n\nFinally, we prove the Lipschitz continuity of the solution map.\nWe have\n\\begin{align*}\n \\|\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}-\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}\\|_{\\mc X_R^k(s_0)}\n&=\\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f})-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})\\|_{\\mc\n X_R^k(s_0)} \\\\\n&\\leq \\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f})-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})\\|_{\\mc\n X_R^k(s_0)}+\n\\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})\\|_{\\mc\n X_R^k(s_0)} \\\\\n&\\lesssim \\delta \\|\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}-\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}\\|_{\\mc X_R^k(s_0)}\n+\\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})\\|_{\\mc\n X_R^k(s_0)}\n\\end{align*}\nand, since\n\\[ \\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})(s)-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal\nS(s-s_0)(\\mathbf} \\newcommand{\\mc}{\\mathcal I-\\mathbf} \\newcommand{\\mc}{\\mathcal P)(\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal g), \\]\nTheorem \\ref{thm:S} yields\n\\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g})(s)-\\mathbf} \\newcommand{\\mc}{\\mathcal K_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}(\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal\n g})(s)\\|_{\\mc H_R^k}\\lesssim e^{-\\omega_0 s}\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal g\\|_{\\mc\n H_R^k} \\]\nfor all $s\\geq s_0$.\nIn summary, $\\|\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}-\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}\\|_{\\mc X_R^k(s_0)}\\lesssim \n\\delta \\|\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal f}-\\Phi_{\\mathbf} \\newcommand{\\mc}{\\mathcal g}\\|_{\\mc X_R^k(s_0)}+\\|\\mathbf} \\newcommand{\\mc}{\\mathcal f-\\mathbf} \\newcommand{\\mc}{\\mathcal\ng\\|_{\\mc H_R^k}$ and if $\\delta>0$ is sufficiently small, the claimed\nLipschitz bound follows.\n\\end{proof}\n\n\\section{Proof of the main result}\n\n\\noindent We are now in a position to prove Theorem \\ref{thm:main}\n\n\\subsection{Construction of the data on the hyperboloid}\n\nAs a first step, we evolve the data prescribed at $t=0$ using the\nstandard Cauchy theory. \nFor this we use the following local existence result.\n\n\\begin{definition}\n For $\\epsilon>0$ we define the spacetime region\n $\\Lambda_\\epsilon\\subset \\R^{1,5}$ by\n \\[ \\Lambda_\\epsilon:=[-4\\epsilon,4\\epsilon]\\times \\R^5 \\cup\n \\{(t,x)\\in \\R^{1,5}: -|x|+\\epsilon\\leq t\\leq |x|-\\epsilon\\}, \\]\n see Fig.~\\ref{fig:Lambda}.\n\\end{definition}\n\n\\begin{definition}\n For $\\delta,\\epsilon>0$ and $m\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ we set\n \\[ \\mc B_{\\delta,\\epsilon}^m:=\\{(f,g)\\in C^\\infty(\\R^5)\\times\n C^\\infty(\\R^5)\\mbox{ radial}: \\mathrm{supp}(f,g)\\subset \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_\\epsilon,\n \\|(f,g)\\|_{H^m(\\R^5)\\times H^{m-1}(\\R^5)}\\leq \\delta\\}. \\]\n\\end{definition}\n\n\\begin{lemma}\n \\label{lem:locwm}\n Let $m\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$ and $m\\geq 8$. Then there exists an $\\epsilon>0$ such\n that for any pair of functions $(f,g)\\in \\mc B_{1,\\epsilon}^m$ there\n exists a unique solution $u=u_{f,g}\\in C^\\infty(\\Lambda_\\epsilon)$\n in $\\Lambda_\\epsilon$ to the Cauchy problem\n \\begin{equation}\n \\label{eq:Cauchyinit}\n \\left \\{\\begin{array}{l}\n (\\partial_t^2-\\Delta_x)u(t,x)=\\frac{2|x|u(t,x)-\\sin(2|x|u(t,x))}{|x|^3} \\\\\n u(0,x)=u_1^*(0,x)+f(x),\\quad \\partial_0 u(0,x)=\\partial_0 u_1^*(0,x)+g(x).\n \\end{array} \\right.\n \\end{equation}\n Furthermore, for any multi-index $\\alpha\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0^6$ of length\n $|\\alpha|\\leq m-3$, we have the estimate\n \\[ \\sup_{(t,x)\\in \\Lambda_\\epsilon}|\\partial^\\alpha\n u_{f,g}(t,x)-\\partial^\\alpha u_1^*(t,x)|\\lesssim\n \\|(f,g)\\|_{H^m(\\R^5)\\times H^{m-1}(\\R^5)}. \\]\n \\end{lemma}\n\n\\begin{proof}\n Thanks to Lemma \\ref{lem:wmsmooth}, Theorems \\ref{thm:LWP},\n \\ref{thm:uniq}, and\n \\ref{thm:reg} apply to the Cauchy problem\n \\eqref{eq:Cauchyinit}. From Theorem \\ref{thm:LWP} we obtain an\n $\\epsilon \\in (0,\\frac19)$ and the existence of a solution $u$ to\n the Cauchy problem \\eqref{eq:Cauchyinit} in the truncated lightcone\n $\\bigcup_{t\\in [-4\\epsilon,4\\epsilon]}\\{t\\}\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_{1-|t|}$ for\n any $(f,g)\\in \\mc B_{1,1}^m$. In particular, this existence result\n holds for all $(f,g)\\in \\mc B_{1,\\epsilon}^m\\subset \\mc B_{1,1}^m$.\n Let $(f,g)\\in \\mc B_{1,\\epsilon}^m$. Since the support of $(f,g)$\n is contained in the ball $\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}^5_\\epsilon$, it follows from finite\n speed of propagation (Theorem \\ref{thm:uniq}) that the unique\n solution $u$ to Eq.~\\eqref{eq:Cauchyinit} in the domain\n $\\{(t,x)\\in \\R^{1,5}: -|x|+\\epsilon\\leq t\\leq |x|-\\epsilon\\}$ is\n $u=u_1^*$. In summary, we obtain a solution $u$ in\n $\\Lambda_\\epsilon$ and by Theorem \\ref{thm:reg},\n $u\\in C^\\infty(\\Lambda_\\epsilon)$.\n\n Furthermore, from Theorem \\ref{thm:LWP} we have the Lipschitz bound\n \\begin{align*}\n \\sup_{t\\in\n [-4\\epsilon,4\\epsilon]}\n &\\|(u(t,\\cdot),\\partial_t u(t,\\cdot))-(u_1^*(t,\\cdot), \\partial_t\n u_1^*(t,\\cdot))\\|_{H^m(\\R^5)\\times H^{m-1}(\\R^5)} \\\\\n &= \\sup_{t\\in\n [-4\\epsilon,4\\epsilon]}\\|(u(t,\\cdot),\\partial_t u(t,\\cdot))-(u_1^*(t,\\cdot), \\partial_t\n u_1^*(t,\\cdot))\\|_{H^m(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1-|t|}^5)\\times H^{m-1}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_{1-|t|}^5)} \\\\\n &\\lesssim \\|(u(0,\\cdot),\\partial_0\n u(0,\\cdot))-(u_1^*(0,\\cdot),\\partial_0\n u_1^*(0,\\cdot))\\|_{H^m(\\R^5)\\times H^{m-1}(\\R^5)} \\\\\n &=\\|(f,g)\\|_{H^m(\\R^5)\\times H^{m-1}(\\R^5)}\n \\end{align*}\n and thus, by Sobolev embedding,\n \\[ \\sup_{(t,x)\\in \\Lambda_\\epsilon}|\\partial^\\alpha\n u(t,x)-\\partial^\\alpha u_1^*(t,x)|\\lesssim \\|(f,g)\\|_{H^m(\\R^5)\\times\n H^{m-1}(\\R^5)}\n \\]\n for all multi-indices $\\alpha\\in \\{0,1\\}\\times \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}_0^5$ of length\n $|\\alpha|\\leq m-3$. Time derivatives of higher order are estimated\n by using the equation to translate them into spatial derivatives.\n This way, the stated bound follows.\n\\end{proof}\n\nNow we obtain the initial data for the hyperboloidal evolution by\nevaluating the solution from Lemma \\ref{lem:locwm} on a suitable\nhyperboloid. Recall from Section \\ref{sec:pert} that in terms of the\nvariable\n\\begin{align*}\n \\Phi(s)(y)&:=e^{-s}\\begin{pmatrix} v(s,y) \\\\ \\partial_s\n v(s,y) \\end{pmatrix}:=e^{-s}\\begin{pmatrix}\\widetilde\n u(T+e^{-s}h(y),e^{-s}y) \\\\ \\partial_s \\widetilde\n u(T+e^{-s}h(y),e^{-s}y) \\end{pmatrix} \\\\\n &:=e^{-s} \\begin{pmatrix}\n u(T+e^{-s}h(y),e^{-s}y)-u_T^*(T+e^{-s}h(y),e^{-s}y) \\\\\n \\partial_s u(T+e^{-s}h(y),e^{-s}y)-\\partial_s\n u_T^*(T+e^{-s}h(y),e^{-s}y)\n \\end{pmatrix},\n\\end{align*}\nEq.~\\eqref{eq:mainu} reads\n\\begin{equation}\n \\label{eq:Phifull}\n \\partial_s\\Phi(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal L \\Phi(s)+\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s)). \n\\end{equation}\nThis motivates the definition of the following \\emph{initial data\n operator}.\n\n\\begin{definition}\n Let $R\\geq\\frac12$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$. For $\\epsilon>0$\n sufficiently small, we define a map\n $\\mathbf} \\newcommand{\\mc}{\\mathcal U: \\mc B^{k+4}_{1,\\epsilon}\\times [1-\\epsilon,1+\\epsilon]\\to\n \\mc H_R^k$\n as follows. For $(f,g)\\in \\mc B_{1,\\epsilon}^{k+4}$ let\n $u_{f,g}\\in C^\\infty(\\Lambda_\\epsilon)$ be the corresponding unique\n solution to the Cauchy problem \\eqref{eq:Cauchyinit} from Lemma\n \\ref{lem:locwm}. Then we set\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)(y):=e^{-s}\\left .\\begin{pmatrix}\n u_{f,g}(T+e^{-s}h(y),e^{-s}y)-u_T^*(T+e^{-s}h(y),e^{-s}y) \\\\\n \\partial_s u_{f,g}(T+e^{-s}h(y),e^{-s}y)-\\partial_s\n u_T^*(T+e^{-s}h(y),e^{-s}y)\n \\end{pmatrix}\\right |_{s=\\log(-\\frac{h(0)}{1+2\\epsilon})}.\n \\]\n\\end{definition}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics{Lambda}\n \\caption{A spacetime diagram illustrating the construction of the\n initial data for the hyperboloidal evolution. The shaded region\n depicts the domain $\\Lambda_\\epsilon$. The solid curved line\n represents the hyperboloid parametrized by the map\n $y\\mapsto (T+e^{-s}h(y), e^{-s}y)$ with $T=1$ and\n $s=\\log(-\\frac{h(0)}{1+2\\epsilon})$. The dotted lines are the\n corresponding translated hyperboloids with the same $s$ but\n $T=1-\\epsilon$ and $T=1+\\epsilon$, respectively. The zigzag\n segment represents the support of the initial data $(f,g)$.}\n \\label{fig:Lambda}\n\\end{figure}\n\nConsequently, our goal is now to solve Eq.~\\eqref{eq:Phifull} for\n$s\\geq s_0=\\log(-\\frac{h(0)}{1+2\\epsilon})$ with initial data\n$\\Phi(s_0)=\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)$ and a suitable\n$T \\in [1-\\epsilon,1+\\epsilon]$.\n\n\\subsection{Properties of the initial data operator}\nFirst, we prove mapping properties for the initial data operator\n$\\mathbf} \\newcommand{\\mc}{\\mathcal U$.\n\n\\begin{lemma}\n \\label{lem:U}\n Let $R\\geq\\frac12$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$. Then there exists an\n $\\epsilon>0$ such that the initial data operator\n $\\mathbf} \\newcommand{\\mc}{\\mathcal U: \\mc B^{k+4}_{1,\\epsilon}\\times [1-\\epsilon,1+\\epsilon]\\to\n \\mc H_R^k$\n is well-defined and for any $(f,g)\\in \\mc B_{1,\\epsilon}^{k+4}$, the\n map $\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),\\cdot): [1-\\epsilon,1+\\epsilon]\\to\\mc H_R^k$ is\n continuous. Furthermore, there exists a $\\gamma_\\epsilon\\in \\R$ such\n that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)=\\gamma_\\epsilon (T-1)\\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*+\\mathbf} \\newcommand{\\mc}{\\mathcal V((f,g),T), \\]\n where $\\mathbf} \\newcommand{\\mc}{\\mathcal V((f,g),T)$ satisfies the bound\n \\[ \\|\\mathbf} \\newcommand{\\mc}{\\mathcal V((f,g),T)\\|_{\\mc H^k_R}\\lesssim\n \\|(f,g)\\|_{H^{k+4}(\\R^5)\\times H^{k+3}(\\R^5)}+|T-1|^2 . \\]\n\\end{lemma}\n\n\\begin{proof}\n According to Lemma \\ref{lem:locwm}, there exists an $\\epsilon>0$\n such that the operator is well-defined since the hyperboloids on\n which the function $u_{f,g}$ is evaluated lie entirely inside of\n $\\Lambda_\\epsilon$, see Fig.~\\ref{fig:Lambda}. The statement about\n the continuity is a simple consequence of the fact that\n $u_{f,g}, u_T^* \\in C^\\infty(\\Lambda_\\epsilon)$. To prove the last\n assertion, we rewrite $\\mathbf} \\newcommand{\\mc}{\\mathcal U$ as\n \\begin{align*}\n \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)(y)\n &=\n e^{-s}\\left .\\begin{pmatrix}\n u_{f,g}(T+e^{-s}h(y),e^{-s}y)-u_1^*(T+e^{-s}h(y),e^{-s}y) \\\\\n \\partial_s u_{f,g}(T+e^{-s}h(y),e^{-s}y)-\\partial_s\n u_1^*(T+e^{-s}h(y),e^{-s}y)\n \\end{pmatrix}\\right |_{s=\\log(-\\frac{h(0)}{1+2\\epsilon})} \\\\\n &\\quad +e^{-s}\\left .\\begin{pmatrix}\n u_1^*(T+e^{-s}h(y),e^{-s}y)-u_T^*(T+e^{-s}h(y),e^{-s}y) \\\\\n \\partial_s u_1^*(T+e^{-s}h(y),e^{-s}y)-\\partial_s\n u_T^*(T+e^{-s}h(y),e^{-s}y)\n \\end{pmatrix}\\right |_{s=\\log(-\\frac{h(0)}{1+2\\epsilon})}.\n \\end{align*}\n Now note that $u_{T+t_0}^*(t+t_0,x)=u_T^*(t,x)$ for any $t_0\\in \\R$\n and thus, by expanding around $T=1$, we infer\n \\begin{align*}\n u_1^*(T+e^{-s}h(y),e^{-s}y)\n &=u_{2-T}^*(1+e^{-s}h(y),e^{-s}y) \\\\\n &=u_1^*(1+e^{-s}h(y),e^{-s}y) \\\\\n &\\quad +\\partial_T u_{2-T}^*(1+e^{-s}h(y),e^{-s}y)|_{T=1}(T-1)+\\varphi_T(s,y),\n \\end{align*}\n where\n $\\|(\\varphi_T(s,\\cdot),\\partial_s\\varphi_T(s,\\cdot))\\|_{\\mc\n H_R^k}\\leq C_s |T-1|^2$. Since\n \\[ \\partial_T u_T^*(t,x)=-\\frac{2}{(T-t)^2+|x|^2}, \\]\n see the proof of Lemma \\ref{lem:sigmapL}, we obtain\n \\[ \\partial_T u_{2-T}^*(t,x)|_{T=1}=-\\partial_T\n u_T^*(t,x)|_{T=1}=\\frac{2}{(1-t)^2+|x|^2} \\] and thus,\n \\begin{align*}\n u_1^*(T+e^{-s}h(y),e^{-s}y)\n &=u_1^*(1+e^{-s}h(y),e^{-s}y)+2\\frac{e^{2s}}{|y|^2+h(y)^2}(T-1)+\\varphi_T(s,y)\n \\\\\n &=u_T^*(T+e^{-s}h(y),e^{-s}y)+2e^{2s}f_{1,1}^*(y)(T-1)+\\varphi_T(s,y) \\\\\n \\partial_s u_1^*(T+e^{-s}h(y),e^{-s}y)\n &=\\partial_s u_T^*(T+e^{-s}h(y),e^{-s}y)+2e^{2s}f_{1,2}^*(y)(T-1)+\\partial_s\\varphi_T(s,y) .\n \\end{align*}\n This yields the stated representation. The bound on\n$\\mathbf} \\newcommand{\\mc}{\\mathcal V((f,g),T)$\n follows easily by Sobolev embedding and the estimate from Lemma\n \\ref{lem:locwm}.\n\\end{proof}\n\n\\FloatBarrier\n\n\\subsection{Hyperboloidal evolution}\n\nThe last step in the proof of Theorem \\ref{thm:main} consists of the\nhyperboloidal evolution.\n\n\\begin{proposition}\n \\label{prop:hyp}\n Let $R\\geq\\frac12$ and $k\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $k\\geq 4$. Then there exists an\n $M>0$ and $\\delta,\\epsilon>0$ such that for any pair\n $(f,g)\\in \\mc B_{\\delta\/M,\\epsilon}^{k+4}$ there exists a\n $T_{f,g}\\in [1-\\delta,1+\\delta]$ and a unique function\n $\\Phi_{f,g}\\in C([s_0,\\infty),\\mc H_R^k)$ that satisfies\n \\[ \\Phi_{f,g}(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T_{f,g})+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal\n S(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi_{f,g}(s'))ds',\\qquad s_0:=\\log\\left\n (-\\frac{h(0)}{1+2\\epsilon}\\right ) \\]\nand\n $\\|\\Phi_{f,g}(s)\\|_{\\mc H^k_R}\\leq \\delta e^{-\\omega_0 s}$ for all\n $s\\geq s_0$.\n\\end{proposition}\n\n\\begin{proof}\n Let $c,\\delta>0$ be as in Proposition \\ref{prop:mod} and choose\n $\\epsilon>0$ so small that\n $\\mathbf} \\newcommand{\\mc}{\\mathcal U: \\mc B_{1,\\epsilon}^{k+4}\\times [1-\\epsilon,1+\\epsilon]\\to\\mc\n H_R^k$\n is well-defined, see Lemma \\ref{lem:U}. By choosing $M_0\\geq 1$\n sufficiently large, we obtain $\\frac{\\delta}{M_0}\\leq \\epsilon$ and\n Lemma \\ref{lem:U} yields the bound\n $\\|\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)\\|_{\\mc H_R^k}\\leq \\frac{\\delta}{c}$ for all\n $(f,g)\\in \\mc B_{\\delta\/M_0^2,\\epsilon}^{k+4}$ and\n $T\\in [1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]$. Consequently,\n Proposition \\ref{prop:mod} implies that for any\n $(f,g)\\in \\mc B_{\\delta\/M_0^2,\\epsilon}^{k+4}$ and\n $T\\in [1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]$, there exists a\n unique function $\\Phi_{f,g,T}\\in C([s_0,\\infty),\\mc H_R^k)$\n satisfying\n \\[ \\Phi_{f,g,T}(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\left [\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)-\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}\\big\n (\\Phi_{f,g,T}, \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)\\big )\\right ]+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal\n S(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N\\big (\\Phi_{f,g,T}(s')\\big )ds',\n \\]\n where $s_0:=\\log(-\\frac{h(0)}{1+2\\epsilon})$. Furthermore, we have\n the bound\n $\\|\\Phi_{f,g,T}(s)\\|_{\\mc H_R^k}\\leq \\delta e^{-\\omega_0 s}$ for all\n $s\\geq s_0$. Thus, our goal is to show that there exists a\n $T_{f,g}\\in [1-\\frac{\\delta}{M_0}, 1+\\frac{\\delta}{M_0}]$ such that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}\\big (\\Phi_{f,g,T_{f,g}}, \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T_{f,g})\\big\n )=\\mathbf} \\newcommand{\\mc}{\\mathcal 0. \\]\n To this end, we define a function\n $\\varphi_{f,g}: [1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]\\to \\R$\n by\n \\[ \\varphi_{f,g}(T):=\\Big (\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}\\big (\\Phi_{f,g,T}, \\mathbf} \\newcommand{\\mc}{\\mathcal\n U((f,g),T)\\big ) \\Big | \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^* \\Big )_{\\mc H_R^k}. \\]\n By Proposition \\ref{prop:mod} and Lemma \\ref{lem:U}, $\\varphi_{f,g}$\n is continuous. Recall that\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}\\big (\\Phi_{f,g,T}, \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)\\big ) =\\mathbf} \\newcommand{\\mc}{\\mathcal P\\mathbf} \\newcommand{\\mc}{\\mathcal\n U((f,g),T)+\\mathbf} \\newcommand{\\mc}{\\mathcal P\\int_{s_0}^\\infty e^{s_0-s'}\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi_{f,g,T}(s'))ds' \\]\n and from Lemmas \\ref{lem:N} and \\ref{lem:U} we see that there exists\n a nonzero constant $\\widetilde \\gamma_\\epsilon$ such that\n \\[ \\varphi_{f,g}(T)=\\widetilde\\gamma_\\epsilon(T-1)+\\phi_{f,g}(T) \\]\n with a continuous function\n $\\phi_{f,g}: [1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]\\to \\R$\n satisfying\n $|\\phi_{f,g}(T)|\\lesssim\n \\frac{\\delta}{M_0^2}+\\delta^2$\n for all $T\\in [1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]$.\n Consequently, the condition $\\varphi_{f,g}(T)=0$ is equivalent to\n the fixed point problem\n $T=1-\\widetilde\\gamma_\\epsilon^{-1}\\phi_{f,g}(T)$ and if we choose\n $M_0$ large enough and $\\delta>0$ sufficiently small,\n $T\\mapsto 1-\\widetilde\\gamma_\\epsilon^{-1}\\phi_{f,g}(T)$ becomes a\n continuous self-map of the interval\n $[1-\\frac{\\delta}{M_0},1+\\frac{\\delta}{M_0}]$ which necessarily has\n a fixed point $T_{f,g}$. By Proposition \\ref{prop:P}, $\\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}$\n has values in $\\langle \\mathbf} \\newcommand{\\mc}{\\mathcal f_1^*\\rangle$ and thus,\n \\[ \\mathbf} \\newcommand{\\mc}{\\mathcal C_{s_0}\\big (\\Phi_{f,g,T_{f,g}}, \\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T_{f,g})\\big\n )=\\mathbf} \\newcommand{\\mc}{\\mathcal 0, \\] as desired. The proof is finished by setting $M=M_0^2$.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:main}}\n\nLet $m\\in \\mathbb{N}} \\newcommand{\\R}{\\mathbb{R}$, $m\\geq 8$.\nAccording to Lemmas \\ref{lem:locwm}, \\ref{lem:U}, and Proposition\n\\ref{prop:hyp}, there exists an $\\epsilon>0$ such that for any pair of\nfunctions $(f,g)\\in \\mc B_{\\delta\/M,\\epsilon}^m$ there exists a\n$T\\in [1-\\delta,1+\\delta]$ and a continuous function $\\Phi=(\\phi_1,\\phi_2): [s_0,\\infty)\\to \\mc\nH_R^{m-4}$ that satisfies\n\\[ \\Phi(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal S(s-s_0)\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)+\\int_{s_0}^s \\mathbf} \\newcommand{\\mc}{\\mathcal\nS(s-s')\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s'))ds' \\]\nfor all $s\\geq s_0:=\\log(-\\frac{h(0)}{1+2\\epsilon})$ and\n$\\|\\Phi(s)\\|_{\\mc H^{m-4}_R}\\leq\\delta e^{-\\omega_0 s}$.\nSince the data $\\mathbf} \\newcommand{\\mc}{\\mathcal U((f,g),T)$ are smooth, they belong to $\\mc\nD(\\mathbf} \\newcommand{\\mc}{\\mathcal L)$ and the function $\\Phi$ is a\nclassical solution to the equation\n\\[ \\partial_s \\Phi(s)=\\mathbf} \\newcommand{\\mc}{\\mathcal L\\Phi(s)+\\mathbf} \\newcommand{\\mc}{\\mathcal N(\\Phi(s)). \\]\nBy a simple inductive argument it follows that $\\Phi$ is\nin fact smooth, cf.~the proof of Theorem \\ref{thm:reg}.\nBy construction, the function $u$, given by\n\\[ (u\\circ \\eta_T)(s,y)=(u_T^*\\circ\\eta_T)(s,y)+e^{s}\\phi_1(s)(y), \\]\nsatisfies Eq.~\\eqref{eq:mainu} in the domain\n$\\eta_T([s_0,\\infty)\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$ and we have\n\\[ \\partial_s(u\\circ \\eta_T)(s,y)=\\partial_s (u_T^*\\circ\n\\eta_T)(s,y)+e^{s}\\phi_2(s)(y). \\]\nBy Theorem \\ref{thm:uniq} we have $u(0,\\cdot)=u_1^*(0,\\cdot)+f$ and\n$\\partial_0 u(0,\\cdot)=\\partial_0 u_1^*(0,\\cdot)+g$ and\nthe stated bounds in Theorem \\ref{thm:main} follow\nimmediately from \n\\[ \\|\\Phi(s)\\|_{\\mc H_R^{m-4}}^2=\\|\\phi_1(s)\\|_{H^{m-3}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}^2\n+\\|\\phi_2(s)\\|_{H^{m-4}(\\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)}^2\\leq \\delta^2 e^{-2\\omega_0 s}. \\]\nFinally, $u=u_1^*$ in $\\Omega_{T,b}\\setminus\n\\eta_T([s_0,\\infty)\\times \\mathbb{B}} \\renewcommand{\\S}{\\mathbb{S}_R^5)$ is a consequence of finite speed of\npropagation, Theorem \\ref{thm:uniq}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Evolution on ILSVRC-2012}\\label{sec:imagenet_evolution}\n\n\n\\textbf {Evolution.} We use ResNet-18 as the target network for pruning function evolution on ILSVRC-2012. \nSince only one task is evaluated, we directly use the retrained accuracy of the pruned network as the function's fitness.\nOther evolution settings for population, selection, mutation, and crossover are kept the same as Sec.~4 of the main paper.\n\n\\noindent\\textbf{Evaluation.} We uniformly prune 30\\% of channels in each layer from a pretrained ResNet-18, resulting in a FLOPs reduction of 36.4\\%.\nDue to the constrained computational budget, \nwe only fine-tune it for 4 epochs using the SGD optimizer with Nesterov momentum~\\cite{nesterov1983method}.\nWe use a batch size of 128 and initialize our learning rate at 0.001.\nThe learning rate is multiplied by 0.4 at epoch 1 and 2.\n\n\\noindent\\textbf{Result.} \nWe show the evolution progress in Fig.~\\ref{fig:evo_on_imagenet}.\nDue to the lack of training budget, the pruned net is clearly not well retrained as they only achieve around 63.5\\% accuracy, \nmuch lower than the performance of methods shown in Tab.~5 of the main paper at the similar pruning level.\nSuch inadequate training results in a imprecise function fitness evaluation evidenced in Sec.~6 of the main paper.\nMoreover, the best evolved function from this strategy, $\\xi_{ImageNet}$ (Eqn.~\\ref{eqn:imagenet_evolved}), \nperforms inferior to the co-evolved function $\\xi^*$ when transferred for CIFAR-100 pruning.\nThese results demonstrate the advantage of our small dataset co-evolution strategy in cost-effectiveness.\n\n\\begin{figure}[t]\n\\vspace{-0.3cm}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Figure\/Evolution_On_ImageNet.eps}\n \\caption{Function evolution on ImageNet.}\n \\label{fig:evo_on_imagenet}\n\\end{figure}\n\n\\begin{table}[t]\n\\begin{equation}\\label{eqn:imagenet_evolved}\n\\xi_{ImageNet}(\\mathcal{C}) = (\\frac{\\mathrm{var}_g(\\mathrm{mean}_s(\\mathcal{F}^+))}{\\mathrm{std}_g(\\mathrm{tr}(\\mathcal{F}^+)) \\times \\mathrm{mean}_g(\\mathcal{F}^-)})^4 \\div \\mathrm{var}_g(\\mathrm{sqrt}(\\mathcal{F}))\n\\end{equation}\n\\end{table}\n\\section{SOAP Implementation}\\label{sec:SOAP}\n\n\\subsection{Operator Space}\nIn Tab.~\\ref{tab:detailed_operator}, we present the detailed operator space with operators and their abbreviations.\n\n\\begin{table}\n\\centering\n\n \\begin{tabular}{|p{2cm}|c|c|}\n\n \\hline\n \\multirow{8}{1em}{\\bf Elementwise \\\\ operators} & addition & $\\mathrm{add}(+)$ \\\\ \\cline{2-3}\n & subtraction & $\\mathrm{sub}(-)$ \\\\\t\\cline{2-3}\n & multiplication & $\\mathrm{mul}(\\times)$ \\\\\t\\cline{2-3}\n & division & $\\mathrm{div}(\\div)$ \\\\\t\\cline{2-3}\n & absolute value & $\\mathrm{abs} $ \\\\\t\\cline{2-3}\n & square & $\\mathrm{sq}$ \\\\\t\\cline{2-3}\n & square root & $\\mathrm{sqrt}$ \\\\\t\\cline{2-3}\n & adding ridge factor & $\\mathrm{ridge}$ \\\\\t\n \\hline\n \n \\multirow{6}{7.5em}{\\bf Matrix \\\\ operators} & matrix trace & $\\mathrm{tr}$ \\\\ \\cline{2-3}\n & matrix multiplication & $\\mathrm{matmul}$ \\\\ \\cline{2-3}\n & matrix inversion & $\\mathrm{inv}$ \\\\ \\cline{2-3}\n & inner product & $\\mathrm{dot}$ \\\\ \\cline{2-3}\n & outer product & $\\mathrm{outprod}$ \\\\ \\cline{2-3}\n & matrix\/vector transpose & $\\mathrm{tran}$ \\\\ \n \\hline\n \n \\multirow{6}{8.5em}{\\bf Statistics \\\\ operators} & summation & $\\mathrm{sum}_{\\{s,g\\}}$ \\\\ \\cline{2-3}\n & product & $\\mathrm{prod}_{\\{s,g\\}}$ \\\\ \\cline{2-3}\n & mean & $\\mathrm{mean}_{\\{s,g\\}}$ \\\\ \\cline{2-3}\n & standard deviation & $\\mathrm{std}_{\\{s,g\\}}$ \\\\ \\cline{2-3}\n & variance & $\\mathrm{var}_{\\{s,g\\}}$ \\\\ \\cline{2-3}\n & counting measure & $\\mathrm{count}_{\\{s,g\\}}$ \\\\ \n \\hline\n \n \\multirow{3}{9.5em}{\\bf Specialized \\\\ operators} & rbf kernel matrix getter & $\\mathrm{rbf}$ \\\\ \\cline{2-3}\n & geometric median getter & $\\mathrm{geo}$ \\\\ \\cline{2-3}\n & tensor slicer & $\\mathrm{slice}$ \\\\ \n \\hline\n \n \\end{tabular}\n \\caption{Detailed Operator Space}~\\label{tab:detailed_operator}\n\\end{table}\n \n\n\\subsection{SOAP Functions}\n\nWith the abbreviations of operators in Tab.~\\ref{tab:detailed_operator} and the symbols of operands presented in Tab.~1 of the main paper,\nwe can thus give the precise expressions of the functions in SOAP:\n\\begin{itemize}\n\n\\item Filter's $\\ell1$-norm: $\\mathrm{sum}_g(\\mathrm{abs}(\\mathcal{W}_I))$\n\n\\item Filter's $\\ell2$-norm: $\\mathrm{sqrt}(\\mathrm{sum}_g(\\mathrm{sq}(\\mathcal{W}_I)))$\n\n\\item Batch normalization's scaling factor: $\\mathrm{abs}(\\mathrm{slice}(\\mathcal{B}))$\n\n\\item Filter's geometric median: $\\mathrm{sqrt}(\\mathrm{sum}_g(\\mathrm{sq}(\\mathcal{W}_I - \\mathrm{geo}(\\mathcal{W}))))$\n\n\\item Discriminant Information: \n\n$\\mathrm{count}_s(\\mathcal{F}^+)\n\\times\n\\mathrm{matmul}(\n\\mathrm{tran}(\\mathrm{mean}_s(\\mathcal{F}^+) - \\mathrm{mean}_s(\\mathcal{F})), \\\\\n\\mathrm{inv}(\\mathrm{ridge}(\\mathrm{matmul}(\n\\mathrm{tran}(\\mathcal{F} - \\mathrm{mean}_s(\\mathcal{F})), \n\\mathcal{F} - \\mathrm{mean}_s(\\mathcal{F})\n))),\n\\\\\n\\mathrm{mean}_s(\\mathcal{F}^+) - \\mathrm{mean}_s(\\mathcal{F})\n)\n$\n\n\\item Maximum Mean Discrepancy: \n\n$\\mathrm{div}(\\mathrm{sum}_g(\\mathrm{rbf}(\\mathcal{F}^+, \\mathcal{F}^+) ),~ \\mathrm{sq}(\\mathrm{count}_s(\\mathcal{F}^+)))\n\\\\\n+ \\mathrm{div}(\\mathrm{sum}_g(\\mathrm{rbf}(\\mathcal{F}^-, \\mathcal{F}^-)) ,~ \\mathrm{sq}(\\mathrm{count}_s(\\mathcal{F}^-))) \n\\\\\n- \\mathrm{div}(\\mathrm{sum}_g(\\mathrm{rbf}(\\mathcal{F}^+, \\mathcal{F}^-)) ,~ \\mathrm{mul}(\\mathrm{count}_s(\\mathcal{F}^+)), \\mathrm{count}_s(\\mathcal{F}^-))) -\\\\ \\mathrm{div}(\\mathrm{sum}_g(\\mathrm{rbf}(\\mathcal{F}^+, \\mathcal{F}^-)) ,~ \\mathrm{mul}(\\mathrm{count}_s(\\mathcal{F}^+)), \\mathrm{count}_s(\\mathcal{F}^-)))$\n\n\\item Generalized Absolute SNR:\n\n$\\mathrm{div}(\\mathrm{abs}(\\mathrm{mean}_g(\\mathcal{F}^+) - \\mathrm{mean}_g(\\mathcal{F}^-)) ,~\n\\mathrm{std}_g(\\mathcal{F}^+) + \\mathrm{std}_g(\\mathcal{F}^-)\n)$\n\n\\item Generalized Student's T-Test:\n\n$\\mathrm{div}(\n\\mathrm{abs}(\n\\mathrm{mean}_g(\\mathcal{F}^+) - \\mathrm{mean}_g(\\mathcal{F}^-)), \\\\\n\\mathrm{sqrt}(\n\\mathrm{div}(\\mathrm{var}_g(\\mathcal{F}^+), \\mathrm{count}_s(\\mathcal{F}^+))~+ \\\\\n\\mathrm{div}(\\mathrm{var}_g(\\mathcal{F}^-), \\mathrm{count}_s(\\mathcal{F}^-))\n)\n)\n$\n\n\n\\item Generalized Fisher Discriminat Ratio:\n\n$\\mathrm{div}(\\mathrm{sq}(\\mathrm{mean}_g(\\mathcal{F}^+) - \\mathrm{mean}_g(\\mathcal{F}^-)) ,~\n\\mathrm{var}_g(\\mathcal{F}^+) + \\mathrm{var}_g(\\mathcal{F}^-)\n)$\n\n\\item Generalized Symmetric Divergence: \n\n$\\mathrm{div}(\\mathrm{var}_g(\\mathcal{F}^+), \\mathrm{var}_g(\\mathcal{F}^-)) + \n\\mathrm{div}(\\mathrm{var}_g(\\mathcal{F}^-), \\mathrm{var}_g(\\mathcal{F}^+)) \n\\\\\n+ \\mathrm{div}(\\mathrm{sq}(\\mathrm{mean}_g(\\mathcal{F}^+) - \\mathrm{mean}_g(\\mathcal{F}^-)) ,~\n\\mathrm{var}_g(\\mathcal{F}^+) + \\mathrm{var}_g(\\mathcal{F}^-)\n)\n$\n\n\\end{itemize}\n\n\n\n\n\\section{Experimental Details}\\label{sec:eval_details}\n\n\n\n\\subsection{Study on Fitness Combination Scheme}\n\n\\textbf{Preliminary Evolution.} We conduct 10 preliminary experiments, where the variables are: $\\alpha \\in \\{0, 0.3, 0.5, 0.7, 1\\}$ and combination scheme $\\in$ \\{weighted geometric mean, weighted arithmetic mean\\}.\nFor each experiment, we have a population of 15 functions which are evolved for 10 generations.\nThe population is initialized with 10 individuals randomly cloned from SOAP and 5 random expression trees.\nThe tournament size is 3, and the number of the selected functions is 5. \nThe next generation is reproduced only from the selected functions.\nOther settings are the same as the main evolution experiment.\n\n\n\\noindent\\textbf{CIFAR-100 Pruning.} \nWe apply the best evolved functions from each preliminary evolution test to prune a ResNet-38~\\cite{he2016deep} on CIFAR-100~\\cite{krizhevsky2009learning}.\nThe baseline ResNet-38 adopts the bottleneck block structure with an accuracy of 72.3\\%.\nWe use each evolved function to prune 40\\% of channels in all layers uniformly,\nresulting in a 54.7\\%\/52.4\\% FLOPs\/parameter reduction.\nThe network is then fine-tuned by the SGD optimizer with 200 epochs.\nWe use the Nesterov Momentum~\\cite{nesterov1983method} with a momentum of 0.9.\nThe mini-batch size is set to be 128, and the weight decay is set to be 1e-3.\nThe training data is transformed with a standard data augmentation scheme~\\cite{he2016deep}.\nThe learning rate is initialized at 0.1 and divided by 10 at epoch 80 and 160.\n\n\\subsection{Main Evolution Experiment}\n\\textbf{MNIST Pruning.} On MNIST~\\cite{lecun1998gradient} pruning task, \nwe prune a LeNet-5~\\cite{lecun1998gradient} with a baseline accuracy of 99.26\\%\nfrom shape of 20-50-800-500 to 5-12-160-40.\nSuch pruning process reduces 92.4\\% of FLOPs and 98.0\\% of parameters.\nThe pruned network is fine-tuned for 300 epochs with a batch size of 200 and a weight decay of 7e-5.\nWe use the Adam optimizer~\\cite{kingma2014adam} with a constant learning rate of 5e-4.\n\n\\noindent\\textbf{CIFAR-10 Pruning.} For CIFAR-10~\\cite{krizhevsky2009learning} pruning, \nwe adopt the VGG-16 structure from~\\cite{li2016pruning} with a baseline accuracy of 93.7\\%.\nWe uniformly prune 40\\% of the channels from all layers \nresulting in 63.0\\% FLOPs reduction and 63.7\\% parameters reduction.\nThe fine-tuning process takes 200 epochs with a batch size of 128.\nWe set the weight decay to be 1e-3 and the dropout ratio to be 0.3.\nWe use the SGD optimizer with Nesterov momentum~\\cite{nesterov1983method}, where the momentum is set to be 0.9.\nWe augment the training samples with a standard data augmentation scheme~\\cite{he2016deep}.\nThe initial learning rate is set to be 0.006 and multiplied by 0.28 at 40\\% and 80\\% of the total number of epochs.\n\n\\subsection{Transfer Pruning}\n\nWe implement the pruning experiments in TensorFlow~\\cite{abadi2016tensorflow} \nand carry them out with NVIDIA Tesla P100 GPUs.\nCIFAR-100 contains 50,000\/10,000 training\/test samples in 100 classes.\nSVHN is a 10-class dataset \nwhere we use 604,388 training images for network training with a test set of 26,032 images.\nILSVRC-2012 contains 1.28 million training images and 50 thousand validation images in 1000 classes.\nWe adopt the standard data augmentation scheme~\\cite{he2016deep} for CIFAR-100 and ILSVRC-2012.\n\n\\subsection{Channel Scoring}\n\nAs many of our pruning functions require activation maps of the channels to determine channels' importance, \nwe need to feed-forward the input images for channel scoring.\nSpecifically, for pruning experiments on MNIST, CIFAR-10, and CIFAR-100, \nwe use all their training images to compute the channel scores.\nOn SVHN and ILSVRC-2012, \nwe randomly sample 20 thousand and 10 thousand training images for channel scoring, respectively.\n\n\n\n\n\n\n\\section{Function Validity}\\label{sec:func_val}\n\nThe function expressions generated from mutation and crossover can be invalid (non-invertible matrix, dimension inconsistency, etc.) due to the random selections of operators, operands, and nodes in the expression trees. \nTo combat this issue and enlarge our valid function space, \nsome operators are deliberately modified from their standard definition. \nFor instance, whenever we need to invert a positive semi-definite scatter matrix $S$, we automatically add a ridge factor $\\rho I$, and invert the matrix $S + \\rho I$. \nFor dimension inconsistency in elementwise operations, we have two options to pad the operand with a smaller dimension: \n(1) with 0 for $+$ and $-$, and 1 for $\\times$, and $\\div$,\n(2) with its own value if it is a scalar.\nMoreover, \nwe conduct a validity test on the mutated\/crossovered functions every time after the mutation\/crossover process.\nThe invalid expressions are discarded, and the mutation\/crossover operations are repeated until we recover the population size with all valid functions.\nThese methods ensure we generate valid function expressions under our vast design space during the evolution process.\n\n\n\n\\section{Extra Evolved Functions}\\label{sec:more_evolved}\n\nWe present additional evolved functions from our co-evolution strategy:\n\\begin{equation}\\label{eqn:evolved_1}\n \\xi_1(\\mathcal{C}) = \\frac{||\\bar{f} - \\mathrm{var}_g(\\mathcal{F}^-)\\mathbf{1}||_2^2}\n {\\mathrm{var}_g(\\mathcal{F}^+) + \\mathrm{var}_g(\\mathcal{F}^-)} \n + \\mathrm{var}_g(\\mathcal{F}^+)\n\\end{equation}\n\\begin{equation}\\label{eqn:evolved_2}\n\\xi_2(\\mathcal{C}) = \\mathrm{var}_g(\\mathcal{F}^+)\n\\end{equation}\n\\begin{equation}\\label{eqn:evolved_3}\n\\xi_3(\\mathcal{C}) = \\mathrm{var}_g(\\mathcal{W}_I)\n\\end{equation}\n\nEqn.~\\ref{eqn:evolved_1} presents a metric with the concept of SNR for classification, while having a novel way of statistics combination.\nMoreover, our evolution experiments find that measuring the variance across all elements in $\\mathcal{F}^+$ (Eqn.~\\ref{eqn:evolved_2}) and $\\mathcal{W}_I$ (Eqn.~\\ref{eqn:evolved_3}) would help us identify important channels empirically.\nThese two functions are simple and effective yet remain undiscovered from the literature. \n\n\n\n\n\n\n\n\n\n\n\\section{Ablation Study}\\label{sec:ablation_study}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.35\\textwidth]{Figure\/Random_Init_vs_with_SOAP.eps}\n \\vspace{-0.4cm}\n \\caption{Comparing random initial population evolution (dashed line) with the evolution in Sec.~\\ref{sec:co-evolve} (solid line). \n Thanks to the expressiveness of our function space, the evolution with randomly-initialized functions also achieve good pruning fitness.\n However, we observe that it converges very early around the 8th generation and stalls at the plateau for a long period. \n Moreover, its final fitness has a clear performance gap with respect to the one in Sec.~\\ref{sec:co-evolve}. \n }\n \\label{fig:no_SOAP_evolution}\n\\end{figure}\n\n\n\\textbf{Random Initial Population.} \nIn Fig.~\\ref{fig:no_SOAP_evolution}, \nwe conduct a control experiment which initializes all individuals as random expression trees \nto study the effectiveness of initializing our population with SOAP.\nWe also turn off the SOAP function insertion in the reproduction process for the control experiment.\nAll other parameters (number of generations, size of population, $\\alpha$, etc.) are kept to be the same as in Sec.~\\ref{sec:co-evolve} for a fair comparison.\nWe find that evolving with random population also achieves a good pruning fitness, \nwhich indicates that our design space is of powerful expressiveness. \nHowever, we observe early convergence and a final performance gap in the control experiment compared to the main experiment in Sec.~\\ref{sec:co-evolve}, \ndemonstrating the advantage of using SOAP for evolution. \n\n\n\\noindent\\textbf{Evolution on ILSVRC-2012.}\nIn contrast to our evolution strategy with a joint fitness function on MNIST and CIFAR-10, \nwe conduct an evolution on only ILSVRC-2012 as a control experiment. \nWe restrict the total computation budget to be the same as Sec.~\\ref{sec:co-evolve}, i.e. 98 GPU-days, \nand evolve on ResNet-18 with a population size of 40 over 25 generations. \nDue to the constrained budget, \neach pruned net is only retrained for 4 epochs.\nWe include detailed evolution settings and results in Supplementary.\nTwo major drawbacks are found with this evolution strategy:\n(1) \\textbf{Imprecise evaluation.} \nDue to the lack of training epochs, the function's actual effectiveness is not precisely revealed.\nWe take two functions with fitness 63.24 and 63.46 from the last generation, \nand use them again to prune ResNet-18 but fully retrain for 100 epochs.\nWe find that the one with lower fitness in evolution achieves an accuracy of 68.27\\% in the full training,\nwhile the higher one only has an accuracy of 68.02\\%. \nSuch result indicates that the evaluation in this evolution procedure could be inaccurate, \nwhile our strategy ensures a full retraining for precise effectiveness assessment.\n(2) \\textbf{Inferior performance.} \nThe best evolved function with this method, $\\xi_{ImageNet}$ (in Supplementary), \nperforms inferior to $\\xi^*$ shown in Eqn.~\\ref{eqn:best-func} \nwhen transferred to a different dataset.\nIn particular, when applied to pruning 56\\% FLOPs from ResNet-110 on CIFAR-100,\n$\\xi_{ImageNet}$ only achieves an accuracy of 72.51\\% while $\\xi^*$ reaches 73.85\\%.\nThese two issues suggest that evolving on two small datasets would have better cost-effectiveness than using a single large scale dataset like ILSVRC-2012.\n\n\n\n\\noindent\\textbf{Feature Selection.} \nWe further apply $\\xi^*$ to another machine learning task, \nfeature selection, \nto visually understand our evolved function.\nIn particular, we compare $\\xi^*$ (\\textbf{right}) vs. DI~\\cite{kung2019methodical} (\\textbf{middle}) on MNIST feature selection in Fig.~\\ref{fig:visual_select}.\nThe red pixels indicate the important features evaluated by the metrics, \nwhile the blue ones are redundant.\nTaking the average feature values map (\\textbf{left}) for reference, \nwe find that our evolved function tends to select features with higher means, where the MNIST pattern is more robust.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{Figure\/Visual_Selection_2.eps}\n \\vspace{-0.3cm}\n \\caption{Feature selection by DI~\\cite{kung2019methodical} (\\textbf{middle}) and $\\xi^*$ (\\textbf{right}) for MNIST, \n where $\\xi^*$ tends to preserve features with higher means and more robust pattern in reference of the average feature values map (\\textbf{left}). \n }\\label{fig:visual_select}\n\\end{figure}\n\n\n\\section{Acknowledgement}\n\nWe thank a former colleague from Princeton Parallel Group, Yanqi Zhou, \nfor her help with parallel implementation of the evolution.\nThis material is based upon work supported by the National Science Foundation under Grant No. CCF-1822949. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.\n\n\\section{Evolution on MNIST and CIFAR-10}\\label{sec:co-evolve}\n\n\\input{Table\/SVHN_CIFAR_Transfer_Pruning_Table}\n\n\\begin{table*}[t]\n\\centering\n\\begin{minipage}{\\textwidth}\n\\begin{equation}\n\\label{eqn:best-func}\n\\xi^*(\\mathcal{C}) = \\frac{\\mathrm{var}_g(\\mathcal{F}^-)}{\\mathrm{var}_g(\\mathcal{F}^+)} \n+ \\frac{\\mathrm{var}_g(\\mathcal{F}^+)}{\\mathrm{var}_g(\\mathcal{F}^-)} \n+ \\frac{\n||\\mathrm{std}_g(\\bar{f}) \\times \\mathrm{var}_g(\\mathcal{F}^-) \\times \n\\bar{f} + (\\mathrm{var}_g(\\mathcal{F}^+) - \\mathrm{mean}_g(\\mathcal{F}^-))\\mathbf{1}\n||^2_2}\n{\\mathrm{var}_g(\\mathcal{F}^+) + \\mathrm{var}_g(\\mathcal{F}^-) }\n\\end{equation}\n\\end{minipage}\n\\end{table*}\n\n\n{\\bf Experiment Settings.}\nWe conduct the experiment with a population size of 40 individuals over 25 generations. \nThe population is initialized with\n20 individuals chosen by randomly cloning functions from SOAP \nand 20 random expression trees. \nThe size of the selection tournament is 4 and \nwe select 10 functions in each generation.\n24 individuals are reproduced from the selected functions, while 6 individuals are from SOAP or randomly built.\nThe mutation and crossover probability are both set to be 0.75.\nWe prune 92.4\\% of FLOPs from a LeNet-5 (baseline acc: 99.26\\%) and 63.0\\% of FLOPs from a VGG-16 (baseline acc: 93.7\\%), respectively. \nSuch aggressive pruning schemes help us better identify functions' effectiveness.\nWe use the weighted geometric mean in Eqn.~\\ref{eqn:geometric} to combine two validation accuracies with $\\alpha = 0.5$.\nOur codes are implemented with DEAP~\\cite{DEAP_JMLR2012} and TensorFlow~\\cite{abadi2016tensorflow} for the genetic operations and the neural network pruning.\nThe experiments are carried out on\na cluster with SLURM job scheduler~\\cite{yoo2003slurm} for workload parallelization.\n\n\n\\noindent\\textbf{Experiment Result.} Our evolution progress is shown in Fig.~\\ref{fig:evo_results},\nwhere the red curve denotes the functions with the maximum fitness \nwhile the green curve plots the ones with the top 25 percentile fitness.\nBoth curves increase monotonically over generations, \nindicating that the quality of both the best function and the entire population improves over time. \nThis demonstrates the effectiveness of our scheme.\nSpecifically, the best pruned LeNet-5\/VGG-16 in the first generation \nhave accuracies of 99.15\\%\/93.55\\% while the best accuracies in the last generation are 99.25\\%\/94.0\\%.\nAs the first generation is initialized with SOAP functions, \nsuch results suggest that the algorithm derives metrics that outperform handcrafted functions in SOAP.\nThe whole evolution takes 98 GPU-days on P100, which is a reasonable amount of computation for modern evolution learning. \nWhile this is a pioneering work\\footnote{Compared to initial works on NAS, which take 2000 GPU-days~\\cite{zoph2018learning} and 3000 GPU-days~\\cite{real2017large}, \nwe are 20\/30x faster.}, we envision that future work could further reduce the evolution computation.\n\n\n\n\\noindent\\textbf{Evolved Function.}\nWe present the winning function in Eqn.~\\ref{eqn:best-func}, \nwhere $\\bar{f} = \\mathrm{mean}_s(\\mathcal{F})$ denotes sample average of the feature maps and $\\mathbf{1}$ is a vector with all entries set to be 1.\nThe first two terms of the function award a high score to channels with class-diverged feature maps whose $\\mathrm{var}_g(\\mathcal{F}^+)$ or $\\mathrm{var}_g(\\mathcal{F}^-)$ is significantly smaller than the other.\nChannels with these feature maps contain rich class information as it generates distinguishable responses to different classes.\nThe third term's \ndenominator computes the sum of the feature maps variances while\nits numerator \ndraws statistics from the average feature maps and the distance between $\\mathcal{F}^+$ and $\\mathcal{F}^-$, which resembles the concept of signal-to-noise ratio. Two points worth mentioning for this function:\n(1) it identifies important statistical concepts from human-designed metrics, where it learns from Symmetric Divergence~\\cite{mak2006solution} to measure the divergence of class feature maps. \n(2) it contains unique math concepts that are empirically good for channel importance measurement, \nwhich is shown in the novel statistics combination of the feature maps in the third term's numerator.\nOur visual result in Sec.~\\ref{sec:ablation_study} shows $\\xi^*$ can be further applied to feature selection, \nwhich represents another machine learning task.\n\n\n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nIn this work, we propose a novel paradigm integrating evolutionary learning with channel pruning, \nwhich first learns novel channel pruning functions from small datasets,\nand then transfers them to larger and more challenging datasets. \nWe develop an end-to-end genetic programming framework to \nautomatically search for transferable pruning functions \nover our novel function design space without any manual modification after evolution. \nWe present and analyze a closed-form evolved function which \ncan offer strong pruning performance and further streamline the design of our pruning strategy.\nThe learned pruning function exhibits \nremarkable generalizability to datasets different from those in the evolution process.\nSpecifically, on SVHN, CIFAR-100, and ILSVRC-2012, we achieve state-of-the-art pruning results.\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nConvolutional neural networks (CNNs) have demonstrated superior performance on various computer vision tasks~\\cite{deng2009imagenet,goodfellow2014generative,ren2015faster,dong2015image}.\nHowever, CNNs require huge storage space, high computational budget, and large memory utilization, \nwhich could far exceed the resource limit of edge devices like mobile phones and embedded gadgets.\nAs a result,\nmany methods have been proposed to reduce their cost, such as \nweight quantization~\\cite{chen2015compressing, courbariaux2016binarized,han2015deep}, \ntensor factorization~\\cite{jaderberg2014speeding, lebedev2014speeding}, \nweight pruning~\\cite{han2015learning,zhang2018systematic},\nand channel pruning~\\cite{ he2019filter,liu2021content}. \nAmong them all, channel pruning is the preferred approach to learn dense compact models, which has been receiving increased focus from the research community. \n\nChannel pruning is usually achieved in three steps: \n(1) score channels' importance with a hand-crafted pruning function; \n(2) remove redundant channels based on the scores;\n(3) retrain the network. \nThe performance of channel pruning largely depends on the pruning function used in step (1). \nCurrent scoring metrics are mostly handcrafted to extract crucial statistics from channels' feature maps~\\cite{he2017channel,yu2018nisp} or kernel parameters~\\cite{li2016pruning,he2019filter} \nin a labelless~\\cite{liu2017learning,he2018soft} or label-aware~\\cite{zhuang2018discrimination,kung2019methodical} manner.\nHowever, the design space of pruning functions is so large that \nhand-crafted metrics are usually sub-optimal, \nand enumerating all functions with human labor under the space is impossible.\nWhile prior evolutionary learning works aim to automate the design for the structure of the network directly~\\cite{suganuma2017genetic,sinha2021evolving},\nno attempts, to the best of our knowledge, have been made to evolve the pruning metrics.\nThese raise the question: can we leverage evolutionary strategies to automatically develop strong pruning functions to advance channel pruning?\n\nTo this end, we take the first step to adopt genetic programming (GP) to learn transferable pruning functions, \nas shown in Fig.~\\ref{fig:general_idea}. \nIn particular, a population of functions is evolved by applying them to pruning tasks of small image classification datasets, \nand the evolved functions can later be transferred to larger and more challenging datasets.\nOur closed-form, explainable, learned functions are transferable and generalizable:\n(1) They are applicable to pruning tasks of different image datasets and networks, and can also be used for other machine learning tasks, e.g., feature selection;\n(2) they demonstrate competitive pruning performance on datasets and networks that are different from those used in the evolution process.\nSuch transferability and generalizability provides a unique advantage to our method, \nwhere prior meta-pruning methods like MetaPruning~\\cite{liu2019metapruning} and LFPC~\\cite{he2020learning} are learned and evolved on the same tasks with no transferability and perform inferior to our approach.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{Figure\/General_Idea_Introduction.eps}\n\\vspace{-0.7cm}\n\\caption{Illustration of our approach. \nCompared to conventional methods which mainly use handcrafted pruning functions,\nwe aim to learn the pruning functions automatically via an evolution strategy, genetic programming.\nThe evolved functions are transferable and generalizable, further enhancing the pruning performance.\n}~\\label{fig:general_idea}\n\\end{figure*}\n\n\nSpecifically, we encode pruning functions using expression trees\nwhere we carefully design our search space to allow transferability of the evolved functions.\nFor example, we propose a uni-tree search space for label-aware pruning metrics, which makes them applicable to different datasets.\nSuch a design space ensures an end-to-end evolution process,\nwhere the learned functions are transferable to other datasets without any manual modification after evolution.\nMoreover, under our encoding space, we are able to build a group of competitive hand-crafted pruning functions, \nwhich we name as SOAP (state-of-the-art population), \nand we find the use of SOAP considerably improves the evolution performance.\nThe populations of the functions are evolved with two different pruning tasks, \nLeNet on MNIST and VGGNet on CIFAR-10.\nWe observe that evolving on two tasks produces better functions than only evolving on one of them,\nand more surprisingly, our scheme can even produce more effective pruning functions than direct evolution on a large dataset, \ne.g., ILSVRC-2012, under the same computational budget.\nWe analyze the merits of an evolved function both mathematically and visually\nand transfer it to three larger datasets, CIFAR-100, SVHN, and ILSVRC-2012, \nwhere it exceeds the state-of-the-art pruning results on all of them.\n\nOur main contributions are summarized as follows:\n\\begin{itemize}\n \\item We propose a novel paradigm where we leverage genetic programming to learn transferable channel pruning functions which advance pruning efficacy.\n \\item We develop a novel design space to allow an end-to-end co-evolution process for searching transferable pruning functions. \n Such a space also enables us to express SOAP, which helps improve the effectiveness of the evolution.\n \\item We provide an analysis on our closed-form evolved functions, \n which could further streamline the design of pruning metrics.\n The evolved functions also show generalizability to other machine learning tasks, e.g., feature selection.\n \\item When transferred to datasets unseen by the evolution,\n our evolved functions achieve state-of-the-art pruning results.\n For example, with 26.9\\% and 53.4\\% FLOPs\\footnote{Number of floating points operations for an image inference.} reduction from MobileNet-V2, \n we achieve top-1 accuracies of 71.90\\% and 69.16\\% on ILSVRC-2012, outperforming the state of the art.\n\\end{itemize}\n\n\n\n\n\n\n\\section{Methodology} \\label{sec:methodology}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{Figure\/Evolution_Learning_Flowchart.eps}\n \\vspace{-0.7cm}\n \\caption{ Illustration of our approach to evolve channel pruning functions. \n A population of functions is applied to conduct pruning tasks on two datasets, MNIST and CIFAR-10.\n Each function receives a fitness value by combining its pruned networks' accuracies. \n The population will then go through a natural selection process to improve the functions' effectiveness.\n }\n \\label{fig:overall_flowchart}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:overall_flowchart},\nwe present our evolution framework,\nwhich leverages\ngenetic programming~\\cite{koza1992genetic} to \nlearn high-quality channel pruning functions.\nWe first describe the design space to encode channel scoring functions.\nNext, we discuss the pruning tasks to evaluate the functions' effectiveness.\nLastly, genetic operators are defined to traverse the function space for competitive solutions.\n\n\n\\subsection{Function Design Space }\n\n\\input{Table\/function_design_space}\n\n{\\bf Expression Tree.} In channel pruning, a pruning function $\\xi : \\mathcal{C} \\longmapsto \\mathbb{R}$ scores the channels to determine their importance\/redundancy,\nwhere $\\mathcal{C}$ denotes feature maps, filters, and their statistics associated with the channels.\nThis scoring process can be viewed as a series of operations with operators (addition, matrix multiplication, etc.) and operands (feature maps, filters, etc.).\nWe thus adopt an expression tree encoding scheme to represent a pruning function\nwhere inner nodes are operators, and leaves are operands. \n\nAs shown in Tab.~\\ref{tab:operand_space} and~\\ref{tab:operator_space}, our function design space includes two types of operands (6 operands in total) \nand four types of operators (23 operators in total), \nvia which a vast number of pruning functions can be expressed.\nThe statistics operators can compute the statistics of an operand in two dimensions, \nnamely, global dimension (subscript with `g') and sample dimension (subscript with `s'). \nThe global\ndimension operators flatten operands into a 1D sequence and extract corresponding statistics, \nwhile the sample dimension operators compute statistics on the axis of samples.\nFor example, $\\mathrm{sum}_g(\\mathcal{W})$ returns the summation of all entries of a kernel tensor, \nwhile $\\mathrm{mean}_s(\\mathcal{F})$ returns $\\bar{f} \\in \\mathbb{R}^{H \\times W}$, which is the sample average of all feature maps.\nWe also include specialized operators \nwhich allow us to build complicated but competitive metrics like maximum mean discrepancy (MMD)~\\cite{gretton2012kernel} and filter's geometric median~\\cite{he2019filter}.\n\n\n\\noindent{\\bf Function Encoding.} \nThe channel scoring functions can be categorized into two types: \nlabelless metrics and label-aware metrics.\nFor labelless functions like filter's $\\ell 1$-norm, \nwe adopt a direct encoding scheme as \n$\\mathrm{sum}_g(\\mathrm{abs}(\\mathcal{W}_I))$\nwith the expression tree shown in Fig.~\\ref{fig:func_encoding}. \n\nFor label-aware metrics such as the one in~\\cite{kung2019methodical} and MMD~\\cite{gretton2012kernel},\nwhich measure class discrepancy of the feature maps,\nwe observe a common computation graph among them, as shown in Fig.~\\ref{fig:func_encoding}:\n(1)~partition the feature maps in a labelwise manner;\n(2) apply the same operations on each label partition and all feature maps; \n(3) average\/sum the scores of all partitions to obtain a single scalar.\nThese metrics can be naively encoded as $C$-branch trees ($C$: number of class labels in the dataset).\nHowever, directly using the naive encoding scheme will result in data-dependent non-transferable metrics because: \n(1) $C$ varies from dataset to dataset (e.g., metrics for CIFAR-10 is not transferable to CIFAR-100); \n(2) mutating the subtrees differently could make the metric overfit to a specific label numbering scheme \n(e.g., for a metric with different subtrees on class-1 and class-2, \nrenumbering the labels would mean the metric would compute something different, which is undesirable).\n\nTo combat the above issues, \nwe express a label-aware function by a uni-tree which encodes the \ncommon operations that are applied to each label partition, as explained in Fig.~\\ref{fig:func_encoding}.\nInstead of directly encoding the operands from a specific label partition, \nlike $\\mathcal{F}^{1+}$ (feature maps with labels equal to 1) and $\\mathcal{F}^{1-}$ (feature maps with labels not equal to 1), \nwe use a symbolic representation of $\\mathcal{F^+}$ and $\\mathcal{F^-}$ to generically encode the partition concept.\nIn the actual scoring process, \nthe uni-tree is compiled back to a $C$-branch computation graph, \nwith $\\mathcal{F^+}$ and $\\mathcal{F^-}$ converted to the specific map partitions.\nSuch uni-tree encoding allows us to evolve label-aware metrics independent of $C$ and label numbering schemes, \nwhich ensures their transferability to datasets unseen by the evolution process. \n\n\\noindent\\textbf{SOAP.} Using the above described function encoding, \nwe can implement a broad range of competitive pruning functions:\nfilter's $\\ell1$-norm~\\cite{li2016pruning},\nfilter's $\\ell2$-norm~\\cite{he2018soft},\nbatch norm's scaling factor~\\cite{liu2017learning}, \nfilter's geometric median~\\cite{he2019filter},\nDiscriminant Information~\\cite{kung2019methodical},\nMaximum Mean Discrepancy~\\cite{gretton2012kernel},\nAbsolute SNR~\\cite{golub1999molecular},\nStudent's T-Test~\\cite{lehmann2006testing},\nFisher Discriminant Ratio~\\cite{pavlidis2001gene},\nand Symmetric Divergence~\\cite{mak2006solution}.\nFor the last four metrics, we adopt the scheme in~\\cite{liu2020rethinking} for channel scoring. \nWe name this group of functions the state-of-the-art population (SOAP), \nwhich helps our evolution in many aspects. \nFor instance, in Sec.~\\ref{sec:ablation_study}, we find that initializing the population with SOAP evolves better pruning functions than random initialization. \nDetailed implementation of SOAP is included in Supplementary.\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Figure\/Encoding_Explanation.eps}\n\\vspace{-0.25cm}\n\\caption{Illustration of the pruning function encoding.\n\\textbf{Left}: For labelless scoring metrics like filter's $\\ell$1-norm, \nwe adopt a direct tree encoding scheme.\n\\textbf{Right}: For label-aware scoring metrics, \nwe encode the $C$-subtree computation graph by a uni-tree ($C$: number of class labels). \nThe uni-tree encodes the common operations (\\textit{op}) on each label partition ($\\mathcal{F}^+, \\mathcal{F}^-$) and all feature maps ($\\mathcal{F}$).\nThis scheme allows transferable function evolution.\n}\\label{fig:func_encoding}\n\\end{figure*}\n\n\\subsection{Function Effectiveness Evaluation}\n\nThe encoded functions are applied to empirical pruning tasks to evaluate their effectiveness.\nTo avoid overfitting on certain data patterns and increase the generality of the evolved functions, \nwe evolve the population of functions on two different pruning tasks, LeNet-5~\\cite{lecun1998gradient} on MNIST~\\cite{lecun1998gradient} and VGG-16~\\cite{simonyan2014very} on CIFAR-10~\\cite{krizhevsky2009learning}.\nIn both pruning tasks, we adopt a one-shot pruning scheme and report the retrained accuracies on validation sets.\nFor each pruning task, we keep the pruning settings (layers' pruning ratios, target pruning layers, etc.)\nand the retraining hyper-parameters (learning rate, optimizer, weight decay factor, etc.) the same for all evaluations\nthroughout the evolution process.\nThis guarantees a fair effectiveness comparison over different functions in all generations \nand ensures we are evolving better functions rather than better hyper-parameters. \nIn this way, we can meta-learn powerful functions that perform well on both MNIST and CIFAR-10 and are generalizable to other datasets.\nNot surprisingly, evolving with both tasks produce stronger pruning functions than evolving on only one of them, shown in Sec.~\\ref{sec:fitness_combination}.\nMoreover, in Sec.~\\ref{sec:ablation_study}, we find our strategy enjoys better cost-effectiveness compared to direct evolution on a large dataset, e.g., ILSVRC-2012.\n\n\n\\subsection{Function Fitness}\\label{sec:fitness_combination}\n\nAfter evaluation, each encoded function receives two accuracies, $\\mathrm{Acc_{MNIST}}$ and $\\mathrm{Acc_{CIFAR}}$, from the pruning tasks.\nWe investigate two accuracy combination schemes,\nweighted arithmetic mean (Eqn.~\\ref{eqn:arthimetic}) and weighted geometric mean (Eqn.~\\ref{eqn:geometric}), \nto obtain the joint fitness of a function.\nA free parameter $\\alpha \\in [0, 1]$ is introduced to control the weights of different tasks.\n \\begin{equation}\n \\label{eqn:arthimetic}\n \\mathrm{Fitness} = \\alpha \\times \\mathrm{Acc_{MNIST}} + (1 - \\alpha) \\times \\mathrm{Acc_{CIFAR}}\n \\end{equation} \n \\vspace{-0.6cm}\n \\begin{equation}\n \\label{eqn:geometric}\n \\mathrm{Fitness} = (\\mathrm{Acc_{MNIST}})^{\\alpha} \\times (\\mathrm{Acc_{CIFAR}})^{1 - \\alpha} \n \\end{equation}\n\n\\noindent\\textbf{Ablation Study.} \nTo decide the fitness combination scheme for the main experiments,\nwe conduct 10 small preliminary evolution tests\nusing a grid of $\\alpha \\in \\{0, 0.3, 0.5, 0.7, 1\\}$ with both combination schemes.\nNote that when $\\alpha \\in \\{0, 1\\}$, the process degenerates to single dataset evolution.\nWe empirically evaluate the generalizability of the best evolved functions from \neach test by applying them to prune a ResNet-38 on CIFAR-100. \nNote CIFAR-100 is not used in the evolution process, and thus the performance on it speaks well for evolved functions' generalizability.\nIn Fig.~\\ref{fig:combination_scheme}, \nwe find that solely evolving on MNIST ($\\alpha = 1$) would be the least effective \noption for CIFAR-100 transfer pruning.\nIn addition, we find that functions evolved on two datasets ($\\alpha \\in [0.3, 0.5, 0.7]$) \ngenerally perform better than the ones that just evolve on a single dataset ($\\alpha \\in [0, 1]$).\nWe observe that setting\n$\\alpha = 0.5$ with weighted geometric mean leads to the best result,\nwhich we adopt in the main experiments.\n\n\\begin{figure*}[t]\n\\begin{minipage}{0.4\\textwidth}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Figure\/fitness_combination.eps}\n\\vspace{-0.3cm}\n\\caption{Preliminary evolution tests on the choice of fitness combination scheme. \nThe best evolved function from each scheme is applied to conduct a pruning test on CIFAR-100 with ResNet-38, and their accuracies are plotted.}\n\\label{fig:combination_scheme}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.57\\textwidth}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Figure\/Evolution_Result.eps}\n\\vspace{-0.3cm}\n\\caption{Progress of the evolution experiment. Each dot represents an individual function evaluation. \nThe red curve shows functions with the best fitness over generations, \nwhile the green curve shows the individuals at the 25 percentile fitness.\nThe effectiveness of the best function and the population's overall quality are both monotonically increasing.\n}\n\\label{fig:evo_results}\n\\end{minipage}\n\\end{figure*}\n\n\\subsection{Genetic Operations}\n\n\\textbf{Selection.} After evaluation, the population will undergo a selection process, \nwhere we adopt tournament selection~\\cite{goldberg1991comparative} to choose a subset of competitive functions. \n\n\\noindent\\textbf{Diversity Maintenance.} This subset of functions is then used to reproduce individuals for the next generation.\nHowever,\nwe observe shrinkage of the population's genetic diversity when all children are reproduced from parents, \nas the selected parents only represent a small pool of genomes.\nSuch diversity shrinkage would result in premature convergence of the evolution process.\nTo combat this issue, \nwe reserve a slot in the next generation and produce individuals in the slots by randomly cloning functions from SOAP or building random trees.\nWe find this adjustment empirically useful to help the evolution proceed longer.\n\n\\noindent\\textbf{Mutation and Crossover.} \nWe conduct mutation and crossover on the reproduced population to traverse the function design space for new expressions. \nWe adopt the conventional scheme of random tree mutation and one point crossover~\\cite{banzhaf1998genetic}.\nAfter mutation and crossover, the population will go through the next evolution iteration. \n\n\n\\noindent\\textbf{Function Validity.} \nThe function expressions generated from mutation and crossover can be invalid (non-invertible matrix, dimension inconsistency, etc.) due to the random selections of operators, operands, and nodes in the expression trees. \nTo combat this issue and enlarge our valid function space, \nsome operators are deliberately modified from their standard definition. \nFor instance, whenever we need to invert a positive semi-definite scatter matrix $S$, we automatically add a ridge factor $\\rho I$, and invert the matrix $S + \\rho I$. \nFor dimension inconsistency in elementwise operations, we have two options to pad the operand with a smaller dimension: \n(1) with 0 for $+$ and $-$, and 1 for $\\times$, and $\\div$,\n(2) with its own value if it is a scalar.\nMoreover, \nwe conduct a validity test on the mutated\/crossovered functions every time after the mutation\/crossover process.\nThe invalid expressions are discarded, and the mutation\/crossover operations are repeated until we recover the population size with all valid functions.\nThese methods ensure we generate valid function expressions under our vast design space during the evolution process.\n\n\n\n\\section{Related Work} \\label{sec:related_work}\n\n\n{\\bf Hand-Crafted Channel Pruning.} \nChannel pruning~\\cite{yu2018nisp,kung2019methodical,zhuang2018discrimination,he2019filter} is generally realized by using a handcrafted pruning function to score channels' saliency and remove redundant ones. \nBased on the scoring procedure, \nit can be categorized into \nlabelless pruning and label-aware pruning.\n\nLabelless channel pruning typically adopts the norm-based property of \nthe channel's feature maps or associated filters as pruning criterion~\\cite{li2016pruning, liu2017learning, he2017channel, luo2017thinet, louizos2017learning, he2018soft,ye2018rethinking, he2019filter,li2019exploiting}.\nFor example, \nLiu et al.~\\cite{liu2017learning} and Ye et al.~\\cite{ye2018rethinking} use the absolute value of scaling factors in the batch-norm,\nwhile $\\ell$1-norm and $\\ell$2-norm of \nchannels' associated filters are computed in~\\cite{li2016pruning,he2018soft,li2019exploiting} as channels' importance. \nOn the other hand, researchers have designed metrics to \nevaluate class discrepancy of channels' feature maps for label-aware pruning~\\cite{zhuang2018discrimination,kung2019methodical,liu2020rethinking}.\nZhuang et al.~\\cite{zhuang2018discrimination} insert discriminant losses in the network and \nremove channels that are the least correlated to the losses after iterative optimization.\nKung et al.~\\cite{kung2019methodical} and Liu et al.~\\cite{liu2020rethinking, liu2021class} adopt closed-form discriminant functions to accelerate the scoring process. \n\nWhile these works use handcrafted scoring metrics,\nwe learn transferable and generalizable pruning functions automatically. \n\n\\noindent\\textbf{Meta-Learning.} Our work falls into the category of meta-learning, \nwhere prior works have attempted to optimize machine learning components, \nincluding \nhyper-parameters~\\cite{bergstra2013making, snoek2015scalable,feurer2015efficient}, \noptimizers~\\cite{chen2017learning, wichrowska2017learned,bello2017neural}, \nand neural network structures~\\cite{zoph2016neural,zoph2018learning,tan2019efficientnet,liu2018progressive,xie2017genetic,real2017large,liu2017hierarchical,real2019regularized}.\n\nPrior works on neural architecture search (NAS) have leveraged reinforcement learning (RL) to discover high-performing network structures~\\cite{zoph2016neural,baker2016designing,zoph2018learning,cai2018proxylessnas,tan2019mnasnet,tan2019efficientnet}.\nRecently, NAS has also been adopted to find efficient network structures~\\cite{tan2019efficientnet,tan2019mnasnet}.\nAnother line of works adopts evolution strategies (ES) to explore the space of network structures~\\cite{fernando2016convolution,\nsuganuma2017genetic,\nxie2017genetic,real2017large,\nliu2017hierarchical,\nreal2019regularized,\ndai2019chamnet,\nmiikkulainen2019evolving,\nstanley2019designing,\no2020neural,\nsinha2021evolving, templier2021geometric},\nwhich demonstrates competitive performance to RL methods.\nThis notion is pioneered by neuro-evolution~\\cite{stanley2002evolving,floreano2008neuroevolution, stanley2009hypercube}\nwhich evolves the topology of small neural networks.\nIn the era of deep learning, Suganuma et al.~\\cite{suganuma2017genetic} leverage Cartesian genetic programming to find competitive network structures.\nReal et al.~\\cite{real2019regularized} evolve networks that improve over the ones found by RL-based NAS~\\cite{zoph2018learning}.\nDai et al.~\\cite{dai2019chamnet} apply ES to design efficient and deployable networks for mobile platforms.\nTemplier et al.~\\cite{templier2021geometric} propose a geometric encoding scheme for more efficient parameter search.\n\nCompared to prior works, \nwe employ evolutionary learning from a new angle for efficient network design, \nwhere we learn transferable pruning functions that produce state-of-the-art pruning results.\nOur work is orthogonal to prior works, for example, our evolved functions can be potentially applied on evolutionary NAS-learned networks to further enhance their efficiency.\n\n\n\\noindent\\textbf{Meta-Pruning.} \nPrior works~\\cite{huang2018learning,he2018amc,liu2019metapruning,chin2020towards,he2020learning} have also adopted a similar notion of learning to prune a CNN. \nWe note that an evolution strategy is used in LeGR~\\cite{chin2020towards} and MetaPruning~\\cite{liu2019metapruning} \nto search for a pair of pruning parameters and network encoding vectors, respectively.\nHowever,\nour approach is drastically different from them in terms of search space and search candidates,\nwhere we search for effective combinations of operands and operators to build transferable and powerful pruning functions.\nHe et al. propose LFPC~\\cite{he2020learning} to learn network pruning criteria (functions) across layers by training a differentiable criteria sampler.\nHowever, rather than learning new pruning functions, \ntheir goal is to search within a pool of existing pruning criteria and find candidates that are good for a certain layer's pruning.\nOn the contrary, \nour evolution recombines the operands and operators \nand produces novel pruning metrics, \nwhich are generally good for all layers.\n\nWe also notice that MetaPruning~\\cite{liu2019metapruning}, LFPC~\\cite{he2020learning}, and other methods~\\cite{huang2018learning,he2018amc,chin2020towards} are all learned on one task (dataset and network) and applied only on the same task with no transferability.\nIn contrast, we only need one evolution learning process, which outputs evolved functions \nthat are transferable across multiple tasks and demonstrate competitive performance on all of them.\n\n\n\n\n\\begin{comment}\nStanley \\cite{stanley2002evolving} used a sequence of node genes and connect genes to encode a multi-layer perceptron. \nOther work has used different representation schemes \\cite{floreano2008neuroevolution, gruau1993genetic, , pugh2013evolving, , yao1999evolving}. \nResearchers have also evolved large modern neural networks like CNNs and LSTMs~\\cite{real2017large,miikkulainen2019evolving,xie2017genetic,suganuma2017genetic,jozefowicz2015empirical}. In our work, we focus on the evolution for image classification. \nXie et al. \\cite{xie2017genetic} proposed to use binary representation to encode the connections of a CNN. \nReal et al.~\\cite{real2017large} has evolved CNNs for CIFAR dataset classification. These works aim to evolve high-accuracy network structures and use accuracy as the fitness function. \n\nThese encoding methods allow a network structure to be represented by a concise {\\it chromosome}. \nIn a genetic algorithm, every chromosome will be evaluated by a {\\it fitness function}. The fitness functions of network evolutions are task dependent, and they can be the reward of pole balancing problem \\cite{stanley2002evolving}, the output of a real control system \\cite{yao1999evolving}, and so on.\nA population, i.e., a set of chromosomes, is initialized to seed the evolution process. The population is evolved via bio-inspired operators such as mutation, crossover, and selection. \n\\end{comment}\n\n\n\\section{Transfer Pruning}\n\n\\noindent\\textbf{Benchmarks.} To show the generalizability of our evolved pruning function,\nwe apply $\\xi^*$ in Eqn.~\\ref{eqn:best-func} to more challenging datasets that are not used in the evolution process:\nCIFAR-100~\\cite{krizhevsky2009learning}, \nSVHN~\\cite{netzer2011reading}, \nand ILSVRC-2012~\\cite{deng2009imagenet}.\nWe compare our method with \nmetrics from SOAP, e.g., \nL1~\\cite{li2016pruning}, FPGM~\\cite{he2019filter}, \nG-SD~\\cite{liu2020rethinking},\nand DI~\\cite{kung2019methodical}, \nwhere $\\xi^*$ outperforms all these handcrafted metrics.\nWe also include other ``learn to prune\" methods like Meta~\\cite{liu2019metapruning} and\nLFPC~\\cite{he2020learning} and other state-of-the-art methods like \nDSA~\\cite{ning2020dsa} and CC~\\cite{li2021towards} for comparison.\nThe results are summarized in Tab.~\\ref{tab:SVHN},~\\ref{tab:CIFAR-100}, and~\\ref{tab:ILSVRC-2012},\nwhere the accuracies are shown as ``baseline acc. $\\rightarrow$ pruned acc.\" and the numbers for all other methods are copied from their papers.\nOn ILSVRC-2012,\nwe report our pruned models at different FLOPs reduction levels and add a suffix specifying their FLOPs pruning ratios (e.g., \\ourmethod\\ 60\\%-pruned).\nThis is because different prior arts report their compressed models at different rates, \nand we want to make a fair comparison to all of them.\nWe find that our evolved function achieves state-of-the-art results on all datasets.\n\n\n\\noindent\\textbf{Settings.} We adopt a one-shot pruning scheme with a uniform pruning ratio across layers for our transfer pruning \nand use the SGD optimizer with Nesterov Momentum~\\cite{nesterov1983method} for retraining.\nThe weight decay factor and the momentum are set to be 0.0001 and 0.9, respectively.\nOn SVHN\/CIFAR-100, \nwe use a batch size of 32\/128 to fine-tune the network with 20\/200 epochs. \nThe learning rate is initialized at 0.05 and multiplied by 0.14 at 40\\% and 80\\% of the total number of epochs.\nOn ILSVRC-2012, we use a batch size of 128 to fine-tune VGG-16\/ResNet-18\/MobileNet-V2\nfor 30\/100\/100 epochs.\nFor VGG-16\/ResNet-18, the learning rate is started at 0.0006 and multiplied by 0.4 at 40\\% and 80\\% of the total number of epochs.\nWe use a cosine decay learning rate schedule for MobileNet-V2~\\cite{sandler2018mobilenetv2} with an initial rate of 0.03.\n\n\\input{Table\/ImageNet_Transfer_Pruning_Table}\n\n\\noindent\\textbf{SVHN.} We first evaluate $\\xi^*$ on SVHN with ResNet-164. \n\\ourmethod\\ outperforms SLIM~\\cite{liu2017learning} by 0.1\\% in accuracy\nwith significant hardware resource savings: \n32.1\\% more FLOPs saving\nand 48.5\\% more parameters saving,\nwhich well demonstrates the effectiveness of $\\xi^*$.\n\n\\noindent\\textbf{CIFAR-100.}\nOn VGG-19, our pruned model achieves an accuracy gain of 0.35\\%\nwith respect to G-SD~\\cite{liu2020rethinking}.\nCompared to LFPC~\\cite{he2020learning} and LeGR~\\cite{chin2020towards}, \nour pruned ResNet-56 achieves an accuracy gain of 0.87\\% and 0.66\\%, respectively, while having 5\\% less FLOPs.\nOn ResNet-110, \nour method outperforms\nFPGM~\\cite{he2019filter} and TAS~\\cite{dong2019network} \nby 1.30\\% and 0.69\\% in terms of accuracy \nwith 4\\% less FLOPs.\nIn comparison with \nLCCL~\\cite{dong2017more}, SLIM~\\cite{liu2017learning}, \nand DI~\\cite{kung2019methodical},\nour pruned ResNet-164 achieves an accuracy of 77.77\\% with 63.2\\% FLOPs reduction which advances all prior methods.\n\n\\noindent\\textbf{ILSVRC-2012.} \nOn VGG-16, our approach improves over the baseline by nearly 1.1\\% in top-1 accuracy with 2.4$\\times$ acceleration.\nOur 3.3$\\times$-accelerated model advances the state of the art by achieving top-1\/top-5 accuracies of 71.64\\%\/90.60\\%. \nOn ResNet-18, our approach reduces 16.8\\% of the FLOPs without top-1 accuracy loss.\nCompared to LCCL~\\cite{dong2017more}, \nour method achieves a 2.72\\% top-1 accuracy gain with a higher FLOPs reduction ratio.\nWe demonstrate top-1 accuracy gains of 1.75\\% and 1.50\\% with respect to SFP~\\cite{he2018soft} and DCP~\\cite{zhuang2018discrimination} with over 40\\% FLOPs reduction.\nWe finally show our performance on a much more compact network, MobileNet-V2, \nwhich is specifically designed for mobile deployment.\nWhen 26.9\\% of FLOPs are pruned, our approach outperforms AMC~\\cite{he2018amc}, Meta~\\cite{liu2019metapruning}, and LeGR~\\cite{chin2020towards} with a top-1 accuracy of 71.90\\%.\nAt a higher pruning ratio, \nour method advances DCP~\\cite{zhuang2018discrimination} and Meta~\\cite{liu2019metapruning}\nby top-1 accuracies of 4.94\\% and 0.96\\%,\nwith 53.4\\% FLOPs reduction.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nIn this paper we consider complex hyperbolic triangle groups, i.e. groups of isometries of the complex hyperbolic plane\ngenerated by three complex reflections in complex geodesics.\nWe will focus on the case of ultra-parallel groups, that is, the case where the complex geodesics are pairwise disjoint.\nIf the pairwise distances between the complex geodesics are \\(m_1\\), \\(m_2\\) and \\(m_3\\),\nthen we say that the group is an ultra-parallel complex hyperbolic triangle group of type \\([m_1,m_2,m_3]\\).\nUnlike real reflections, complex reflections can be of arbitrary order.\nIf an ultra-parallel complex hyperbolic triangle group is generated by reflections of orders \\(n_1,n_2,n_3\\)\nin complex geodesics \\(C_1,C_2,C_3\\) with the distance between~\\(C_{k-1}\\) and~\\(C_{k+1}\\) equal to~\\(m_k\\) for~\\(k=1,2,3\\),\nthen we say that the group is of type \\([m_1,m_2,m_3;n_1,n_2,n_3]\\).\nUltra-parallel triangle groups of types \\([m,m,0;2,2,2]\\) and \\([m,m,2m;2,2,2]\\) have been considered previously in \\cite{WyssGall},\nwhile groups of type \\([m_1,m_2,0;2,2,2]\\) have been considered in \\cite{MPP} and \\cite{Mo}.\n\nIn this paper, we will study discreteness of ultra-parallel complex hyperbolic triangle groups of type \\([m,m,0;3,3,2]\\).\nThe deformation space of such groups for given \\(m\\) is of real dimension one.\nSuch a group is determined up to an isometry by the angular invariant \\(\\alpha\\in[0,2\\pi]\\),\nsee section~\\ref{background}.\nWe will determine some conditions on the angular invariant that ensure that an \\([m,m,0;3,3,2]\\)-triangle group is discrete or non-discrete respectively.\n\nThe main discreteness result of the paper is the following proposition obtained using a compression property:\n\n\\begin{proposition}\n\\label{prop1}\nA complex hyperbolic ultra-parallel triangle group of type \\([m,m,0;3,3,2]\\)\nwith angular invariant \\(\\alpha\\) is discrete if \n\\[\\cos(\\alpha)\\le-\\frac{1}{2}\\quad\\mbox{and}\\quad\\cosh\\left(\\frac{m}{2}\\right)\\ge\\frac{2}{\\sqrt{3}}.\\]\n\\end{proposition}\n\nWe contrast this proposition with the following non-discreteness result obtained using Shimizu's lemma:\n\n\\begin{proposition}\n\\label{prop2}\nA complex hyperbolic ultra-parallel triangle group of type \\([m,m,0;3,3,2]\\)\nwith angular invariant \\(\\alpha\\) is non-discrete if\n\\[\\cos(\\alpha)>1-\\frac{1}{36\\cosh^2\\left(\\frac{m}{2}\\right)}.\\]\n\\end{proposition}\n\nCombining these results, we see that there is a gap between the intervals of discreteness and non-discreteness as illustrated\nin Figure~\\ref{fig-gap}.\nFurther study is needed to close this gap.\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}\n\\foreach \\Point in {(0.5,1), (1.91,1), (3.25,1), (4.84,1), (5.5,1)}{\\node at \\Point {\\textbullet};}\n\\draw[line width=.35mm, blue] (0.57,1)--(1.8375,1);\n\\draw[line width=.35mm, red] (4.9125,1)--(5.427,1);\n\\draw[line width=.35mm, red] (5.573,1)--(5.98,1);\n\\path[draw] (1.9,1)--(4.84,1);\n\\path[draw] (5.85,0.9)--(6,1.002);\n\\path[draw] (5.85,1.1)--(6,0.998);\n\\path[draw, dashed] (1.91,1)--(1.91,0.2);\n\\path[draw, dashed] (4.84,1)--(4.84,0.2);\n\\node at (0.5,0.7) {$-1$};\n\\node at (5.5,0.7) {$1$};\n\\node at (3.25,0.7) {$0$};\n\\node at (1.875,-0.3) {$-\\frac{1}{2}$};\n\\node at (4.7,-0.3) {$1-\\frac{1}{36\\cosh^2\\left(\\frac{m}{2}\\right)}$};\n\\node at (6.65,1) {$\\cos(\\alpha)$};\n\\path[clip] (1.4,0)--(5.35,0)--(5.35,2)--(1.4,2)--cycle;\n\\draw(5.75,1) circle (0.9cm);\n\\draw(1,1) circle (.9cm);\n\\end{tikzpicture}\n\\end{center}\n\\caption{Gap between discreteness (blue) and non-discreteness (red) results}\n\\label{fig-gap}\n\\end{figure}\n\n\\section{Background}\n\n\\label{background}\n\nIn this section we will give a brief introduction to complex hyperbolic geometry, for further details see \\cite{Gold}.\n\n\\subsection{Complex hyperbolic plane:}\nLet $\\mathbb C^{2,1}$ be the $3$-dimensional complex vector space\nequipped with a Hermitian form $\\<\\cdot,\\cdot\\>$ of signature $(2,1)$,\ne.g.\n\\[\\=z_1\\bar{w}_1+z_2\\bar{w}_2-z_3\\bar{w}_3.\\]\nIf $z\\in\\mathbb C^{2, 1}$ then we know that $\\$ is real.\nThus we define subsets $V_-$, $V_0$ and $V_+$ of $\\mathbb C^{2,1}$ as follows\n\\begin{align*}\n V_-&=\\{z\\in\\mathbb C^{2,1}\\,\\,\\big|\\,\\,\\<0\\},\\\\\n V_0&=\\{z\\in\\mathbb C^{2,1}\\backslash\\{0\\}\\,\\,\\big|\\,\\,\\=0\\},\\\\\n V_+&=\\{z\\in\\mathbb C^{2,1}\\,\\,\\big|\\,\\,\\>0\\}.\n\\end{align*}\nWe say that $z\\in\\mathbb C^{2,1}$ is {\\it negative\\\/}, {\\it null\\\/} or {\\it positive\\\/} if $z$ is in $V_-$, $V_0$ or $V_+$ respectively. Define a projection map $\\mathbb P$ on the points of $\\mathbb C^{2,1}$ with $z_3\\ne0$ as \n\\[\\mathbb P : z=\\begin{bmatrix} z_1\\\\ z_2\\\\ z_3\\end{bmatrix}\\mapsto\\begin{pmatrix} z_1\/z_3\\\\ z_2\/z_3\\end{pmatrix}\\in\\mathbb P(\\mathbb C^{2,1}).\\]\nThat is, provided $z_3\\ne0$, \n\\[z=(z_1, z_2, z_3)\\mapsto[z]=[z_1:z_2:z_3]=\\left[\\frac{z_1}{z_3}:\\frac{z_2}{z_3}:1\\right].\\]\nThe {\\it projective model\\\/} of the complex hyperbolic plane is defined to be the collection of negative lines in $\\mathbb C^{2,1}$\nand its boundary is defined to be the collection of null lines.\nThat is\n\\[H_{\\c}^2=\\mathbb P(V_-)\\quad\\text{and}\\quad{\\partial\\chp}=\\mathbb P(V_0).\\]\nThe metric on $H_{\\c}^2$, called the {\\it Bergman metric\\\/}, is given by the distance function~$\\rho$ defined by the formula\n\\[\n \\cosh^2\\left(\\frac{\\rho([z],[w])}{2}\\right)\n =\\frac{\\langle{z, w\\rangle}\\langle{w, z\\rangle}}{\\langle{z, z\\rangle}\\langle{w, w\\rangle}},\n\\]\nwhere $[z]$ and $[w]$ are the images of~$z$ and $w$ in $\\mathbb C^{2,1}$ under the projectivisation map~$\\mathbb P$. \nThe group of holomorphic isometries of~$H_{\\c}^2$ with respect to the Bergman metric\ncan be identified with the projective unitary group $\\PU(2,1)$.\n\n\\subsection{Complex geodesics:}\nA {\\it complex geodesic\\\/} is a projectivisation of a 2-dimensional complex subspace of $\\mathbb C^{2,1}$.\nAny complex geodesic is isometric to \\[\\{[z:0:1]\\,\\,\\big|\\,\\, z\\in\\mathbb C\\}\\] in the projective model.\nAny positive vector $c\\in V_+$ determines a two-dimensional complex subspace\n\\[\\{z\\in\\mathbb C^{2,1}\\,\\,\\big|\\,\\, \\=0\\}.\\]\nProjecting this subspace we obtain a complex geodesic\n\\[\\mathbb P\\left(\\{z\\in\\mathbb C^{2,1}\\,\\,\\big|\\,\\, \\=0\\}\\right).\\]\nConversely, any complex geodesic is represented by a positive vector $c\\in V_+$, \ncalled a {\\it polar vector\\\/} of the complex geodesic.\nA polar vector is unique up to multiplication by a complex scalar.\nWe say that the polar vector~$c$ is {\\it normalised\\\/} if $\\=1$.\n\n\\bigskip\nLet $C_1$ and $C_2$ be complex geodesics with normalised polar vectors~$c_1$ and~$c_2$ respectively.\nWe call $C_1$ and $C_2$ {\\it ultra-parallel\\\/} if they have no points of intersection in $H_{\\c}^2$,\nin which case $|\\|\\ge1$ and\n\\[|\\|=\\cosh\\left(\\frac{1}{2}\\dist(C_1, C_2)\\right),\\]\nwhere $\\dist(C_1, C_2)$ is the distance between $C_1$ and~$C_2$.\nThis includes pairs~$C_1$ and~$C_2$ intersecting on the boundary ${\\partial\\chp}$,\nin which case $|\\|=1$ and $\\dist(C_1, C_2)=0$.\n\n\\subsection{Complex reflections:}\nFor a given complex geodesic $C$, a {\\it minimal complex hyperbolic reflection of order~$n$} in~$C$\nis the isometry $\\iota_C$ in $\\PU(2,1)$ of order~$n$ with fixed point set~$C$ given by\n\\[\\iota(z) = -z+(1-\\mu)\\frac{\\}{\\}c,\\]\nwhere $c$ is a polar vector of~$C$ and $\\mu=\\exp(2\\pi i\/n)$.\n\n\\subsection{Complex hyperbolic triangle groups:}\nA {\\it complex hyperbolic triangle\\\/} is a triple \\((C_1, C_2, C_3)\\) of complex geodesics in $H_{\\c}^2$.\nA triangle \\((C_2, C_2, C_3)\\) is a {\\it complex hyperbolic ultra-parallel \\([m_1, m_2, m_3]\\)-triangle\\\/} if the complex geodesics are ultra-parallel at distances $m_k=\\dist(C_{k-1}, C_{k+1})$ for $k=1,2,3$.\nA {\\it complex hyperbolic ultra-parallel \\([m_1,m_2,m_3;n_1,n_2,n_3]\\)-triangle group\\\/}\nis a subgroup of \\(\\PU(2,1)\\) generated by complex reflections \\(\\iota_k\\) of order \\(n_k\\) in the sides \\(C_k\\)\nof a complex hyperbolic ultra-parallel \\([m_1, m_2, m_3]\\)-triangle \\((C_2, C_2, C_3)\\).\n\n\\subsection{Angular invariant:}\nFor each fixed triple \\(m_1, m_2, m_3\\) the space of \\([m_1, m_2, m_3]\\)-triangles is of real dimension one.\nWe can describe a parametrisation of the space of complex hyperbolic triangles in $H_{\\c}^2$ by means of an angular invariant ${\\alpha}$.\nWe define the {\\it angular invariant\\\/} ${\\alpha}$ of the triangle \\((C_1, C_2, C_3)\\) by\n\\[{\\alpha}=\\arg\\left(\\prod_{k=1}^3 \\\\right),\\]\nwhere $c_k$ is the normalised polar vector of the complex geodesic~$C_k$.\nWe use the following proposition, given in \\cite{Pra}, which gives criteria for the existence of a triangle group\nin terms of the angular invariant.\n\n\\begin{proposition}\n\\label{traingle-existence}\nAn $[m_1, m_2, m_3]$-triangle in $H_{\\c}^2$ is determined uniquely up to isometry\nby the three distances between the complex geodesics and the angular invariant~${\\alpha}$.\nFor any ${\\alpha}\\in[0, 2\\pi]$, an $[m_1, m_2, m_3]$-triangle with angular invariant~${\\alpha}$ exists if and only if\n\\[\\cos({\\alpha})<\\frac{r_1^2+r_2^2+r_3^2-1}{2r_1r_2r_3},\\]\nwhere $r_k=\\cosh(m_k\/2)$.\n\\end{proposition}\n\n\\noindent\nFor $m_3=0$ we have $r_3=1$ and the right hand side of the inequality in Propositionl~\\ref{traingle-existence} is \n\\[\\frac{r_1^2+r_2^2}{2r_1r_2}\\ge1,\\]\nso the condition on~${\\alpha}$ is always satisfied,\ni.e.\\ for any ${\\alpha}\\in[0, 2\\pi]$ there exists an $[m_1, m_2, m_3]$-triangle with angular invariant~${\\alpha}$.\n\n\\subsection{Heisenberg group:}\nThe boundary of the complex hyperbolic space can be identified with the {\\it Heisenberg space\\\/} \n\\[\\mathcal{N}=\\mathbb C\\times\\mathbb R\\cup\\{\\infty\\}=\\{(\\zeta,\\nu)\\,\\,\\big|\\,\\,\\zeta\\in\\mathbb C,\\nu\\in\\mathbb R\\}\\cup\\{\\infty\\}.\\]\nOne homeomorphism taking ${\\partial\\chp}$ to $\\mathcal{N}$ is given by the stereographic projection:\n\\[\n [z_1:z_2:z_3]\\mapsto\\left(\\frac{z_1}{z_2+z_3}, \\Im\\left(\\frac{z_2-z_3}{z_2+z_3}\\right)\\right)~\\text{if}~z_2+z_3\\ne0,\n \\quad\n [0:z:-z]\\mapsto\\infty.\n\\]\n\n\\subsection{Chains:}\nA complex geodesic in~$H_{\\c}^2$ is homeomorphic to a disc,\nits intersection with the boundary of the complex hyperbolic plane is homeomorphic to a circle.\nCircles that arise as the boundaries of complex geodesics are called {\\it chains\\\/}.\n\n\\bigskip\nThere is a bijection between chains and complex geodesics. We can therefore, without loss of generality, talk about reflections in chains instead of reflections in complex geodesics. \n\n\\bigskip\nChains can be represented in the Heisenberg space, for more details see \\cite{Gold}.\nChains passing through~$\\infty$ are represented by vertical straight lines defined by $\\zeta = \\zeta_0$.\nSuch chains are called {\\it vertical\\\/}.\nThe vertical chain $C_{\\zeta_0}$ defined by $\\zeta=\\zeta_0$ has a polar vector\n\\[c_{\\zeta_0}=\\begin{bmatrix}1\\\\ -\\bar{\\zeta_0}\\\\ \\bar{\\zeta_0}\\end{bmatrix}.\\]\nA chain not containing~$\\infty$ is called {\\it finite\\\/}.\nA finite chain is represented by an ellipse whose vertical projection $\\mathbb C\\times\\mathbb R\\rightarrow\\mathbb C$ is a circle in~$\\mathbb C$.\nThe finite chain with centre $(\\zeta_0,\\nu_0)\\in\\mathcal{N}$ and radius $r_0 > 0$ has a polar vector\n\\[\\begin{bmatrix}2\\zeta_0 \\\\ 1+r_0^2-\\zeta_0\\bar{\\zeta_0}+i\\nu_0 \\\\ 1-r_0^2+\\zeta_0\\bar{\\zeta_0}-i\\nu_0 \\end{bmatrix}\\]\nand consists of all $(\\zeta,\\nu)\\in\\mathcal{N}$ satisfying the equations\n\\[|\\zeta-\\zeta_0|=r_0,\\quad\\nu=\\nu_0-2\\Im(\\zeta\\bar{\\zeta}_0).\\]\n\n\\subsection{Heisenberg isometries:}\nThe metric induced on~$\\mathcal{N}$ from $H_{\\c}^2$ via the identification of~$\\mathcal{N}$ and ${\\partial\\chp}$ is the {\\it Cygan metric\\\/}\n\\[\n \\rho_0\\left((\\zeta_1, \\nu_2), (\\zeta_2, \\nu_2)\\right)\n = \\Big|\\abs{\\zeta_1-\\zeta_2}^2-i(\\nu_1-\\nu_2)-2i\\Im(\\zeta_1\\bar{\\zeta_2})\\Big|^{1\/2}.\n\\]\n\n\\bigskip\\noindent\nA {\\it Heisenberg translation\\\/}~$T_{(\\xi, \\nu)}$ by $(\\xi, \\nu)\\in\\mathcal{N}$ is given by\n\\[(\\zeta,\\omega)\\mapsto(\\xi, \\nu)+(\\zeta,\\omega)=(\\xi+\\zeta,\\omega+\\nu+2\\Im(\\xi\\bar{\\zeta}))\\]\nand corresponds to the following element in $\\PU(2,1)$\n\\[\n \\begin{pmatrix}\n 1 & \\xi & \\xi \\\\\n -\\bar{\\xi} & 1-\\frac{|\\xi|^2-i\\nu}{2} & -\\frac{|\\xi| ^2-i\\nu}{2}\\\\ \n \\bar{\\xi} & \\frac{|\\xi|^2-i\\nu}{2} & 1+\\frac{|\\xi|^2-i\\nu}{2} \n \\end{pmatrix}.\n\\]\n\n\\bigskip\\noindent\nA {\\it rotation\\\/}~$R_{\\mu}$ by $\\mu\\in\\mathbb C$, $|\\mu|=1$ is given by\n\\[(\\zeta,\\omega)\\mapsto(\\mu\\cdot\\zeta,\\omega)\\]\nand corresponds to the following element in $\\PU(2,1)$\n\\[\n \\begin{pmatrix}\n \\mu & 0 & 0 \\\\\n 0 & 1 & 0\\\\ \n 0 & 0 & 1\n \\end{pmatrix}.\n\\]\n\n\\bigskip\\noindent\nA minimal complex reflection~$\\iota_{C_{\\varphi}}$ of order~$n$ in a vertical chain~$C_{\\varphi}$ with polar vector \n\\[c_{\\varphi}=\\begin{bmatrix}1\\\\ -\\bar{\\varphi}\\\\ \\bar{\\varphi}\\end{bmatrix}\\]\n is given by\n\\[(\\zeta,\\omega)\\mapsto(\\mu\\zeta+(1-\\mu)\\varphi,\\omega+2|\\varphi|^2\\Im(1-\\mu)+2\\Im((1-\\mu)\\bar{\\varphi}\\zeta)\\]\nand corresponds to the following element in~$\\PU(2,1)$\n\\[\n \\begin{pmatrix}\n -\\mu & -(1-\\mu)\\varphi & -(1-\\mu)\\varphi\\\\ \n -(1-\\mu)\\bar{\\varphi} & (1-\\mu)|\\varphi|^2-1 & (1-\\mu)|\\varphi|^2\\\\ \n (1-\\mu)\\bar{\\varphi} & -(1-\\mu)|\\varphi|^2 & -(1-\\mu)|\\varphi|^2-1\n \\end{pmatrix},\n\\] \nwhere $\\mu=\\exp(2\\pi i\/n)$.\nThe complex reflextion $\\iota_{C_{\\varphi}}$ can also be described as a composition of a Heisenberg translation and a rotation:\n\\[\\iota_{C_{\\varphi}}=R_{\\mu}\\circ T_{(\\xi,\\nu)},\\]\nwhere \n\\[\\xi=(\\bar{\\mu}-1)\\varphi \\quad\\text{and}\\quad \\nu=2|\\varphi|^2\\cdot\\Im(1-\\mu)=-4|\\varphi|^2\\sin\\left(\\frac{2\\pi}{n}\\right).\\]\n\n\n\\subsection{Products of reflections in chains:}\nWhat effect does the minimal complex reflection of order~$n$ in the vertical chain~$C_\\zeta$ have on another vertical chain, $C_\\xi$, which intersects $\\mathbb C\\times\\{0\\}$ at~$\\xi$?\n\n\\bigskip\nWe calculate \n\\begin{align*}\n \\begin{pmatrix}\n -\\mu & -(1-\\mu)\\zeta & -(1-\\mu)\\zeta \\\\\n -(1-\\mu)\\bar{\\zeta} & (1-\\mu)|\\zeta|^2-1 & (1-\\mu)|\\zeta|^2 \\\\\n (1-\\mu)\\bar{\\zeta} & -(1-\\mu)|\\zeta|^2 & -(1-\\mu)|\\zeta|^2-1\n \\end{pmatrix}\n \\begin{bmatrix}1\\\\ -\\bar{\\xi}\\\\ \\bar{\\xi}\\end{bmatrix}\n =\n \\begin{bmatrix} -\\mu\\\\ -(1-\\mu)\\bar{\\zeta}+\\bar{\\xi}\\\\ (1-\\mu)\\bar{\\zeta} -\\bar{\\xi}\\end{bmatrix}.\n\\end{align*}\nThis vector is a multiple of\n\\[\n \\begin{bmatrix} \n 1\\\\ \n (1-\\mu)\\bar{\\mu}\\bar{\\zeta}-\\bar{\\mu}\\bar{\\xi}\\\\\n -(1-\\mu)\\bar{\\mu}\\bar{\\zeta} +\\bar{\\mu}\\bar{\\xi}\n\\end{bmatrix}\n=\\begin{bmatrix} 1 \\\\ -\\overline{\\left(\\mu\\xi-(\\mu-1)\\zeta\\right)} \\\\ \\overline{\\left(\\mu\\xi-(\\mu-1)\\zeta\\right)} \\end{bmatrix}\n\\]\nwhich is the polar vector of the vertical chain that intersects $\\mathbb C\\times\\{0\\}$ at $\\mu\\xi-(\\mu-1)\\zeta$. \nThis corresponds to rotating $\\xi$ around $\\zeta$ through $\\frac{2\\pi}{n}$.\nSo if we have a vertical chain $C_{\\xi}$, the minimal complex reflection of order~$n$ in another vertical chain~$C_{\\zeta}$ rotates $C_{\\xi}$ as a set around $C_{\\zeta}$ through $\\frac{2\\pi}{n}$\n(but not point-wise as there is also vertical translation on the chain).\n\n\\subsection{Bisectors and spinal spheres:}\nUnlike in the real hyperbolic space, there are no totally geodesic real hypersurfaces in $H_{\\c}^2$.\nAn acceptable substitute are the metric bisectors.\nLet $z_1, z_2\\inH_{\\c}^2$ be two distinct points.\nThe {\\it bisector equidistant\\\/} from~$z_1$ and~$z_2$ is defined as\n\\[\\{z\\inH_{\\c}^2\\,\\,\\big|\\,\\, \\rho(z_1,z)=\\rho(z_2,z)\\}.\\]\nThe intersection of a bisector with the boundary of~$H_{\\c}^2$ is a smooth hypersurface in~${\\partial\\chp}$ called a {\\it spinal sphere\\\/}, which is diffeomorphic to a sphere. \nAn example is the bisector\n\\[\\mathcal{C}=\\{[z:it:1]\\inH_{\\c}^2\\,\\,\\big|\\,\\, |z|^2<1-t^2,~z\\in\\mathbb C,~t\\in\\mathbb R\\}.\\]\nIts boundary, the {\\it unit spinal sphere\\\/}, can be described as\n\\[U=\\{(\\zeta,\\nu)\\in\\mathcal{N}\\,\\,\\big|\\,\\, |\\zeta|^4+\\nu^2=1\\}.\\]\n\n\\section{Parametrisation of complex hyperbolic triangle groups of type $[m_1, m_2, 0;n_1,n_2,n_3]$}\n\nFor $r_1, r_2 \\ge1$ and ${\\alpha}\\in(0,2\\pi)$, let $C_1$, $C_2$ and $C_3$ be the complex geodesics with respective polar vectors\n\\[\n c_1 = \\begin{bmatrix}1 \\\\ -r_2e^{-i\\theta} \\\\ r_2e^{-i\\theta} \\end{bmatrix},\\quad\n c_2 = \\begin{bmatrix}1 \\\\ r_1e^{i\\theta} \\\\ -r_1e^{i\\theta} \\end{bmatrix}\n \\quad\\mbox{and}\\quad\n c_3 = \\begin{bmatrix}0 \\\\ 1 \\\\ 0 \\end{bmatrix},\n\\]\nwhere $\\theta=(\\pi-{\\alpha})\/2\\in(-\\pi\/2,\\pi\/2)$.\nThe type of triangle formed by $C_1,C_2,C_3$ is an ultra-parallel $[m_1, m_2,0]$-triangle with angular invariant ${\\alpha}$,\nwhere $r_k=\\cosh(m_k\/2)$ for~$k=1,2$.\n\n\\bigskip\nLet $\\iota_k$ be the minimal complex reflection of order $n_k$ in the chain $C_k$ for $k=1,2,3$.\nThe group $\\<\\iota_1,\\iota_2,\\iota_3\\>$ generated by these three complex reflections\nis an ultra-parallel complex hyperbolic triangle groups of type $[m_1, m_2, 0;n_1,n_2,n_3]$.\nLooking at the arrangement of the chains $C_1$, $C_2$ and $C_3$ in the Heisenberg space $\\mathcal{N}$,\nthe finite chain~$C_3$ is the (Euclidean) unit circle in $\\mathbb C\\times\\{0\\}$,\nwhereas $C_1$ and~$C_2$ are vertical lines\nthrough the points $\\varphi_1 = r_2e^{i\\theta}$ and $\\varphi_2=-r_1e^{-i\\theta}$ respectively.\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n \\path[clip] (-8,-5)--(-8,5)--(8,5)--(8,-5)--(-8,-5)--cycle;\n \\draw (-7,-4)--(5,-4)--(7,4)--(-5,4)--(-7,-4)--cycle;\n \\draw (-5.5,0)--(5.5,0);\n \\draw (-1,-3.5)--(1,3.5);\n \\draw (-1,-3.5)--(1,3.5);\n \\draw[line width=.35mm, red] (-2.5,1.5)--(-2.5,5);\n \\draw[line width=.35mm, red] (3,1.8)--(3,5);\n \\draw[line width=.35mm, red,dashed] (-2.5,1.5)--(-2.5,-4);\n \\draw[line width=.35mm, red,dashed] (3,1.8)--(3,-4);\n \\draw[line width=.35mm, red] (0,0) ellipse (2cm and 0.5cm);\n \n \\path[draw,dotted] (0,0)--(-2.5,1.5);\n \\path[draw,dotted] (0,0)--(3,1.8);\n \\foreach \\Point in {(0,0), (-2.5,1.5), (3,1.8)}{\\node at \\Point {$\\bullet$};}\n \\node at (1.1,-0.8) {$C_3$};\n \\node at (-2.9,3) {$C_2$};\n \\node at (3.4,3) {$C_1$};\n \\node at (3.7,1.8) {$r_2 e^{i\\theta}$};\n \\node at (-3.5,1.5) {$-r_1 e^{-i\\theta}$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{Chains $C_1$, $C_2$ and $C_3$}\n\\label{fig-chains}\n\\end{figure}\n\n\n\\noindent\nThe reflections $\\iota_k$ for $k=1,2$ are given by \n\\[\n (\\zeta,\\omega)\n \\mapsto\n (\\mu_k\\zeta+(1-\\mu_k)\\varphi_k,\\omega+2|\\varphi_k|^2\\Im(1-\\mu_k)+2\\Im((1-\\mu_k)\\bar{\\varphi_k}\\zeta),\n\\]\nwhere $\\mu_k=\\exp(2\\pi i\/n_k)$.\nAs discussed in the previous section, $\\iota_k$ rotates any vertical chain as a set through $\\frac{2\\pi}{n_k}$ around~$C_k$.\n\n\\section{Compression property}\n\n\\label{sec-compression}\n\nLet $C_1,C_2,C_3$ be chains in~$\\mathcal{N}$ as in the previous section.\nLet $\\iota_k$ be the minimal complex reflection of order~$n_k$ in the chain~$C_k$ for $k=1,2,3$.\nWe will assume that $n_3=2$.\nTo prove the discreteness of the group $\\<\\iota_1,\\iota_2,\\iota_3\\>$ we will use the following discreteness criterion discussed in \\cite{WyssGall}:\n\n\\begin{proposition}\n\n\\label{criterion}\n\nIf there exist subsets $U_1$, $U_2$ and~$V$ in~$\\mathcal{N}$ with $U_1\\cap U_2=\\varnothing$ and $V\\subsetneq U_1$ such that \n\\begin{enumerate}\n\\item\n$\\iota_3(U_1) = U_2$;\n\\item\n$g(U_2)\\subsetneq V$ for all~$g\\ne\\Id$ in $\\<\\iota_1,\\iota_2\\>$\n\\end{enumerate}\nthen the group $\\<\\iota_1,\\iota_2,\\iota_3\\>$ is a discrete subgroup of $\\PU(2,1)$.\nGroups with properties~(1) and~(2) are called {\\it compressing\\\/}.\n\\end{proposition}\n\n\\bigskip\nProjecting the actions of complex reflections~$\\iota_1$ and~$\\iota_2$ to~$\\mathbb C\\times\\{0\\}$ we obtain\nrotations~$j_1$ and $j_2$ of~$\\mathbb C$ around $\\varphi_1=r_2e^{i\\theta}$ and $\\varphi_2=-r_1e^{-i\\theta}$\nthrough $\\frac{2\\pi}{n_1}$ and $\\frac{2\\pi}{n_2}$ respectively.\nWe will use Proposition~\\ref{criterion} to prove the following Lemma:\n\n\\begin{lemma}\n\\label{f(0)}\nIf $|f(0)|\\ge2$ for all $f\\ne\\Id$ in $\\$ and $|h(0)|\\ge2$ for all vertical translations $h\\ne\\Id$ in $\\<\\iota_1,\\iota_2\\>$,\nthen the group $\\<\\iota_1,\\iota_2,\\iota_3\\>$ is discrete.\n\\end{lemma}\n\n\\begin{proof}\nConsider the unit spinal sphere \n\\[U=\\{(\\zeta, \\nu)\\in\\mathcal{N}\\,\\,\\big|\\,\\,|\\zeta|^4+\\nu^2=1\\}.\\]\nThe complex reflection~$\\iota_3$ in~$C_3$ is given by\n\\[\\iota_3([z_1:z_2:z_3])=[-z_1:z_2:-z_3]=[z_1:-z_2:z_3].\\]\nThe complex reflection~$\\iota_3$ preserves the bisector\n\\[\\mathcal{C}=\\{[z:it:1]\\inH_{\\c}^2\\,\\,\\big|\\,\\, |z|^2<1-t^2, z\\in\\mathbb C, t\\in\\mathbb R\\}\\]\nand hence preserves the unit spinal sphere~$U$ which is the boundary of the bisector~$\\mathcal{C}$.\nThe complex reflection~$\\iota_3$ interchanges the points $[0:1:1]$ and $[0:-1:1]$ in~$H_{\\c}^2$,\nwhich correspond to the points $(0,0)$ and $\\infty$ in $\\mathcal{N}$.\nTherefore, $\\iota_3$ leaves $U$ invariant and switches the inside of $U$ with the outside.\n\n\\bigskip\nLet $U_1$ be the part of $\\mathcal{N}\\backslash U$ outside~$U$, containing~$\\infty$,\nand let $U_2$ be the part inside~$U$, containing the origin.\nClearly\n\\[U_1\\cap U_2=\\varnothing\\quad\\text{and}\\quad \\iota_3(U_1)=U_2.\\]\nTherefore, if we find a subset $V\\subsetneq U_1$ such that $g(U_2)\\subsetneq V$\nfor all elements $g\\ne\\Id$ in $\\<\\iota_1,\\iota_2\\>$,\nthen we will show that $\\<\\iota_1,\\iota_2,\\iota_3\\>$ is discrete.\nLet\n\\[W=\\{(\\zeta,\\nu)\\in\\mathcal{N}\\,\\,\\big|\\,\\,|\\zeta|=1\\}\\]\nbe the cylinder consisting of all vertical chains through $\\zeta\\in\\mathbb C$ with $|\\zeta|=1$.\nLet\n\\[\n W_1=\\{(\\zeta,\\nu)\\in\\mathcal{N}\\,\\,\\big|\\,\\,|\\zeta|>1\\}\n \\quad\\text{and}\\quad\n W_2=\\{(\\zeta,\\nu)\\in\\mathcal{N}\\,\\,\\big|\\,\\,|\\zeta|<1\\}\n\\]\nbe the parts of $\\mathcal{N}\\backslash W$ outside and inside the cylinder~$W$ respectively. \nWe have $U_2\\subset W_2$ and so $g(U_2)\\subset g(W_2)$ for all~$g\\in\\<\\iota_1,\\iota_2\\>$.\nThe set $W_2$ is a union of vertical chains.\nWe know that elements of $\\<\\iota_1,\\iota_2\\>$ map vertical chains to vertical chains.\nThere is also a vertical translation on the chain itself.\nTherefore, we look at both the intersection of the images of~$W_2$ with $\\mathbb C\\times\\{0\\}$ and the vertical displacement of~$W_2$. \n\n\\bigskip\nElements of $\\<\\iota_1,\\iota_2\\>$ move the intersection of $W_2$ with $\\mathbb C\\times\\{0\\}$ by rotations $j_1$ and $j_2$\naround \\(r_2e^{i\\theta}\\) and \\(-r_1e^{-i\\theta}\\) through $\\frac{2\\pi}{n_1}$ and $\\frac{2\\pi}{n_2}$ respectively.\nProvided that the interior of the unit circle is mapped completely off itself under all non-identity elements in $\\$,\nthen the same is true for $W_2$ and hence for $U_2$ under all elements in $\\<\\iota_1,\\iota_2\\>$ that are not vertical translations.\n\n\\bigskip\nVertical translations $h$ are Heisenberg translations by $(0,\\nu)$ for some~$\\nu\\in\\mathbb R$:\n\\[(\\zeta,\\omega) \\mapsto(\\zeta,\\omega+\\nu).\\]\nSuch translations will shift~$W_2$ and its images $g(W_2)$ vertically by the same distance.\nHence, the same is true for $U_2$ and its images $g(U_2)$.\n\n\\bigskip\nWe choose $V$ to be the union of all the images of $U_2$ under all non-vertical elements of $\\<\\iota_1,\\iota_2\\>$.\nThis subset will satisfy the compressing conditions\nassuming that the interior of the unit circle is mapped off itself by any non-identity element in $\\$\nand that the interior of the unit spinal sphere $U$ is mapped off itself\nby any non-identity vertical translation in $\\<\\iota_1,\\iota_2\\>$.\nSince the radius of the unit circle is preserved under rotations,\nwe need to show that the origin is moved the distance of at least twice the radius of the circle:\n\\[|f(0)|\\ge2\\quad\\text{for all}~f\\in\\,~f\\ne\\Id.\\]\nSince vertical translations shift the spinal spheres vertically,\nwe need to show that they shift by at least the height of the spinal sphere:\n\\[|h(0)|\\ge2\\quad\\text{for all vertical translations}~h\\in\\<\\iota_1,\\iota_2\\>,~h\\ne\\Id.\\]\nWe see that the conditions of this Lemma ensure that the sets $U_1$, $U_2$ and~$V$ satisfy the conditions of Proposition~\\ref{criterion}.\n\\end{proof}\n\nWe will now focus on the case of $[m,m,0;3,3,2]$-groups, i.e.\\ $m_1=m_2=m$, $n_1=n_2=3$ and $n_3=2$.\nFrom now on we will consider the following configuration of chains in~$\\mathcal{N}$:\n$C_3$ is the (Euclidean) unit circle in $\\mathbb C\\times\\{0\\}$, whereas $C_1$ and~$C_2$ are vertical lines\nthrough the points $\\varphi_1 = re^{i\\theta}$ and $\\varphi_2=-re^{-i\\theta}$ respectively,\nwhere $r=\\cosh(m\/2)$ and $\\theta\\in(-\\pi\/2,\\pi\/2)$.\nThe type of triangle formed by $C_1,C_2,C_3$ is an ultra-parallel $[m,m,0]$-triangle with angular invariant ${\\alpha}=\\pi-2\\theta\\in(0,2\\pi)$.\nWe will consider the ultra-parallel $[m,m,0;3,3,2]$-triangle group ${\\Gamma}=\\<\\iota_1,\\iota_2,\\iota_3\\>$\ngenerated by the minimal complex reflections $\\iota_1,\\iota_2,\\iota_3$ of orders~$3,3,2$ in the chains~$C_1,C_2,C_3$ respectively.\n\n\\section{Proof of Proposition~\\ref{prop1}}\n\n\\noindent\nLet ${\\Gamma}=\\<\\iota_1,\\iota_2,\\iota_3\\>$ be an ultra-parallel $[m,m,0;3,3,2]$-triangle group as described at the end of section~\\ref{sec-compression}.\nIn this section we will use Lemma~\\ref{f(0)} to find conditions for the group ${\\Gamma}$ to be discrete.\n\n\\begin{proof}\nLet us consider the structure of the group~$\\<\\iota_1,\\iota_2\\>$ in more detail.\nWe can write every element~$f$ in $\\<\\iota_1,\\iota_2\\>$ as a word in the generators~$\\iota_1^{\\pm1}$ and~$\\iota_2^{\\pm1}$.\nLet $\\iota_{k_1k_2k_3}=\\iota_{k_1}\\iota_{k_2}\\iota_{k_3}$.\nThe elements $\\iota_{112}$, $\\iota_{121}$, $\\iota_{122}$, $\\iota_{211}$, $\\iota_{212}$ and $\\iota_{221}$ are Heisenberg translations.\nThe group~$\\mathcal{T}$ generated by these elements is the subgroup of Heisenberg translations in the group~$\\<\\iota_1,\\iota_2\\>$.\nUsing the relations $\\iota_1^3=\\iota_2^3=\\Id$ we see that all these elements can be expressed in terms of\n\\[T_1=\\iota_{212}\\quad\\text{and}\\quad T_2=\\iota_{112}\\]\nas $\\iota_{221}=T_2^{-1}$, $\\iota_{122}=T_2T_1^{-1}$, $\\iota_{211}=T_1T_2^{-1}$\nand $\\iota_{121}=T_2T_1^{-1}T_2^{-1}$.\nHence the group~$\\mathcal{T}$ is generated by two Heisenberg translations~$T_1$ and~$T_2$.\nDirect computation shows that $T_k$ for $k=1,2$ is a Heisenberg translation by $(v_k,t_k)$, where\n\\begin{align*}\n (v_1,t_1)&=(2\\sqrt{3}r\\cos(\\theta)\\cdot i,12\\sqrt{3}r^2\\cos^2(\\theta)),\\\\\n (v_2,t_2)&=(r\\cos(\\theta)\\cdot(3+i\\sqrt{3}),12r^2\\sin(\\theta)\\cos(\\theta)).\n\\end{align*}\nAny vertical translation in~$\\<\\iota_1,\\iota_2\\>$ belongs to the subgroup~$\\mathcal{T}=\\$.\nCalculation shows that the commutator\n\\[H=[T_1,T_2]=T_1^{-1}T_2^{-1}T_1T_2\\]\nis the vertical Heisenberg translation by $(0,4\\Im(v_1\\bar{v}_2))$, where\n\\begin{align*}\n 4\\Im(v_1\\bar{v}_1)\n =4\\Im\\left(2\\sqrt{3}r^2\\cos^2(\\theta)\\cdot(3i+\\sqrt{3})\\right)\n =24\\sqrt{3}r^2\\cos^2(\\theta).\n\\end{align*}\nRecall that elements of the form $(\\zeta,\\nu)$ with~$\\zeta=0$ are central in the group~$\\mathcal{N}$,\nhence the vertical translation~$H$ commutes with any other Heisenberg translation.\nThe group~$\\mathcal{T}$ has the presentation\n\\[\\mathcal{T}=\\\\]\nand is isomorphic to the uniform lattice ${\\Gamma}_1=\\mathbb Z[i]\\times\\mathbb Z$ in~$\\mathcal{N}$\nvia $T_1\\mapsto(1,0)$, $T_2\\mapsto(i,0)$, $H\\mapsto(0,1)$.\nUsing the identities $T_1H=HT_1$, $T_2H=HT_2$ and $T_1T_2=T_2T_1H$,\nevery element of~$\\mathcal{T}$ can be written in the form $T_1^xT_2^yH^n$ for some $x,y,n\\in\\mathbb Z$.\nProjecting to~$\\mathbb C\\times\\{0\\}$, the element $T_1^xT_2^yH^n$ acts as the translation by $x v_1+y v_2$.\nHence the element $T_1^xT_2^yH^n$ is a vertical translation if and only if $x=y=0$, i.e.\\ if it is a power of~$H$.\n\n\\bigskip\nWe will now check that the conditions of Lemma~\\ref{f(0)} are satisfied.\nWe first check that $|h(0)|\\ge2$ for all vertical translations $h\\ne\\Id$ in $\\<\\iota_1,\\iota_2\\>$.\nThe only vertical translations in $\\<\\iota_1,\\iota_2\\>$ are powers of~$H$, the vertical Heisenberg translation\nby $(0,24\\sqrt{3}r^2\\cos^2(\\theta))$.\nWe need the displacement of each vertical translation~$H^n$, $n\\ne0$, to be at least the height of the spinal sphere, i.e.\n\\[24\\sqrt{3}r^2\\cos^2(\\theta)\\ge2\\Longleftrightarrow r^2\\cos^2(\\theta)\\ge\\frac{\\sqrt{3}}{36}.\\]\nBy assumption ${\\alpha}\\in(0,2\\pi)$ and $\\cos({\\alpha})\\le-\\frac{1}{2}$, hence $\\frac{2\\pi}{3}\\le{\\alpha}\\le\\frac{4\\pi}{3}$.\nUsing ${\\alpha}=\\pi-2\\theta$, we have $|\\theta|\\le\\frac{\\pi}{6}$.\nFor $\\cos(\\theta)\\ge\\frac{\\sqrt{3}}{2}$ and $r\\ge\\frac{2}{\\sqrt{3}}$ we have\n\\[\n r^2\\cos^2(\\theta)\\ge\n \n 1>\\frac{\\sqrt{3}}{36},\n\\]\nhence the condition $|h(0)|\\ge2$ is satisfied for all vertical translations $h\\ne\\Id$ in $\\<\\iota_1,\\iota_2\\>$.\n\n\\bigskip\nWe will now check that $|f(0)|\\ge2$ for all $f\\ne\\Id$ in $\\$.\nFor $k=1,2$, projecting $\\iota_k$ to~$\\mathbb C$,\nwe obtain a rotation~$j_k$ of $\\mathbb C\\times\\{0\\}$ through~$\\frac{2\\pi}{3}$ around $\\varphi_k$.\nWe can write every element~$f$ in $\\$ as a word in the generators~$j_1^{\\pm1}$ and~$j_2^{\\pm1}$.\nUsing the relations $j_1^{-1}=j_1^2$ and $j_2^{-1}=j_2^2$ we can rewrite~$f$ as a word in just~$j_1$ and~$j_2$.\nFigure~\\ref{fig2} shows the points $f(0)$ for all words~$f$ of length up to~$6$ in the case $r=1$ and $\\theta=0$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{order.png}\n\\caption{Points \\(f(0)\\) for all words \\(f\\) up to length \\(6\\).}\n\\label{fig2}\n\\end{figure}\n\nProjecting $\\iota_{k_1k_2k_3}$ to~$\\mathbb C$ we obtain Euclidean translations $j_{k_1k_2k_3}=j_{k_1}j_{k_2}j_{k_3}$ on~$\\mathbb C$:\n\\[\n j_{k_1k_2k_3}(z)=z+(1-\\mu)(\\mu^2\\varphi_{k_3}+\\mu\\varphi_{k_2}+\\varphi_{k_1}),\n \\quad\\text{where}~\\mu=\\exp(2\\pi i\/3).\n\\]\nTranslations $j_{k_1k_2k_3}$ generate the subgroup of all translations in the group~$\\$.\nThis subgroup can be generated by two translations $j_{212}$ and $j_{112}$ by $v_1$ and~$v_2$ respectively,\nwhere, as above,\n\\begin{align*}\n v_1\n \n &=\\mu\\cdot(1-\\mu)\\cdot(\\varphi_1-\\varphi_2)\n =2r\\sqrt{3}\\cos(\\theta)\\cdot i,\\\\\n v_2\n \n &=(1-\\mu^2)\\cdot(\\varphi_1-\\varphi_2)\n =r\\cos(\\theta)\\cdot(3+i\\sqrt{3}).\n\\end{align*}\nUsing translations $j_{k_1k_2k_3}$,\nwe are able to break down any element~$f$ of $\\$, written as a word in the generators~$j_1$ and~$j_2$,\ninto a sequence of translations by~$v_1$ and~$v_2$, followed by a word of length at most~$2$, so that\nevery point in the orbit of~$0$ under $\\$ is of the form $p+xv_1+yv_2$,\nwhere $x,y\\in\\mathbb Z$ and $p=w(0)$ for $w\\in\\{\\Id,j_1,j_2,j_1^2,j_2^2,j_1j_2,j_2j_1\\}$.\nThese transformations are given by\n\\begin{align*}\n &j_k(z)=\\mu\\cdot z+(1-\\mu)\\cdot\\varphi_k,\\quad k\\in\\{1,2\\},\\\\\n \n \n \n &j_kj_l(z)=\\mu^2\\cdot z+(1-\\mu)\\cdot(\\varphi_k+\\mu\\cdot\\varphi_l),\\quad k,l\\in\\{1,2\\}.\n \n \n \n\\end{align*}\nThis structure of the orbit is reflected in the three translational symmetries that can be observed in Figure~\\ref{fig2}, \nvertically, at the angle $\\frac{\\pi}{6}$ and at the angle $-\\frac{\\pi}{6}$ to the horizontal axis. \nUsing\n\\[|v_1|^2=|v_2|^2=2\\Re(v_1\\bar{v}_2)=12r^2\\cos^2(\\theta),\\]\nwe calculate\n\\begin{align*}\n &|p+xv_1+yv_2|^2\\\\\n &=x^2|v_1|^2+y^2|v_2|^2+2xy\\Re(v_1\\bar{v}_2)+2x\\Re(p\\bar{v}_1)+2y\\Re(p\\bar{v}_2)+|p|^2\\\\\n &=12r^2\\cos^2(\\theta)\\cdot(x^2+xy+y^2)+2x\\Re(p\\bar{v}_1)+2y\\Re(p\\bar{v}_2)+|p|^2.\n\\end{align*}\nWe make a coordinate change $u=y-x$ and $v=x+y$, that is $x=(v-u)\/2$ and $y=(u+v)\/2$.\nPoints $(x,y)\\in\\mathbb Z^2$ are mapped to points $(u,v)\\in\\mathbb Z^2$ with $u\\equiv v\\mod 2$.\nWe obtain\n\\begin{align*}\n |p+xv_1+yv_2|^2\n &=3r^2\\cos^2(\\theta)\\cdot(u^2+3v^2-2au-6bv+a^2+3b^2)\\\\\n &=3r^2\\cos^2(\\theta)\\cdot((u-a)^2+3(v-b)^2),\\\\\n\\end{align*}\nwhere\n\\begin{align*}\n a\n =\\frac{\\Re(p(\\bar{v}_1-\\bar{v}_2))}{6r^2\\cos^2(\\theta)}\n =-\\frac{\\Re\\left(p(3+i\\sqrt{3})\\right)}{6r\\cos(\\theta)},~\n b=-\\frac{\\Re(p(\\bar{v}_1+\\bar{v}_2))}{18r^2\\cos^2(\\theta)}\n =-\\frac{\\Re\\left(p(1-i\\sqrt{3})\\right)}{6r\\cos(\\theta)}\n\\end{align*}\nand\n\\[a^2+3b^2=\\frac{|p|^2}{3r^2\\cos^2(\\theta)}.\\]\nOur aim is to show that $|p+xv_1+yv_2|^2\\ge3r^2$ for all~$(x,y)\\in\\mathbb Z^2$\nexcluding the case $p=0$, $x=y=0$ that corresponds to $f=\\Id$. \nThis is equivalent to $(u-a)^2+3(v-b)^2\\ge\\sec^2(\\theta)$ for all~$(u,v)\\in\\mathbb Z^2$ with $u\\equiv v\\mod 2$\nexcluding the case $a=b=u=v=0$.\nNote that this inequality is always satisfied if $|u-a|\\ge\\sec(\\theta)$ or $|v-b|\\ge\\sec(\\theta)\/\\sqrt{3}$,\nso we only need to check that\n\\[g(u,v)=(u-a)^2+3(v-b)^2-\\sec^2(\\theta)\\ge0\\]\nfor all $(u,v)\\in\\mathbb Z^2$ with $u\\equiv v\\mod 2$ inside the bounding box\n\\[\\left(a-\\sec(\\theta),a+\\sec(\\theta)\\right)\\times\\left(b-\\frac{\\sec(\\theta)}{\\sqrt{3}},b+\\frac{\\sec(\\theta)}{\\sqrt{3}}\\right).\\]\nIn the following table we list the values of $a$, $b$ and $a^2+3b^2$\nin terms of $t=\\tan(\\theta)$ and $\\mu=\\exp(2\\pi i\/3)=-\\frac{1-i\\sqrt{3}}{2}$\nfor $w\\in\\{\\Id,j_1,j_2,j_1^2,j_2^2,j_1j_2,j_2j_1\\}$:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c| }\n\\hline\n\\(w\\) & \\(p=w(0)\\) & \\(a\\) & \\(b\\) & \\(a^2+3b^2\\) \\\\\n\\hline\n\\(\\Id\\) & \\(0\\) & \\(0\\) & \\(0\\) & \\(0\\) \\\\\n\\hline\n\\(j_1\\) & $(1-\\mu)\\cdot\\varphi_1$ & \\(-1\\) & \\(-\\frac{t}{\\sqrt{3}}\\) & \\(t^2+1\\) \\\\\n\\hline\n\\(j_2\\) & $(1-\\mu)\\cdot\\varphi_2$ & \\(1\\) & \\(-\\frac{t}{\\sqrt{3}}\\) & \\(t^2+1\\) \\\\\n\\hline\n\\(j_1^2\\) & $(1-\\bar\\mu)\\cdot\\varphi_1$ & \\(\\frac{1}{2}(t\\sqrt{3}-1)\\) & \\(-\\frac{1}{6}(3+t\\sqrt{3})\\) & \\(t^2+1\\) \\\\\n\\hline\n\\(j_2^2\\) & $(1-\\bar\\mu)\\cdot\\varphi_2$ & \\(\\frac{1}{2}(t\\sqrt{3}+1)\\) & \\(\\frac{1}{6}(3-t\\sqrt{3})\\) & \\(t^2+1\\) \\\\\n\\hline\n\\(j_1j_2\\) & $(1-\\mu)(\\varphi_1+\\mu\\varphi_2)$ & \\(-\\frac{1}{2}(3-t\\sqrt{3})\\) & \\(\\frac{1}{6}(3-t\\sqrt{3})\\) & \\((t-\\sqrt{3})^2\\) \\\\\n\\hline\n\\(j_2j_1\\) & $(1-\\mu)(\\varphi_2+\\mu\\varphi_1)$ & \\(\\frac{1}{2}(3+t\\sqrt{3})\\) & \\(-\\frac{1}{6}(3+t\\sqrt{3})\\) & \\((t+\\sqrt{3})^2\\) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\n\\bigskip\\noindent\nUnder the assumption $|\\theta|\\le\\frac{\\pi}{6}$\nwe have $t=\\tan(\\theta)\\in[-d,d]$ and $\\sec(\\theta)\\in[1,2d]$, where $d=1\/\\sqrt{3}\\approx0{.}577$.\nIn each of the seven cases we list the bounds on $a$ and~$b$ and the size of the bounding box\n\\[(\\min(a)-2d,\\max(a)+2d)\\times(\\min(b)-2\/3,\\max(b)+2\/3).\\]\nWe then calculate\n\\begin{align*}\n g(u,v)\n &=(u-a)^2+3(v-b)^2-\\sec^2(\\theta)\\\\\n &=u^2+3v^2-2au-6bv+(a^2+3b^2)-(t^2+1)\n\\end{align*}\nand check that $g(u,v)\\ge0$ for all $(u,v)\\in\\mathbb Z^2$ with $u=v\\mod2$ and inside the bounding box.\n\\begin{enumerate}[$\\bullet$]\n\\item\n$w=\\Id$, $a=b=0$:\nThe bounding box~$(-2d,2d)\\times(-2\/3,2\/3)\\subset(-2,2)\\times(-1,1)$\ncontains only one point~$(u,v)\\in\\mathbb Z^2$ with $u=v\\mod2$, the point $(u,v)=(0,0)$,\nwhich corresponds to the excluded case $f=\\Id$.\n\\item\n$w=j_1$, $a=-1$, $b=-t\/\\sqrt{3}\\in[-1\/3,1\/3]$:\nThe bounding box\n\\[(-1-2d,-1+2d)\\times(-1,1)\\subset(-3,1)\\times(-1,1)\\]\ncontains points $(0,0)$ and $(-2,0)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2+2u+2tv\\sqrt{3}+(t^2+1)-(t^2+1)\\\\\n &=u^2+3v^2+2u+2tv\\sqrt{3}\n\\end{align*}\nis non-negative: $g(0,0)=g(-2,0)=0$.\n\\item\n$w=j_2$, $a=1$, $b=-t\/\\sqrt{3}\\in[-1\/3,1\/3]$:\nThe bounding box\n\\[(1-2d,1+2d)\\times(-1,1)\\subset(-1,3)\\times(-1,1)\\]\ncontains points $(0,0)$ and $(2,0)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2-2u+2tv\\sqrt{3}+(t^2+1)-(t^2+1)\\\\\n &=u^2+3v^2-2u+2tv\\sqrt{3}\n\\end{align*}\nis non-negative: $g(0,0)=g(2,0)=0$.\n\\item\n$w=j_1^2$, $a=\\frac{1}{2}(t\\sqrt{3}-1)\\in[-1,0]$, $b=-\\frac{1}{6}(3+t\\sqrt{3})\\in[-2\/3,-1\/3]$:\nThe bounding box\n\\[(-1-2d,2d)\\times(-4\/3,1\/3)\\subset(-3,2)\\times(-2,1)\\]\ncontains points $(1,-1)$, $(0,0)$, $(-1,-1)$ and $(-2,0)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2-u(t\\sqrt{3}-1)+v(3+t\\sqrt{3})+(t^2+1)-(t^2+1)\\\\\n &=u^2+3v^2+u+3v-(u-v)t\\sqrt{3}\n\\end{align*}\n is non-negative:\n\\begin{align*}\n &g(0,0)=g(-1,-1)=0,~\n g(1,-1)=2-2t\\sqrt{3}\\ge0,~\n g(-2,0)=2+2t\\sqrt{3}\\ge0.\n\\end{align*}\n\\item\n$w=j_2^2$, $a=\\frac{1}{2}(t\\sqrt{3}+1)\\in[0,1]$, $b=\\frac{1}{6}(3-t\\sqrt{3})\\in[1\/3,2\/3]$:\nThe bounding box\n\\[(-2d,1+2d)\\times(-1\/3,4\/3)\\subset(-2,3)\\times(-1,2)\\]\ncontains points $(2,0)$, $(1,1)$, $(0,0)$ and $(-1,1)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2-u(t\\sqrt{3}+1)-v(3-t\\sqrt{3})+(t^2+1)-(t^2+1)\\\\\n &=u^2+3v^2-u-3v-(u-v)t\\sqrt{3}\n\\end{align*}\n is non-negative:\n\\begin{align*}\n &g(0,0)=g(1,1)=0,~\n g(-1,1)=2+2t\\sqrt{3}\\ge0,~\n g(2,0)=2-2t\\sqrt{3}\\ge0.\n\\end{align*}\n\\item\n$w=j_1j_2$, $a=-\\frac{1}{2}(3-t\\sqrt{3})\\in[-2,-1]$, $b=\\frac{1}{6}(3-t\\sqrt{3})\\in[1\/3,2\/3]$:\nThe bounding box\n\\[(-2-2d,-1+2d)\\times(-1\/3,4\/3)\\subset(-4,1)\\times(-1,2)\\]\ncontains points $(0,0)$, $(-1,1)$, $(-2,0)$ and $(-3,1)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2+u(3-t\\sqrt{3})-v(3-t\\sqrt{3})+(t-\\sqrt{3})^2-(t^2+1)\\\\\n &=u^2+3v^2+3(u-v)+2-(u-v+2)t\\sqrt{3}\n\\end{align*}\n is non-negative:\n\\begin{align*}\n &g(-1,1)=g(-2,0)=0,~\n g(-3,1)=2+2t\\sqrt{3}\\ge0,~\n g(0,0)=2-2t\\sqrt{3}\\ge0.\n\\end{align*}\n\\item\n$w=j_2j_1$, $a=\\frac{1}{2}(3+t\\sqrt{3})\\in[1,2]$, $b=-\\frac{1}{6}(3+t\\sqrt{3})\\in[-2\/3,-1\/3]$:\nThe bounding box\n\\[(1-2d,2+2d)\\times(-4\/3,1\/3)\\subset(-1,4)\\times(-2,1)\\]\ncontains points $(0,0)$, $(1,-1)$, $(2,0)$ and $(3,-1)$.\nThe function\n\\begin{align*}\n g(u,v)\n &=u^2+3v^2-u(3+t\\sqrt{3})+v(3+t\\sqrt{3})+(t+\\sqrt{3})^2-(t^2+1)\\\\\n &=u^2+3v^2-3(u-v)+2-(u-v-2)t\\sqrt{3}\n\\end{align*}\n is non-negative:\n\\begin{align*}\n &g(1,-1)=g(2,0)=0,~\n g(0,0)=2+2t\\sqrt{3}\\ge0,~\n g(3,-1)=2-2t\\sqrt{3}\\ge0.\n\\end{align*}\n\\end{enumerate}\nIn all cases we have shown that $g(u,v)\\ge0$, hence $|p+xv_1+yv_2|^2\\ge3r^2$.\nUnder the assumption $r\\ge\\frac{2}{\\sqrt{3}}$ we have $3r^2\\ge4$.\nTherefore $|f(0)|\\ge2$ for all $f\\ne\\Id$ in $\\$.\nHence all conditions of Lemma~\\ref{f(0)} are satisfied\nand we can conclude that the group $\\<\\iota_1,\\iota_2,\\iota_3\\>$ is discrete\nif $\\cos({\\alpha})\\le-\\frac{1}{2}$ and $r\\ge\\frac{2}{\\sqrt{3}}$.\n\\end{proof}\n\n\\section{Proof of Proposition~\\ref{prop2}}\n\n\\noindent\nLet ${\\Gamma}=\\<\\iota_1,\\iota_2,\\iota_3\\>$ be an ultra-parallel $[m,m,0;3,3,2]$-triangle group as described at the end of section~\\ref{sec-compression}.\nIn this section we will use the following complex hyperbolic version of Shimizu's Lemma introduced in \\cite{Par97}\nto find conditions for the group ${\\Gamma}$ not to be discrete.\n\n\\begin{lemma}\n\\label{shimizu}\nLet ${\\Gamma}$ be a discrete subgroup of $\\PU(2,1)$.\nLet $g\\in{\\Gamma}$ be a Heisenberg translation by $(\\xi, \\nu)$ and $h=(h_{ij})_{1\\le i,j\\le 3}\\in{\\Gamma}$\nbe an element that satisfies $h(\\infty)\\ne\\infty$,\nthen \n\\[r_h^2 \\le\\rho_0(g(h^{-1}(\\infty)),h^{-1}(\\infty))\\rho_0(g(h(\\infty)), h(\\infty)) +4\\abs{\\xi}^2,\\]\nwhere $\\rho_0$ is the Cygan metric on~$\\mathcal{N}$\nand \n\\[r_h = \\sqrt{\\frac{2}{\\abs{h_{22}-h_{23}+h_{32}-h_{33}}}}\\]\nis the radius of the isometric sphere of~$h$.\n\\end{lemma}\n\n\\bigskip\\noindent\nWe will now prove Proposition~\\ref{prop2}:\n\n\\begin{proof}\nWe will apply Lemma~\\ref{shimizu} to translations of the form $g=\\iota_{k_1k_2k_3}=\\iota_{k_1}\\iota_{k_2}\\iota_{k_3}$\nand the element $h=\\iota_3$ in ${\\Gamma}=\\<\\iota_1,\\iota_2,\\iota_3\\>$.\n\n\\bigskip\\noindent\nThe matrix of the element $h=\\iota_3=\\iota_3^{-1}$ is\n\\[h=h^{-1}=\\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix}.\\]\nThe radius of the isometric sphere of~$h$ is $r_h=1$.\nTo calculate $h(\\infty)$ we first map $\\infty$ from the Heisenberg space to the boundary of complex hyperbolic 2-space.\nThat is,\n\\[\\infty\\mapsto [0:1:-1]\\in{\\partial\\chp}.\\]\nWe apply $h$ to this point\n\\[\n \\begin{pmatrix} -1 & 0 & 0\\\\ 0 & 1 & 0\\\\ 0 & 0 & -1\\end{pmatrix}\n \\begin{bmatrix} 0\\\\ 1\\\\ -1\\end{bmatrix}\n =\\begin{bmatrix} 0\\\\ 1\\\\ 1\\end{bmatrix}\n = [0:1:1]\\in{\\partial\\chp}.\n\\]\nNote that $h(\\infty)\\ne\\infty$.\nMapping this point back to the Heisenberg space\n\\[[0:1:1]\\mapsto(0,0)\\in\\mathcal{N}.\\]\n\n\\bigskip\\noindent\nFor a Heisenberg translation~$g$ by $(\\xi,\\nu)\\in\\mathcal{N}$,\nwe have $g(h(\\infty))=g(0,0)=(\\xi, \\nu)$.\nNote that $\\rho_0(g(h(\\infty)),h(\\infty))=\\rho_0(g(h^{-1}(\\infty)),h^{-1}(\\infty))$ since $h=h^{-1}$.\nThe distance \\(\\rho_0(g(h(\\infty)),h(\\infty))\\) is equal to\n\\[\\rho_0((\\xi,\\nu), (0, 0))=\\left||\\xi|^2-i\\nu\\right|^\\frac{1}{2}=\\sqrt{\\left||\\xi|^2-i\\nu\\right|}.\\]\nSubstituting these values into the inequality given in Lemma~\\ref{shimizu},\nwe obtain that if the group is discrete then\n\\[1\\le\\sqrt{|\\xi|^4+\\nu^2}+4|\\xi|^2.\\]\nHence the group is not discrete if\n\\[\\sqrt{|\\xi|^4+\\nu^2}<1-4|\\xi|^2.\\]\nIf $4|\\xi|^2>1$ then this inequality is never satisfied.\nIf $4|\\xi|^2\\le1$, we can square both sides and rearrange to obtain that the group is not discrete if\n\\[4|\\xi|^2\\le1\\quad\\text{and}\\quad1-8|\\xi|^2+15|\\xi|^4-\\nu^2>0.\\]\nDirect computation shows that $g=\\iota_{212}$ is a Heisenberg translation by\n\\[(\\xi,\\nu)=\\left(2r\\sqrt{3}\\cos(\\theta)\\cdot i,12\\sqrt{3}r^2\\cos^2(\\theta)\\right).\\]\nSubstituting\n\\[|\\xi|^2=12r^2\\cos^2(\\theta)\\quad\\mbox{and}\\quad \\nu^2=432r^4\\cos^4(\\theta),\\]\nwe obtain that the group~${\\Gamma}$ is not discrete under the following conditions:\n\\begin{align*}\n &\\cos^2(\\theta)\\le\\frac{1}{48r^2}\n \\quad\\text{and}\\quad\n 1728r^4\\cos^4(\\theta)-96r^2\\cos^2(\\theta)+1>0\\\\\n &\\Longleftrightarrow\n \\cos^2(\\theta)\\le\\frac{1}{48r^2}\n \\quad\\text{and}\\quad\n \\bigg(\\cos^2(\\theta)<\\frac{1}{72r^2}\\quad\\text{or}\\quad\\cos^2(\\theta)>\\frac{1}{24r^2}\\bigg)\\\\\n &\\Longleftrightarrow\n \\cos^2(\\theta)<\\frac{1}{72r^2}.\n\\end{align*}\nUsing $\\cos^2(\\theta)=\\frac{1}{2}\\left(\\cos(2\\theta)+1\\right)=\\frac{1}{2}\\left(1-\\cos(\\alpha)\\right)$\nwe conclude that the group~${\\Gamma}$ is not discrete provided that\n\\[\\cos({\\alpha})>1-\\frac{1}{36r^2}=1-\\frac{1}{36\\cosh^2\\left(\\frac{m}{2}\\right)}.\\qedhere\\] \n\\end{proof}\n\n\n\n\n\n\n\n\\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\nLet $O^\\nu_{-e}$ be the parabolic category $\\mathcal{O}$ with parabolic type $\\nu$ of the affine version of the Lie algebra $\\mathfrak{gl}_N$ at level $-N-e$. In \\cite{RSVV}, a categorical representation of the affine Kac-Moody algebra $\\widetilde{\\mathfrak{sl}}_e$ in $O^\\nu_{-e}$ is considered. Roughly, this means that there are exact biadjoint functors $E_i,F_i\\colon O^\\nu_{-e}\\to O^\\nu_{-e}$ for $i\\in[0,e-1]$ which induce a representation of the Lie algebra $\\widetilde{\\mathfrak{sl}}_e$ on the Grothendieck group $[O^\\nu_{-e}]$ of $O^\\nu_{-e}$.\nThe definition of a categorical representation is given in Section \\ref{ch3:subs_categ-action}.\nThe category $O^\\nu_{-e}$ admits a decomposition\n$$\nO^\\nu_{-e}=\\bigoplus_{\\mu\\in\\mathbb{Z}^e}O^\\nu_{\\mu}\n$$\nthat lifts the decomposition of the $\\widetilde{\\mathfrak{sl}}_e$-module $[O^\\nu_{-e}]$ in a direct sum of weight spaces.\n\nThe category $O_\\mu^\\nu$ is Koszul by \\cite{SVV}. Its Koszul dual category is the category $O_{\\nu,+}^\\mu$ defined similarly to $O_\\nu^\\mu$ at a positive level. In particular, the Koszul duality exchanges the parameter $\\nu$ (\\emph{the parabolic type}) with the parameter $\\mu$ (\\emph{the singular type}). The Koszul duality yields an equivalence of bounded derived categories $D^b(O^\\nu_\\mu)\\simeq D^b({O}_{\\nu,+}^\\mu)$. More details about the Koszul duality can be found in \\cite{BGS}.\n\nLet $\\alpha_0,\\cdots,\\alpha_{e-1}$ be the simple roots of $\\widetilde{\\mathfrak{sl}}_e$. We have\n$$\nE_i(O^\\nu_\\mu)\\subset O^\\nu_{\\mu+\\alpha_i}, \\quad F_i(O^\\nu_\\mu)\\subset O^\\nu_{\\mu-\\alpha_i}.\n$$\nThe aim of this paper is to prove that Koszul dual functors\n$$\nD^b(O^\\mu_{\\nu,+})\\to D^b(O^{\\mu+\\alpha_i}_{\\nu,+}),\\quad D^b(O^\\mu_{\\nu,+})\\to D^b(O^{\\mu-\\alpha_i}_{\\nu,+})\n$$\nto the functors\n$$\nE_i\\colon D^b(O_\\mu^\\nu)\\to D^b(O_{\\mu+\\alpha_i}^\\nu),\\quad F_i\\colon D^b(O_\\mu^\\nu)\\to D^b(O_{\\mu-\\alpha_i}^\\nu).\n$$\nare the Zuckerman functors.\n\nUnfortunately, we cannot solve this problem for the full category $O$. But we are able to do this for a subcategory $\\mathbf{A}$ of $O$.\n\n\n\n\n\nBy definition, the Zuckerman functor is a composition of a parabolic inclusion functor with a parabolic truncation functor. Thus it is natural to try to decompose the functors $E_i$ and $F_i$ in \"smaller\" functors.\nWe want to find such a decomposition for $F$ in the following way.\n\nLet $\\overline O^\\nu_{\\mu}$, $\\overline E_i$, $\\overline F_i$ be defined in the same way as $O^\\nu_{\\mu}$, $E_i$, $F_i$ with $e$ replaced by $e+1$. Let $\\overline\\alpha_0,\\cdots,\\overline\\alpha_e$ be the simple roots of $\\widetilde{\\mathfrak{sl}}_{e+1}$. Fix $k\\in[0,e-1]$. For an $e$-tuple $\\mu=(\\mu_1,\\cdots,\\mu_e)$ we set\n\n$$\n\\overline\\mu=\n\\left\\{\\begin{array}{ll}\n(\\mu_1,\\cdots,\\mu_k,0,\\mu_{k+1},\\cdots,\\mu_e) & \\mbox{ if }k\\ne 0,\\\\\n(0,\\mu_1,\\cdots,\\mu_e) & \\mbox{ if }k=0.\n\\end{array}\\\\\n\\right.\n$$\nNote that we have $\\overline{(\\mu-\\alpha_k)}=\\overline\\mu-\\overline\\alpha_k-\\overline\\alpha_{k+1}$.\n\nBy \\cite{Fie-str}, there is an equivalence of categories $\\theta\\colon O^\\nu_{\\mu}\\to \\overline O^\\nu_{\\overline\\mu}$. The direct sum of such equivalences identifies the category $O_{-e}^\\nu$ with a direct factor of the category $\\overline O_{-e-1}^\\nu$. We want to compare the $\\widetilde{\\mathfrak{sl}}_{e}$-action on $O_{-e}^\\nu$ with the $\\widetilde{\\mathfrak{sl}}_{e+1}$-action on $\\overline O_{-e-1}^\\nu$. More precisely, we want to prove the following conjecture.\n\n\\smallskip\n\\begin{conj}\n\\label{ch3:conj-intro-art}\nThe following diagram of functors is commutative.\n\\begin{equation}\n\\label{ch3:diag-conj-intro}\n\\begin{diagram}\n\\node{\\overline O^\\nu_{\\overline\\mu}} \\arrow{e,t}{\\overline F_k}\n\\node{\\overline O^\\nu_{\\overline\\mu-\\overline\\alpha_k}} \\arrow{e,t}{\\overline F_{k+1}}\n\\node{\\overline O^\\nu_{\\overline\\mu-\\overline\\alpha_k-\\overline\\alpha_{k+1}}} \\arrow{s,r}{\\theta^{-1}} \\\\\n\\node{O^\\nu_{\\mu}} \\arrow{n,l}{\\theta}\n\\arrow[2]{e,b}{F_k} \\node[2]{O^\\nu_{\\mu-\\alpha_k}}\n\\end{diagram}\n\\end{equation}\n\\end{conj}\n\n\\smallskip\nOur motivation is the following. Assume that the conjecture holds. Then we can prove in the same way as in \\cite{MOS} that the functor $\\overline F_k$ is Koszul dual to the parabolic inclusion functor and the functor $\\overline F_{k+1}$ is Koszul dual to the parabolic truncation functor. Then we can deduce that $F_k$ is Koszul dual to the Zuckerman functor (which is the composition of the parabolic inclusion functor and the parabolic truncation functor). Thus the problem is reduced to the proof of this conjecture.\n\nIt is not hard to see that the diagram from Conjecture \\ref{ch3:conj-intro-art} is commutative at the level of Grothendieck groups. In the case of the category $\\mathcal{O}$ of ${\\mathfrak{gl}}_N$ (instead of affine ${\\mathfrak{gl}}_N$) this is already enough to prove the analogue of Conjecture \\ref{ch3:conj-intro-art}, using the theory of projective functors. Indeed, \\cite[Thm.~3.4]{BG} implies that two projective functors are isomorphic if their actions on the Grothendieck group coincide. Unfortunately, there is no satisfactory theory of projective functors for the affine case (an attempt to develop such a theory was given in \\cite{FrMal}).\n\nWe choose another strategy to prove this conjecture. The idea is to relate the notion of a categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ with the notion of a categorical representation of $\\widetilde{\\mathfrak{sl}}_{e+1}$. This can be done using the following inclusion of Lie algebras $\\widetilde{\\mathfrak{sl}}_e\\subset\\widetilde{\\mathfrak{sl}}_{e+1}$\n$$\ne_r\\mapsto\n\\left\\{\\begin{array}{rl}\ne_r &\\mbox{ if }r\\in[0,k-1],\\\\\n{[e_k,e_{k+1}]} &\\mbox{ if }r=k,\\\\\ne_{r+1} &\\mbox{ if }r\\in[k+1,e-1],\n\\end{array}\\right.\n$$\n$$\nf_r\\mapsto\n\\left\\{\\begin{array}{rl}\nf_r &\\mbox{ if }r\\in[0,k-1],\\\\\n{[f_{k+1},f_k]} &\\mbox{ if }r=k,\\\\\nf_{r+1} &\\mbox{ if }r\\in[k+1,e-1].\n\\end{array}\\right.\n$$\n\nFirst, we recall the notion of a categorical representation. Let $\\bfk$ be a field. Let $\\mathcal{C}$ be an abelian $\\mathrm{Hom}$-finite $\\bfk$-linear category that admits a direct sum decomposition $\\mathcal{C}=\\bigoplus_{\\mu\\in\\mathbb{Z}^{e}}\\mathcal{C}_\\mu$. A categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ in $\\mathcal{C}$ is a pair of biadjoint functors $E_i,F_i\\colon \\mathcal{C}\\to\\mathcal{C}$ for $i\\in[0,e-1]$ satisfying a list of axioms. The main axiom is that for each $d\\in\\mathbb{N}$ there is an algebra homomorphism $R_d\\to \\mathrm{End}(F^d)^{\\rm op}$, where $R_d$ is the KLR algebra of rank $d$ associated with the quiver $A_{e-1}^{(1)}$.\n\n\nNow we explain our main result about categorical representations. Let $\\overline\\mathcal{C}$ be an abelian $\\mathrm{Hom}$-finite $\\bfk$-linear category. Assume that $\\overline\\mathcal{C}=\\bigoplus_{\\mu\\in\\mathbb{Z}^{e+1}}\\overline\\mathcal{C}_\\mu$ has a structure of a categorical representation of $\\widetilde{\\mathfrak{sl}}_{e+1}$ with respect to functors $\\overline E_i,\\overline F_i$ for $i\\in[0,e]$.\nAssume additionally that the subcategory $\\overline\\mathcal{C}_\\mu$ is zero whenever $\\mu$ has a negative entry.\nFor each $e$-tuple $\\mu\\in\\mathbb{N}^e$ we consider the $(e+1)$-tuple $\\overline\\mu$ as above and we set\n$\\mathcal{C}_\\mu=\\overline\\mathcal{C}_{\\overline\\mu}$,\n$$\n\\mathcal{C}=\\bigoplus_{\\mu\\in \\mathbb{N}^e}\\mathcal{C}_{\\mu}.\n$$\nNext, consider the endofunctors of $\\mathcal{C}$ given by\n$$\nE_i=\n\\left\\{\n\\begin{array}{lll}\n\\restr{\\overline E_i}{\\mathcal{C}} &\\mbox{ if } 0\\leqslant i2$ and $\\nu=(\\nu_1,\\cdots,\\nu_l)$ satisfies $\\nu_r>|\\alpha|$ for each $r\\in[1,l]$. There exists a dimension vector $\\beta$ for $\\overline\\Gamma$ such that for each dimension vector $\\alpha$ for $\\Gamma$ there are equivalences of categories $\\theta'_{\\alpha}\\colon \\mathbf{A}^\\nu[\\alpha]\\to \\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]$ and $\\theta'_{\\alpha+\\alpha_k}\\colon \\mathbf{A}^\\nu[\\alpha+\\alpha_k]\\to \\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k+\\overline\\alpha_{k+1}]$ such that the following diagram is commutative\n$$\n\\begin{CD}\n\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]@>{\\overline F_{k+1}\\overline F_k}>> \\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k+\\overline\\alpha_{k+1}]\\\\\n@A{\\theta'_\\alpha}AA @A{\\theta'_{\\alpha+\\alpha_k}}AA\\\\\n\\mathbf{A}^\\nu[\\alpha]@>{F_k}>> \\mathbf{A}^\\nu[\\beta+\\alpha+\\alpha_k].\n\\end{CD}\n$$\n\\qed\n\\end{thm}\n\n\\smallskip\nThe technique of \\cite{RSVV} uses essentially the deformation argument. To make it applicable in our situation we have to find a version of Theorem \\ref{ch3:thm-isom_KLR_e_e+1_intro} over a local ring. This is done in Lemma \\ref{ch3:lem_morph-Phi-over-ring}.\n\nThe paper has the following structure. In Section \\ref{ch3:sec_KLR-Hecke} we study KLR algebras. In particular, we prove Theorem \\ref{ch3:thm-isom_KLR_e_e+1_intro} and its deformed version over a local ring. In Section \\ref{ch3:sec_catO} we study categorical representations. We prove our main result about categorical representations (Theorem \\ref{ch3:thm_main-thm-categ-rep-int-art}). Next, we use the categorical representations to decompose the functor $F$ in the category $\\mathbf{A}$ (Theorem \\ref{ch3:thm_intro-main-decomp-functors}). In Section \\ref{ch3:sec_gr-lifts} we prove that in some cases the functors $E$ and $F$ for the category $O$ admit graded lifts. In Section \\ref{ch3:sec-Koszul} we prove that the functors $E$ and $F$ for the category $\\mathbf{A}$ are Koszul dual to Zuckerman functors. We deduce this from the main results of Sections \\ref{ch3:sec_catO}, \\ref{ch3:sec_gr-lifts} using an approach similar to \\cite{MOS}. In Appendix \\ref{ch3:app-gener_categ_lemma} we generalize Lemma \\ref{ch3:lem_categ-e-e+1} to arbitrary symmetric Kac-Moody Lie algebras. In Appendix \\ref{ch3:app-geom_constr} we give a geometric construction of the isomorphism $\\Phi$ in Theorem \\ref{ch3:thm_KLR-e-e+1}.\n\n\n\n\\section{KLR algebras and Hecke algebras}\n\\label{ch3:sec_KLR-Hecke}\n\nFor a noetherian ring $A$ we denote by $\\mathrm{mod}(A)$ the abelian category of left finitely generated $A$-modules. We denote by $\\mathbb{N}$ the set of non-negative integers. By a commutative diagram of functors we always mean a diagram that commutes up to an isomorphism of functors.\n\n\\subsection{Kac-Moody algebras associated with a quiver}\n\\label{ch3:subs_KM-quiv}\nLet $\\Gamma=(I,H)$ be a quiver without $1$-loops with the set of vertices $I$ and the set of arrows $H$. For $i,j\\in I$ let $h_{i,j}$ be the number of arrows from $i$ to $j$ and set also $a_{i,j}=2\\delta_{i,j}-h_{i,j}-h_{j,i}$. Let $\\mathfrak{g}_I$ be the Kac-Moody algebra over $\\mathbb{C}$ associated with the matrix $(a_{i,j})$. Denote by $e_i$, $f_i$ for $i\\in I$ the Serre generators of $\\mathfrak{g}_I$.\n\nFor each $i\\in I$ let $\\alpha_i$, $\\check\\alpha_i$ be the simple root and coroot corresponding to $e_i$ and let $\\Lambda_i$ be the fundamental weight. Set\n$$\nQ_I=\\bigoplus_{i\\in I}\\mathbb{Z}\\alpha_i,\\quad Q^+_I=\\bigoplus_{i\\in I}\\mathbb{N}\\alpha_i, \\quad P_I=\\bigoplus_{i\\in I}\\mathbb{Z}\\Lambda_i,\\quad P^+_I=\\bigoplus_{i\\in I}\\mathbb{N}\\Lambda_i.\n$$\nLet $X_I$ be the free abelian group with basis $\\{\\varepsilon_i;~i\\in I\\}$. Set also\n\\begin{equation}\n\\label{ch3:eq_X+}\nX^+_I=\\bigoplus_{i\\in I}\\mathbb{N}\\varepsilon_i.\n\\end{equation}\n\nFor $\\alpha\\in Q^+_I$ denote by $|\\alpha|$ its height. Set $I^\\alpha=\\{{\\mathbf{i}}=(i_1,\\cdots,i_{|\\alpha|})\\in I^{|\\alpha|};~ \\sum_{r=1}^{|\\alpha|}\\alpha_{i_r}=\\alpha\\}$.\n\nLet $\\Gamma_\\infty=(I_\\infty,H_\\infty)$ be the quiver with the set of vertices $I_\\infty=\\mathbb{Z}$ and the set of arrows $H_\\infty=\\{i\\to i+1;~i\\in I_\\infty\\}$. Assume that $e>1$ is an integer. Let $\\Gamma_e=(I_e,H_e)$ be the quiver with the set of vertices $I_e=\\mathbb{Z}\/e\\mathbb{Z}$ and the set of arrows $H_e=\\{i\\to i+1;~i\\in I_e\\}$. Then $\\mathfrak{g}_{I_e}$ is the Lie algebra $\\widetilde{\\mathfrak{sl}}_e=\\mathfrak{sl}_e\\otimes\\mathbb{C}[t,t^{-1}]\\oplus\\mathbb{C} \\bm{1}$.\n\nAssume that $\\Gamma=(I,H)$ is a quiver whose connected components are of the form $\\Gamma_e$, with $e\\in\\mathbb{N}$, $e>1$ or $e=\\infty$. For $i\\in I$ denote by $i+1$ and $i-1$ the (unique) vertices in $I$ such that there are arrows $i\\to i+1$, $i-1\\to i$.\nLet us also consider the following additive map\n$$\n\\iota\\colon Q_I\\to X_I,\\quad\\alpha_i\\mapsto \\varepsilon_i-\\varepsilon_{i+1}.\n$$\n\nFix a formal variable $\\delta$ and set $X_I^\\delta=X_I\\oplus\\mathbb{Z}\\delta$.\nWe can lift the $\\mathbb{Z}$-linear map $\\iota$ to a $\\mathbb{Z}$-linear map\n$$\n\\iota^\\delta\\colon Q_I\\to X_I^\\delta,\\quad \\alpha_i\\mapsto \\varepsilon_i-\\varepsilon_{i+1}-\\delta.\n$$\nNote that the map $\\iota^\\delta\\colon Q_I\\to X^\\delta_I$ is injective (while $\\iota$ is not injective).\nWe may omit the symbols $\\iota$, $\\iota^\\delta$ and write $\\alpha$ instead of $\\iota(\\alpha)$ or $\\iota^\\delta(\\alpha)$.\n\nWe will sometimes abbreviate\n$$\nQ_e=Q_{I_e},\\quad X_e=X_{I_e},\\quad X^\\delta_e=X^\\delta_{I_e},\\quad P_e=P_{I_e}.\n$$\n\n\n\n\n\n\\subsection{Doubled quiver}\n\\label{ch3:subs_not-quiv-I-Ibar}\nLet $\\Gamma=(I,H)$ be a quiver without $1$-loops. Fix a decomposition $I=I_0\\sqcup I_1$ such that there are no arrows between the vertices in $I_1$. In this section we define a \\emph{doubled quiver} $\\overline\\Gamma=(\\overline I,\\overline H)$ associated with $(\\Gamma,I_0,I_1)$. The idea is to \"double\" each vertex in the set $I_1$ (we do not touch the vertices from $I_0$). We replace each vertex $i\\in I_1$ by a couple of vertices $i^1$ and $i^2$ with an arrow $i^1\\to i^2$. Each arrow entering to $i$ should be replaced by an arrow entering to $i^1$, each arrow coming from $i$ should be replaced by an arrow coming from $i^2$.\n\nNow we describe the construction of $\\overline\\Gamma=(\\overline I,\\overline H)$ formally. Let $\\overline I_0$ be a set that is in bijection with $I_0$. Let $i^0$ be the element of $\\overline I_0$ associated with an element $i\\in I_0$. Similarly, let $\\overline I_1$ and $\\overline I_2$ be sets that are in bijection with $I_1$. Denote by $i^1$ and $i^2$ the element of $\\overline I_1$ and $\\overline I_2$ respectively that correspond to an element $i\\in I_1$. Put $\\overline I=\\overline I_0\\sqcup\\overline I_1\\sqcup \\overline I_2$.\nWe define $\\overline H$ in the following way. The set $\\overline H$ contains $4$ types of arrows:\n\\begin{itemize}\n \\item[\\textbullet] an arrow $i^0\\to j^0$ for each arrow $i\\to j$ in $H$ with $i,j\\in I_0$,\n \\item[\\textbullet] an arrow $i^0\\to j^1$ for each arrow $i\\to j$ in $H$ with $i\\in I_0,j\\in I_1$,\n \\item[\\textbullet] an arrow $i^2\\to j^0$ for each arrow $i\\to j$ in $H$ with $i\\in I_1,j\\in I_0$,\n \\item[\\textbullet] an arrow $i^1\\to i^2$ for each vertex $i\\in I_1$.\n\\end{itemize}\n\n\n\n\nSet $I^\\infty=\\coprod_{d\\in \\mathbb{N}} I^d$, $\\overline I^\\infty=\\coprod_{d\\in \\mathbb{N}} \\overline I^d$, where $I^d$, $\\overline I^d$ are the cartesian products.\nThe concatenation yields a monoid structure on $I^\\infty$ and $\\overline I^\\infty$.\nLet $\\phi\\colon I^\\infty\\to \\overline I^\\infty$ be the unique morphism of monoids such that for $i\\in I\\subset I^\\infty$ we have\n$$\n\\phi(i)=\n\\left\\{\\begin{array}{ll}\ni^0 &\\mbox{ if }i\\in I_0,\\\\\n(i^1,i^2) &\\mbox{ if } i\\in I_1.\n\\end{array}\\right.\n$$\n\nThere is a unique $\\mathbb{Z}$-linear map $\\phi\\colon Q_I\\to Q_{\\overline I}$ such that $\\phi(I^\\alpha)\\subset I^{\\phi(\\alpha)}$ for each $\\alpha\\in Q^+_I$. It is given by\n\\begin{equation}\n\\label{ch3:eq_phi(alpha)}\n\\phi(\\alpha_{i})=\n\\left\\{\\begin{array}{ll}\n\\alpha_{i^0} &\\mbox{ if }i\\in I_0,\\\\\n\\alpha_{i^1}+\\alpha_{i^2}&\\mbox{ if }i\\in I_1.\n\\end{array}\\right.\n\\end{equation}\nLet $\\phi$ denote also the unique additive embedding\n\\begin{equation}\n\\label{ch3:eq_phi(mu)}\n\\phi\\colon X_I\\to X_{\\overline I}, \\quad\\varepsilon_i\\mapsto \\varepsilon_{i'},\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{ch3:eq_i'}\ni'=\n\\left\\{\\begin{array}{ll}\ni^0 &\\mbox{ if }i\\in I_0,\\\\\ni^1&\\mbox{ if }i\\in I_1.\n\\end{array}\\right.\n\\end{equation}\n\n\n\\subsection{KLR algebras}\n\\label{ch3:subs_KLR}\n\nLet $\\bfk$ be a field. Let $\\Gamma=(I,H)$ be a quiver without $1$-loops. For $r\\in[1,d-1]$ let $s_r$ be the transposition $(r,r+1)\\in\\mathfrak{S}_d$. For ${\\mathbf{i}}=(i_1,\\cdots,i_d)\\in I^d$ set $s_r({\\mathbf{i}})=(i_1,\\cdots,i_{r-1},i_{r+1},i_r,i_{r+2},\\cdots,i_d)$.\nFor $i,j\\in I$ we set\n$$\nQ_{i,j}(u,v)=\n\\left\\{\\begin{array}{ll}\n0& \\mbox{ if }i=j,\\\\\n(v-u)^{h_{i,j}}(u-v)^{h_{j,i}}& \\mbox{ else}.\n\\end{array}\\right.\n$$\n\n\\smallskip\n\\begin{df}\n\\label{ch3:def_KLR}\nThe \\emph{KLR-algebra} $R_{d,\\bfk}(\\Gamma)$ is the $\\bfk$-algebra with the set of generators $\\tau_1,\\cdots,\\tau_{d-1},x_1,\\cdots,x_d,e({\\mathbf{i}})$ where ${\\mathbf{i}}\\in I^d$,\nmodulo the following defining relations\n\\begin{itemize}\n \\item[\\textbullet] $e({\\mathbf{i}})e({\\mathbf{j}})=\\delta_{{\\mathbf{i}},{\\mathbf{j}}}e({\\mathbf{i}})$,\n \\item[\\textbullet] $\\sum_{{\\mathbf{i}}\\in I^d}e({\\mathbf{i}})=1$,\n \\item[\\textbullet] $x_re({\\mathbf{i}})=e({\\mathbf{i}})x_r$,\n \\item[\\textbullet] $\\tau_re({\\mathbf{i}})=e(s_r({\\mathbf{i}}))\\tau_r$,\n \\item[\\textbullet] $x_rx_s=x_sx_r$,\n \\item[\\textbullet] $\\tau_rx_{r+1}e({\\mathbf{i}})=(x_r\\tau_r+\\delta_{i_r,i_{r+1}})e({\\mathbf{i}})$,\n \\item[\\textbullet] $x_{r+1}\\tau_re({\\mathbf{i}})=(\\tau_rx_r+\\delta_{i_r,i_{r+1}})e({\\mathbf{i}})$,\n \\item[\\textbullet] $\\tau_rx_s=x_s\\tau_r$, if $s\\ne r,r+1$,\n \\item[\\textbullet] $\\tau_r\\tau_s=\\tau_s\\tau_r$, if $|r-s|>1$,\n \\item[\\textbullet] $\\tau_r^2e({\\mathbf{i}})=\n\\left\\{\n\\begin{array}{ll}\n0 &\\mbox{if } i_r=i_{r+1},\\\\\nQ_{i_r,i_{r+1}}(x_r,x_{r+1})e({\\mathbf{i}}) &\\mbox{else},\n\\end{array}\n\\right.\n$\n \\item[\\textbullet] $(\\tau_r\\tau_{r+1}\\tau_r-\\tau_{r+1}\\tau_r\\tau_{r+1})e({\\mathbf{i}})=$\n\n$\\left\\{\n\\begin{array}{ll}\n(x_{r+2}-x_{r})^{-1}(Q_{i_{r},i_{r+1}}(x_{r+2},x_{r+1})-Q_{i_r,i_{r+1}}(x_{r},x_{r+1}))e({\\mathbf{i}}) &\\mbox{if }i_r=i_{r+2},\\\\\n0 &\\mbox{else},\n\\end{array}\n\\right.$\n\\end{itemize}\nfor each ${\\mathbf{i}}$, ${\\mathbf{j}}$, $r$ and $s$. We may write $R_{d,\\bfk}=R_{d,\\bfk}(\\Gamma)$. The algebra $R_{d,\\bfk}$ admits a $\\mathbb{Z}$-grading such that $\\deg e({\\mathbf{i}})=0$, $\\deg x_r=2$, $\\deg\\tau_se({\\mathbf{i}})=-a_{i_s,i_{s+1}}$, for each $1\\leqslant r\\leqslant d$, $1\\leqslant s< d$ and ${\\mathbf{i}}\\in I^d$.\n\\end{df}\n\n\\smallskip\nFor each $\\alpha\\in Q^+_I$ such that $|\\alpha|=d$ set $e(\\alpha)=\\sum_{{\\mathbf{i}}\\in I^\\alpha} e({\\mathbf{i}})\\in R_{d,\\bfk}$. It is a homogeneous central idempotent of degree zero. We have the following decomposition into a sum of unitary $\\bfk$-algebras\n$R_{d,\\bfk}=\\bigoplus_{|\\alpha|=d}R_{\\alpha,\\bfk}$, where $R_{\\alpha,\\bfk}=e(\\alpha)R_{d,\\bfk}$.\n\nLet $\\bfk^{(I)}_d$ be the direct sum of copies of the ring $\\bfk_d[x]:=\\bfk[x_1,\\cdots,x_d]$ labelled by $I^d$.\nWe write\n\\begin{equation}\n\\label{ch3:eq_k^I}\n\\bfk^{(I)}_d=\\bigoplus_{{\\mathbf{i}}\\in I^d}\\bfk_d[x]e({\\mathbf{i}}),\n\\end{equation}\nwhere $e({\\mathbf{i}})$ is the idempotent of the ring $\\bfk^{(I)}_d$ projecting to the component ${\\mathbf{i}}$. A polynomial in $\\bfk_d[x]$ can be considered as an element of $\\bfk^{(I)}_d$ via the diagonal inclusion.\nFor each $i,j\\in I$ fix a polynomial $P_{i,j}(u,v)$ such that we have $Q_{i,j}(u,v)=P_{i,j}(u,v)P_{j,i}(v,u)$.\nThen by \\cite[Sec.~3.2]{Rouq-2KM} the algebra $R_{d,\\bfk}$ has a representation in the vector space $\\bfk^{(I)}_d$ such that the element $e({\\mathbf{i}})$ acts by the projection to $\\bfk^{(I)}_de({\\mathbf{i}})$, the element $x_r$ acts by multiplication by $x_r$ and such that for $f\\in\\bfk_d[x]$ we have\n\\begin{equation}\n\\label{ch3:eq_action-on-polyn}\n\\tau_r\\cdot fe({\\mathbf{i}})=\n\\left\\{\\begin{array}{ll}\n(x_r-x_{r+1})^{-1}(s_r(f)-f)e({\\mathbf{i}}) &\\mbox{ if }i_r=i_{r+1},\\\\\nP_{i_r,i_{r+1}}(x_{r+1},x_r)s_r(f)e(s_r({\\mathbf{i}}))&\\mbox{ otherwise}.\\\\\n\\end{array}\\right.\n\\end{equation}\n\nWe will always choose $P_{i,j}$ in the following way:\n$$\nP_{i,j}(u,v)=(u-v)^{h_{j,i}}.\n$$\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_basis-KLR}\nThere is an explicit construction of a basis of a KLR algebra.\nAssume ${\\mathbf{i}},{\\mathbf{j}}\\in I^\\alpha$. Set $\\mathfrak{S}_{{\\mathbf{i}},{\\mathbf{j}}}=\\{w\\in \\mathfrak{S}_d;~ w({\\mathbf{i}})={\\mathbf{j}}\\}$. For each permutation $w\\in \\mathfrak{S}_{{\\mathbf{i}},{\\mathbf{j}}}$ fix a reduced expression $w=s_{p_1}\\cdots s_{p_r}$ and set $\\tau_w=\\tau_{p_1}\\cdots\\tau_{p_r}$.\nThen the vector space $e({\\mathbf{j}})R_{\\alpha,\\bfk} e({\\mathbf{i}})$ has a basis $\\{\\tau_wx_1^{a_1}\\cdots x_d^{a_d}e({\\mathbf{i}});~w\\in\\mathfrak{S}_{{\\mathbf{i}},{\\mathbf{j}}}, a_1,\\cdots,a_d\\in\\mathbb{N}$\\}. Note that the element $\\tau_w$ depends on the reduced expression of $w$.\nMoreover, if we change the reduced expression of $w$, then the element $\\tau_we({\\mathbf{i}})$ changes only by a linear combination of monomials of the form $\\tau_{q_1}\\cdots\\tau_{q_t}x_1^{b_1}\\cdots x_d^{b_d}e({\\mathbf{i}})$\nwith $t<\\ell(w)$.\n\\end{rk}\n\n\\smallskip\nFix a weight $\\Lambda=\\sum_{i\\in I}n_i\\Lambda_i\\in P_I^+$.\n\n\\smallskip\n\\begin{df}\nThe \\emph{cyclotomic KLR-algebra} $R^\\Lambda_{\\alpha,\\bfk}$ is the quotient of $R_{\\alpha,\\bfk}$ by the two-sided ideal generated by\n$x_1^{n_{i_1}}e({\\mathbf{i}})$ for ${\\mathbf{i}}=(i_1,\\cdots,i_d)\\in I^\\alpha$.\n\\end{df}\n\n\\smallskip\nFor each $i\\in I$ there is an inclusion $R_{\\alpha,\\bfk}\\subset R_{\\alpha+\\alpha_i,\\bfk}$ that takes $e({\\mathbf{i}})$ to $e({\\mathbf{i}},i)$, $x_r$ to $x_re({\\mathbf{i}},i)$, $\\tau_r$ to $\\tau_re({\\mathbf{i}},i)$. It factors through a homomorphism $R^\\Lambda_{\\alpha,\\bfk}\\to R^\\Lambda_{\\alpha+\\alpha_i,\\bfk}$. Let\n$$\nF^\\Lambda_i\\colon\\mathrm{mod}(R^\\Lambda_{\\alpha,\\bfk})\\to \\mathrm{mod}(R^\\Lambda_{\\alpha+\\alpha_i,\\bfk}),\\qquad E^\\Lambda_i\\colon\\mathrm{mod}(R^\\Lambda_{\\alpha+\\alpha_i,\\bfk})\\to \\mathrm{mod}(R^\\Lambda_{\\alpha,\\bfk})\n$$\nbe the induction and restriction functors with respect to this algebra homomorphism.\n\n\n\n\\subsection{Balanced KLR algebras}\n\\label{ch3:subs_bal-quot}\nFix a decomposition $I=I_0\\sqcup I_1$ as in Section \\ref{ch3:subs_not-quiv-I-Ibar} and consider the quiver $\\overline\\Gamma=(\\overline I,\\overline H)$ as in Section \\ref{ch3:subs_not-quiv-I-Ibar}. Recall the decomposition $\\overline I=\\overline I_0\\sqcup \\overline I_1\\sqcup \\overline I_2$. In this section we work with the KLR algebra associated with the quiver $\\overline\\Gamma$.\n\nWe say that a sequence ${\\mathbf{i}}=(i_1,i_2,\\cdots,i_d)\\in \\overline I^d$ is \\emph{unordered} if there is an index $r\\in[1,d]$ such that the number of elements from $\\overline I_2$ in the sequence $(i_1,i_2,\\cdots,i_r)$ is strictly greater than the number of elements from $\\overline I_1$. We say that it is \\emph{well-ordered} if for each index $a$ such that $i_a=i^1$ for some $i\\in I_1$, we have $aw(a+1)$. Really, let $J$ be the set of indices $a\\in[1,\\overline d]$ such that $i'_a\\in \\overline I_1$. As ${\\mathbf{j}}'$ is well-ordered, we have $\\sum_{a\\in J}(w(a+1)-w(a))=\\#J$. As $w$ is unbalanced, not all summands in this sum are equal to $1$. Then one of the summands must be negative. Let $a\\in J$ be an index such that $w(a)>w(a+1)$. We can assume that the reduced expression of $w$ is of the form $w=s_{p_1}\\cdots s_{p_k}s_{a}$. In this case the element $\\tau_we({\\mathbf{i}}')$ is zero in $S_{\\overline\\alpha,\\bfk}$ because the idempotent $s_a({\\mathbf{i}}')$ is unordered.\n\nAssume that $w\\in\\mathfrak{S}_{{\\mathbf{i}}',{\\mathbf{j}}'}$ is balanced. Thus there exists some $\\widetilde w\\in \\mathfrak{S}_{{\\mathbf{i}},{\\mathbf{j}}}$ such that $u(\\widetilde w)=w$. We choose an arbitrary reduced expression $\\widetilde w=s_{p_1}\\cdots s_{p_k}$ and we choose the reduced expression of $w=s_{q_1}\\cdots s_{q_r}$ induced by the reduced expression of $\\widetilde w$. Then the element $\\tau_we({\\mathbf{i}}')=\\tau_{q_1}\\cdots\\tau_{q_r}e({\\mathbf{i}}')\\in S_{\\alpha,\\bfk}$ is equal to $\\pm (\\tau_{p_1}\\cdots \\tau_{p_k}e({\\mathbf{i}}))^*$.\n\nThe discussion above shows that the image of an element $b'\\in\\mathbf{B}'$ in $e({\\mathbf{j}}')S_{\\overline\\alpha,\\bfk}e({\\mathbf{i}}')$ is either zero or is of the form $\\pm b^*$ for some $b\\in\\mathbf{B}$. Moreover, each $b^*$ for $b\\in\\mathbf{B}$ can be obtained in such a way.\nNow we get the following.\n\\begin{itemize}\n\\item[\\textbullet] The elements $e(\\phi({\\mathbf{i}}))$, $x^*_r$, $\\tau^*_r$ generate $S_{\\overline\\alpha,\\bfk}$ because the image of each element of $\\mathbf{B}'$ in $e({\\mathbf{j}}')S_{\\overline\\alpha,\\bfk}e({\\mathbf{i}}')$ is either zero or a monomial on $e(\\phi({\\mathbf{i}}))$, $x^*_r$, $\\tau^*_r$.\n\\item[\\textbullet] The representation $\\bfk^{(\\overline I)}_{\\overline \\alpha,\\mathrm{ord}}\/J_{\\overline\\alpha,\\mathrm{ord}}$ of $S_{\\overline\\alpha,\\bfk}$ is faithful because the spanning set $\\{b^*;~b\\in\\mathbf{B}\\}$ of $e({\\mathbf{j}}')S_{\\overline\\alpha,\\bfk}e({\\mathbf{i}}')$ acts on the polynomial representation by linearly independent operators (because the polynomial representation of $R_{\\alpha,\\bfk}$ is faithful).\n\\end{itemize}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe center of the algebra $R_{\\alpha,\\bfk}$ is the ring of symmetric polynomials $\\bfk_d[x]^{\\mathfrak{S}_d}$, see \\cite[Prop.~3.9]{Rouq-2KM}. Thus $S_{\\overline\\alpha,\\bfk}$ is a $\\bfk_d[x]^{\\mathfrak{S}_d}$-algebra under the isomorphism $\\Phi_{\\alpha,\\bfk}$. Let $\\Sigma$ be the polynomial $\\prod_{a1$ or $a=\\infty$.\nNote that there is an action of the symmetric group $\\mathfrak{S}_d$ on $\\bfk^{(I)}_{d}$ permuting the variables and the components of ${\\mathbf{i}}$.\nConsider the following elements in $R_{\\alpha,\\bfk}[\\Sigma^{-1}]$:\n$$\n\\Psi_re({\\mathbf{i}})=\n\\left\\{\\begin{array}{ll}\n((x_r-x_{r+1})\\tau_r+1)e({\\mathbf{i}})&\\mbox{ if } i_{r+1}=i_{r},\\\\\n-(x_r-x_{r+1})^{-1}\\tau_r e({\\mathbf{i}})&\\mbox{ if } i_{r+1}=i_{r}-1,\\\\\n\\tau_r e({\\mathbf{i}})&\\mbox{ else}.\\\\\n\\end{array}\\right.\n$$\nThe element $\\Psi_re({\\mathbf{i}})$ is called \\emph{intertwining operator}. Using the formulas (\\ref{ch3:eq_action-on-polyn}) we can check that $\\Psi_re({\\mathbf{i}})$ still acts on the polynomial representation and the corresponding operator is equal to $s_re({\\mathbf{i}})$. Note also that $\\widetilde\\Psi_r=(x_r-x_{r+1})\\Psi_r$ is an element of $R_{\\alpha,\\bfk}$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_morph-Phi-intert-op}\nThe images of intertwining operators by the morphism $\\Phi_{\\alpha,\\bfk}\\colon R_{\\alpha,\\bfk}\\to S_{\\overline\\alpha,\\bfk}$ can be described in the following way.\nFor ${\\mathbf{i}}\\in I^\\alpha$ such that $i_r-1\\ne i_{r+1}$ we have\n$$\n\\Phi_{\\alpha,\\bfk}(\\Psi_re({\\mathbf{i}}))=\n\\left\\{\\begin{array}{ll}\n\\Psi_{r'}e(\\phi({\\mathbf{i}})),& \\mbox{ if }i_r,i_{r+1}\\in I_0,\\\\\n\\Psi_{r'}\\Psi_{r'+1}e(\\phi({\\mathbf{i}}))& \\mbox{ if }i_r\\in I_1,i_{r+1}\\in I_0,\\\\\n\\Psi_{r'+1}\\Psi_{r'}e(\\phi({\\mathbf{i}}))& \\mbox{ if }i_r\\in I_0,i_{r+1}\\in I_1,\\\\\n\\Psi_{r'+1}\\Psi_{r'+2}\\Psi_{r'}\\Psi_{r'+1}e(\\phi({\\mathbf{i}}))& \\mbox{ if }i_r,i_{r+1}\\in I_1.\\\\\n\\end{array}\\right.\n$$\nFor ${\\mathbf{i}}\\in I^\\alpha$ such that $i_r-1= i_{r+1}$ we have\n$$\n\\Phi_{\\alpha,\\bfk}(\\widetilde\\Psi_re({\\mathbf{i}}))=\n\\left\\{\\begin{array}{ll}\n\\widetilde\\Psi_{r'}e(\\phi({\\mathbf{i}})),& \\mbox{ if }i_r,i_{r+1}\\in I_0,\\\\\n\\widetilde\\Psi_{r'}\\Psi_{r'+1}e(\\phi({\\mathbf{i}}))& \\mbox{ if }i_r\\in I_1,i_{r+1}\\in I_0,\\\\\n\\Psi_{r'+1}\\widetilde\\Psi_{r'}e(\\phi({\\mathbf{i}}))& \\mbox{ if }i_r\\in I_0,i_{r+1}\\in I_1.\\\\\n\\end{array}\\right.\n$$\nHere $r'=r'_{\\mathbf{i}}$ is as in Section \\ref{ch3:subs_comp-pol-reps}.\n\\end{lem}\n\\begin{proof}[Proof]\nThe right hand side in the formulas for $\\Phi_{\\alpha,\\bfk}(\\Psi_re({\\mathbf{i}}))$ is an element $X$ in $S_{\\overline\\alpha,\\bfk}[\\Sigma^{-1}]$ that acts by the same operator that $\\Psi_re({\\mathbf{i}})$ on the polynomial representation. However, since the polynomial representation of $S_{\\overline\\alpha,\\bfk}$ is faithful by Lemma \\ref{ch3:lem_pol-rep-of-S-defined+faith}, and since there is a well-defined element of $S_{\\overline\\alpha,\\bfk}$ which acts in the same way as $X$ in this polynomial representation, we conclude that $X$ makes sense as an element of $S_{\\overline\\alpha,\\bfk}$.\n\n\\end{proof}\n\n\nThe goal of the rest of Section \\ref{ch3:sec_KLR-Hecke} is to obtain a deformed version of the isomorphism from Theorem \\ref{ch3:thm_KLR-e-e+1} over some local ring $R$ (see Section \\ref{ch3:subs_def-ring}). More precisely, the localized Hecke algebra over a field is isomorphic to the localized KLR algebra (see Section \\ref{ch3:subs_isom-KLR-Hecke-gen}).\nWe want to construct an $R$-algebra homomorphism between a localized Hecke algebra and a balanced analogue of a localized Hecke algebra (see Section \\ref{ch3:subs_param-Hecke} for the choice of parameters and Section \\ref{ch3:subs_deform-Phi} for the definition of a balanced analogue of the cyclotomic Hecke algebra) such that this homomorphism is compatible with the localization of the homomorphism from Theorem \\ref{ch3:thm_KLR-e-e+1} over the residue field of $R$ and over its field of fractions.\n\n\\subsection{Special quivers}\n\\label{ch3:subs_not-e-e+1}\nFrom now on we will be interested only in some special types of quivers.\n\n\\medskip\nFirst, consider the quiver $\\Gamma=\\Gamma_{e}$, where $e$ is an integer $>1$. In particular, from now on we fix $I=\\mathbb{Z}\/e\\mathbb{Z}$. Fix $k\\in [0,e-1]$ and set $I_1=\\{k\\}$, $I_0=I\\backslash\\{k\\}$. In this case the quiver $\\overline\\Gamma$ is isomorphic to $\\Gamma_{e+1}$. More precisely,\nthe decomposition $\\overline I=\\overline I_0\\sqcup\\overline I_2\\sqcup\\overline I_2$ is such that $\\overline I_1=\\{k\\}$, $\\overline I_2=\\{k+1\\}$. To avoid confusion, for $i\\in \\overline I$ we will write $\\overline\\alpha_i$, $\\overline\\varepsilon_i$ and $\\overline\\Lambda_i$ for $\\alpha_i$, $\\varepsilon_i$ and $\\Lambda_i$ respectively.\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_def-ordered-Ibar}\nIf $\\Gamma$ is as above, a sequence ${\\mathbf{i}}=(i_1,\\cdots,i_d)\\in \\overline I^d$ is well-ordered if for each index $a$ such that $i_a=k$ we have $a{\\widetilde\\phi}>> Q_{\\widetilde I}\\\\\n@V{\\pi_e}VV @V{\\pi_{e+1}}VV\\\\\nQ_I @>{\\phi}>> Q_{\\overline I}\\\\\n\\end{CD}\n\\qquad\\qquad\n\\begin{CD}\nX_{\\widetilde I} @>{\\widetilde\\phi}>> X_{\\widetilde I}\\\\\n@V{\\pi_e}VV @V{\\pi_{e+1}}VV\\\\\nX_I @>{\\phi}>> X_{\\overline I}\\\\\n\\end{CD}\n\\qquad\\qquad\n\\begin{CD}\n\\widetilde I^{\\widetilde\\alpha} @>{\\widetilde\\phi}>> \\widetilde I^{\\widetilde\\phi(\\widetilde\\alpha)}\\\\\n@V{\\pi_e}VV @V{\\pi_{e+1}}VV\\\\\nI^{\\alpha} @>{\\phi}>> \\overline I^{\\phi(\\alpha)}\\\\\n\\end{CD}\n$$\n\n\\subsection{Deformation rings}\n\\label{ch3:subs_def-ring}\n\nIn this section we introduce some general definitions from \\cite{RSVV} for a later use.\n\nWe call \\emph{deformation ring} $(R,\\kappa,\\tau_1,\\cdots,\\tau_l)$ a regular commutative noetherian $\\mathbb{C}$-algebra $R$ with $1$ equipped with a homomorphism $\\mathbb{C}[\\kappa^{\\pm 1},\\tau_1,\\cdots,\\tau_l]\\to R$. Let $\\kappa,\\tau_1,\\cdots,\\tau_r$ denote also the images of $\\kappa,\\tau_1,\\cdots,\\tau_r$ in $R$.\nA deformation ring is \\emph{in general position} if any two elements of the set\n$$\n\\{\\tau_u-\\tau_v+a\\kappa+b,\\kappa-c;~a,b\\in\\mathbb{Z},c\\in \\mathbb{Q},u\\ne v\\}\n$$\nhave no common non-trivial divisors.\nA \\emph{local deformation ring} is a deformation ring which is a local ring and such that $\\tau_1,\\cdots,\\tau_l, \\kappa-e$ belong to the maximal ideal of $R$.\nNote that each $\\mathbb{C}$-algebra that is a field has a \\emph{trivial} local deformation ring structure, i.e., such that $\\tau_1=\\cdots=\\tau_l=0$ and $\\kappa=e$. We always consider $\\mathbb{C}$ as a local deformation ring with a trivial deformation ring structure.\nA $\\mathbb{C}$-algebra $R$ is called \\emph{analytic} if it is a localization of the ring of germs of holomorphic functions on some compact polydisc $D\\subset \\mathbb{C}^d$ for some $d\\geqslant 1$.\n\n\nWe will write $\\overline\\kappa=\\kappa(e+1)\/e$, $\\overline\\tau_r=\\tau_r(e+1)\/e$.\nWe will abbreviate $R$ for $(R,\\kappa,\\tau_1,\\cdots,\\tau_l)$ and $\\overline R$ for $(R,\\overline\\kappa,\\overline\\tau_1,\\cdots,\\overline\\tau_l)$.\n\n\nLet $R$ be a local analytic deformation ring with residue field $\\bfk$. Consider the elements $q_{e}=\\exp(2\\pi \\sqrt{-1}\/\\kappa)$ and $q_{e+1}=\\exp(2\\pi \\sqrt{-1}\/\\overline\\kappa)$ in $R$. These elements specialize to $\\zeta_{e}=\\exp(2\\pi \\sqrt{-1}\/e)$ and $\\zeta_{e+1}=\\exp(2\\pi \\sqrt{-1}\/(e+1))$ in $\\bfk$.\n\n\\subsection{Hecke algebras}\n\\label{ch3:subs_Hecke}\nLet $R$ be a commutative ring with $1$. Fix an element $q\\in R$.\n\n\\smallskip\n\\begin{df}\nThe \\emph{affine Hecke algebra} $H_{R,d}(q)$ is the $R$-algebra generated by $T_1,\\cdots,T_{d-1}$ and the invertible elements $X_1,\\cdots,X_d$ modulo the following defining relations\n$$\n\\begin{array}{lllll}\nT_{r}T_{s}=T_{s}T_{r} \\mbox{ if }|r-s|>1,& T_{r}T_{r+1}T_{r}=T_{r+1}T_{r}T_{r+1}\\\\\n(T_r-q)(T_r+1)=0,\n&T_rX_r=X_rT_r ~\\mbox{ if }|r-s|>1,\\\\\n T_rX_{r+1}=X_rT_r+(q-1)X_{r+1}, &T_rX_{r}=X_{r+1}T_r-(q-1)X_{r+1}.\\\\\n\\end{array}\n$$\n\\end{df}\n\nFix elements $Q_1,\\cdots,Q_l\\in R$.\n\n\\smallskip\n\\begin{df}\nThe \\emph{cyclotomic Hecke algebra} $H^Q_{d,R}(q)$ is the quotient of $H_{d,R}(q)$ by the two-sided ideal generated by $(X_1-Q_1)\\cdots(X_1-Q_l)$.\n\\end{df}\n\nAssume that $R=\\bfk$ is a field and $q\\ne 0, 1$, $Q_r\\ne 0$.\nThe algebra $H_{d,\\bfk}(q)$ has a faithful representation in the vector space $\\bfk[X^{\\pm 1}_1,\\cdots,X^{\\pm 1}_d]$ such that $X^{\\pm 1}_r$ acts by multiplication by $X^{\\pm 1}_r$ and $T_r$ by\n$$\nT_r(P)=qs_r(P)+(q-1)X_{r+1}(X_{r}-X_{r+1})^{-1}(s_r(P)-P).\n$$\n\nThe following operator acts on $\\bfk[X_1^{\\pm 1},\\cdots,X_d^{\\pm 1}]$ as the reflection $s_r$\n$$\n\\Psi_r=\\frac{X_r-X_{r+1}}{qX_r-X_{r+1}}(T_r-q)+1=(T_r+1)\\frac{X_r-X_{r+1}}{X_r-qX_{r+1}}-1.\n$$\nFor a future use, consider the element $\\widetilde\\Psi_r\\in H_{d,\\bfk}$ given by\n$$\n\\widetilde\\Psi_r=(qX_r-X_{r+1})\\Psi_r=(X_{r}-X_{r+1})T_r+(q-1)X_{r+1}.\n$$\n\nNow,\nconsider the subset $\\mathscr{F}=\\mathscr{F}(q,Q)$ of $\\bfk$\ngiven by\n\\begin{equation}\n\\label{ch3:eq_F(q,Q)}\n\\mathscr{F}(q,Q)=\\bigcup_{r\\in \\mathbb{Z},t\\in[1,l]}\\{q^rQ_t\\}.\n\\end{equation}\nWe can consider $\\mathscr{F}$ as the vertex set of a quiver with an arrow $i\\to j$ if and only if $j=qi$.\nLet $M$ be a finite dimensional $H^Q_{d,\\bfk}(q)$-module. For each ${\\mathbf{i}}=(i_1,\\cdots,i_d)\\in \\mathscr{F}^d$ let $M_{\\mathbf{i}}$ be the generalized eigenspace of $X_1,\\cdots,X_d$ with eigenvalues $i_1,\\cdots,i_d$.\nThe algebra $H^Q_{d,\\bfk}(q)$ contains an idempotent $e({\\mathbf{i}})$ which acts on each finite dimensional $H^Q_{d,\\bfk}(q)$-module $M$ by projection to $M_{\\mathbf{i}}$ with respect to $\\bigoplus_{{\\mathbf{j}}\\ne {\\mathbf{i}}}M_{\\mathbf{j}}$.\nFor $\\alpha\\in Q^+_\\mathscr{F}$ we consider the central idempotent $e(\\alpha)=\\sum_{{\\mathbf{i}}\\in \\mathscr{F}^\\alpha}e({\\mathbf{i}})$. The algebra $H^Q_{d,\\bfk}$ decomposes in a direct sum of blocks $H^Q_{\\alpha,\\bfk}$ with $\\alpha\\in Q^+_\\mathscr{F}$ where $H^Q_{\\alpha,\\bfk}=e(\\alpha)H^Q_{d,\\bfk}$. See \\cite{BK-blocks} for more details.\n\n\n\\subsection{The isomorphism between Hecke and KLR algebras}\n\\label{ch3:subs_isom-KLR-Hecke-gen}\nAssume that $R=\\bfk$ is a field and $q\\ne 0, 1$.\n\nFirst, we define some localized versions of Hecke algebras and KLR algebras.\nLet $\\mathscr{F}$ be a subset of $\\bfk$ such that $q^\\mathbb{Z}\\mathscr{F}\/q^\\mathbb{Z}$ is finite. We view $\\mathscr{F}$ as the vertex set of a quiver with an arrow $i\\to j$ if and only if $j=qi$.\nConsider the algebra\n$$\nA_1=\\bigoplus_{{\\mathbf{i}}\\in \\mathscr{F}^d}\\bfk[X_1^{\\pm 1},\\cdots,X_d^{\\pm 1}][(X_r-X_t)^{-1},(qX_r-X_t)^{-1};~r\\ne t]e({\\mathbf{i}}),\n$$\nwhere $e({\\mathbf{i}})$ are orthogonal idempotents and $X_r$ commutes with $e({\\mathbf{i}})$.\nLet $H_{d,\\bfk}^{\\rm loc}(q)$ be the $A_1$-module given by the extension of scalars from the $\\bfk[X_1^{\\pm 1},\\cdots,X_d^{\\pm 1}]$-module $H_{d,\\bfk}(q)$. It is an $A_1$-algebra such that\n$$\nT_re({\\mathbf{i}})-e(s_r({\\mathbf{i}}))T_r=(1-q)X_{r+1}(X_r-X_{r+1})^{-1}(e({\\mathbf{i}})-e(s_r({\\mathbf{i}})))\n$$\nand\n$$\nZ^{-1}T_r=T_rZ^{-1},\\quad \\mbox{ where } Z=\\prod_{r{\\sim}>> \\mathscr{F}(q_e,Q)\\\\\n@V{\\pi_e}VV @VVV\\\\\nI @>{\\sim}>> \\mathscr{F}(\\zeta_e,Q),\n\\end{CD}\n\\qquad\\qquad\n\\begin{CD}\n\\widetilde I @>{\\sim}>> \\mathscr{F}(q_{e+1}, \\overline Q)\\\\\n@V{\\pi_{e+1}}VV @VVV\\\\\n\\overline I @>{\\sim}>> \\mathscr{F}(\\zeta_{e+1},\\overline Q).\n\\end{CD}\n$$\n\n\nWe will identify\n\\begin{equation}\n\\label{ch3:eq_ident-quivers}\nI\\simeq \\mathscr{F}(\\zeta_e,Q),\\quad \\overline I\\simeq \\mathscr{F}(\\zeta_{e+1}, \\overline Q), \\quad \\widetilde I\\simeq \\mathscr{F}(q_{e},Q), \\quad \\widetilde I\\simeq \\mathscr{F}(q_{e+1},\\overline Q)\n\\end{equation}\nas above.\n\nRecall from Section \\ref{ch3:subs_Hecke} that the cyclotomic Hecke algebra $H^\\nu_{d,\\bfk}(\\zeta_e)$ contains some idempotents $e({\\mathbf{i}})$ for ${\\mathbf{i}}\\in I^d$.\nThese idempotents lift to idempotents in the $R$-algebra $H^\\nu_{d,R}(q_e)$\nbecause this algebra is free over $R$ by \\cite[Thm.~2.2]{Mat} and $R$ is a local algebra, see, e.g., \\cite[Ex.~6.16]{CR}. We also denote these idempotents $e({\\mathbf{i}})$.\n\nBy base change we have the $K$-algebra $H^{\\nu}_{d,K}(q_{e})$\nwhich contains some idempotents $e({\\mathbf{j}})\\in H^\\nu_{d,K}(q_e)$ for ${\\mathbf{j}}\\in \\widetilde I^d$.\nThe idempotent\n$e({\\mathbf{i}})\\in H^\\nu_{d,R}(q_e)$\ndecomposes in $H^\\nu_{d,K}(q_e)$\nin the following way\n$$\ne({\\mathbf{i}})=\\sum_{{\\mathbf{j}}\\in \\widetilde I^d,\\pi_e({\\mathbf{j}})={\\mathbf{i}}}e({\\mathbf{j}})\n$$\n\nWe have the decompositions\n$$\nH^\\nu_{d,R}(q_e)=\\bigoplus_{\\alpha\\in Q^+_{I},|\\alpha|=d}H^\\nu_{\\alpha,R}(q_e),\n$$\n$$\nH^\\nu_{d,K}(q_e)=\\bigoplus_{\\alpha\\in Q^+_{I},|\\alpha|=d}H^\\nu_{\\alpha,K}(q_e),\n$$\nwhere\n$$\nH^\\nu_{\\alpha,K}(q_e)=\\bigoplus_{\\widetilde\\alpha\\in Q^+_{\\widetilde I},\\pi_e(\\widetilde\\alpha)=\\alpha}H^\\nu_{\\widetilde\\alpha,K}(q_e).\n$$\n\nOur goal is to obtain an analogue of Theorem \\ref{ch3:thm_KLR-e-e+1} over the ring $R$.\nFirst, consider the algebras $\\widehat H_{d,\\bfk}(\\zeta_e)$ and $\\widehat H_{d,K}(q_e)$ defined in the same way as in Section \\ref{ch3:subs_isom-KLR-Hecke-gen} with respect to the sets $\\mathscr{F}(\\zeta_e,Q)\\subset \\bfk$ and $\\mathscr{F}(q_e,Q)\\subset K$, where $Q$ is as in Section \\ref{ch3:subs_param-Hecke}. We can consider the $R$-algebra $\\widehat H_{d,R}(q_e)$ defined in a similar way with respect to the same set of idempotents as $\\widehat H_{d,\\bfk}(q_e)$. Sometimes we will write $\\widehat H^\\nu_{d,\\bfk}(\\zeta_e)$, $\\widehat H^\\nu_{d,\\bfk}(q_e)$ and $\\widehat H^\\nu_{d,K}(q_e)$ to specify the parameter $\\nu$ (the $l$-tuple $Q$ in Section \\ref{ch3:subs_param-Hecke} depends on the $l$-tuple $\\nu$). Note that the algebra $\\widehat H_d$ (over $\\bfk$, $R$ or $K$) admits the same block decomposition as the block decomposition for the cyclotomic Hecke algebra $H^\\nu_d$ described above.\n\n\n\nNow, we define the algebra $\\widehat{SH}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})$ that is a Hecke analogue of a localization of the balanced KLR algebra $S_{\\overline\\alpha,\\bfk}$. To do so, consider the idempotent $\\mathbf{e}=\\sum_{{\\mathbf{i}}\\in \\overline I^{\\overline\\alpha}_{\\rm ord}}e({\\mathbf{i}})$ in $\\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})$. We set\n$$\n\\widehat{SH}_{\\overline\\alpha,\\overline\\bfk}(q_{e+1})=\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\/\\sum_{{\\mathbf{j}}\\in \\overline\nI^{\\overline\\alpha}_{\\rm un}}\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})e({\\mathbf{j}})\\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}.\n$$\nWe define the $K$-algebra $\\widehat{SH}_{\\overline\\alpha,\\overline K}(q_{e+1})$ as a similar quotient of $\\mathbf{e} H^{\\overline\\nu}_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}$ by the two-sided ideal generated by $\\{e({\\mathbf{j}});~{\\mathbf{j}}\\in \\widetilde I^{\\overline\\alpha}_{\\rm un})\\}$.\nFinally, we define the $R$-algebra $\\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$ as the image in $\\widehat{SH}_{\\overline\\alpha,\\overline K}(q_{e+1})$ of the following composition of homomorphisms\n$$\n\\mathbf{e}\\widehat{H}^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to\\mathbf{e}\\widehat{H}^{\\overline\\nu}_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}\\to\\widehat{SH}_{\\overline\\alpha,\\overline K}(q_{e+1}).\n$$\n\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_4-isom-KLR-Hecke}\n\nBy Proposition \\ref{ch3:prop-isom_Hekce-KLR-widetilde} we have algebra isomorphisms\n$$\n\\widehat R_{\\alpha,\\bfk}(\\Gamma) \\simeq \\widehat H_{\\alpha,\\bfk}(\\zeta_e),\n\\quad \\widehat R_{\\alpha,K}(\\widetilde \\Gamma)\\simeq \\widehat H_{\\alpha,K}(q_{e}),\n$$\n$$\n\\widehat R_{\\overline\\alpha,\\bfk}(\\overline \\Gamma)\\simeq \\widehat H_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1}), \\quad \\widehat R_{\\overline\\alpha,K}(\\widetilde \\Gamma)\\simeq \\widehat H_{\\overline\\alpha,\\overline K}(q_{e+1}),\n$$\nfrom which we deduce that\n$$\n\\widehat S_{\\overline\\alpha,\\bfk}(\\overline \\Gamma)\\simeq \\widehat {SH}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1}), \\quad \\widehat S_{\\overline\\alpha,K}(\\widetilde \\Gamma)\\simeq \\widehat {SH}_{\\overline\\alpha,\\overline K}(q_{e+1}).\n$$\n\\end{rk}\n\n\\smallskip\nWe may use these isomorphisms without mentioning them explicitly. Using the identifications above between KLR algebras and Hecke algebras, a localization of the isomorphism in Theorem \\ref{ch3:thm_KLR-e-e+1} yields an isomorphism\n\\begin{equation}\n\\label{ch3:eq_Phi-Hecke-k}\n\\Phi_{\\alpha,\\bfk}\\colon \\widehat H_{\\alpha,\\bfk}(\\zeta_e)\\to \\widehat {SH}_{\\overline\\alpha,\\bfk}(\\zeta_{e+1}).\n\\end{equation}\nIn the same way we also obtain an algebra isomorphism\n$$\n\\Phi_{\\widetilde\\alpha,K}\\colon \\widehat H_{\\widetilde\\alpha,K}(q_e)\\to \\widehat{SH}_{\\widetilde\\phi(\\widetilde\\alpha),K}(q_{e+1})\n$$\nfor each $\\widetilde\\alpha\\in Q^+_{\\widetilde I}$.\nTaking the sum over all $\\widetilde\\alpha\\in Q^+_{\\widetilde I}$ such that $\\pi_{e}(\\widetilde\\alpha)=\\alpha$\nyields a isomorphism\n\\begin{equation}\n\\label{ch3:eq_Phi-Hecke-K}\n\\Phi_{\\alpha,K}\\colon \\widehat H_{\\alpha,K}(q_e) \\to \\widehat{SH}_{\\overline\\alpha,K}(q_{e+1}).\n\\end{equation}\n\n\\begin{lem}\nThe homomorphism $\\mathbf{e}\\widehat{H}^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to \\mathbf{e}\\widehat{H}^{\\overline\\nu}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})\\mathbf{e}$\nfactors through a homomorphism $\\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})\\to \\widehat{SH}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})$.\n\\end{lem}\n\n\\begin{proof}[Proof]\nIn Section \\ref{ch3:subs_pol-rep-BKLR} we constructed a faithful polynomial representation of $S_{\\overline\\alpha,\\bfk}$. Let us call it $\\mathcal{P}ol_\\bfk$. It is constructed as a quotient of the standard polynomial representation of $\\mathbf{e} R_{\\overline\\alpha,\\bfk}\\mathbf{e}$. After localization we get a faithful representation $\\widehat\\mathcal{P}ol_\\bfk$ of $\\widehat S_{\\overline\\alpha,\\bfk}$. Thus the kernel of the algebra homomorphism $\\mathbf{e} \\widehat R_{\\overline\\alpha,\\bfk}\\mathbf{e}\\to \\widehat S_{\\overline\\alpha,\\bfk}$ is the annihilator of the representation $\\widehat\\mathcal{P}ol_\\bfk$. We can transfer this to the Hecke side and we obtain that the kernel of the algebra homomorphism $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})\\mathbf{e}\\to \\widehat {SH}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})$ is the annihilator of the representation $\\widehat\\mathcal{P}ol_\\bfk$. Similarly, we can characterize the kernel of the algebra homomorphism $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}\\to \\widehat{SH}_{\\overline\\alpha,\\overline K}(q_{e+1})$ as the annihilator of a similar representation $\\widehat\\mathcal{P}ol_K$.\n\nThe $K$-vector space $\\widehat\\mathcal{P}ol_K$ has an $R$-submodule $\\widehat\\mathcal{P}ol_R$ stable by the action of $\\mathbf{e}\\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}$ such that $\\bfk\\otimes_R \\widehat\\mathcal{P}ol_R=\\widehat\\mathcal{P}ol_\\bfk$ and it is compatible with the algebra homomorphism $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to \\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline \\bfk}(\\zeta_{e+1})\\mathbf{e}$. By definition of $\\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$ and the discussion above, the kernel of the algebra homomorphism $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to \\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$ is formed by the elements that act by zero on $\\widehat\\mathcal{P}ol_K$. Thus each element of this kernel acts by zero on $\\widehat\\mathcal{P}ol_R$. This implies, that an element of the kernel of $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to \\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$ specializes to an element of the kernel of $\\mathbf{e} \\widehat H^{\\overline\\nu}_{\\overline\\alpha,\\overline \\bfk}(q_{e+1})\\mathbf{e}\\to \\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$. This proves the statement.\n\\end{proof}\n\n\n\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_morph-Phi-over-ring}\nThere is a unique algebra homomorphism $\\Phi_{\\alpha,R}\\colon \\widehat H_{\\alpha,R}(q_e)\\to \\widehat{SH}_{\\overline\\alpha,R}(q_{e+1})$ such that the following diagram is commutative\n$$\n\\begin{CD}\n\\widehat H_{\\alpha,\\bfk}(\\zeta_e) @>{\\Phi_{\\alpha,\\bfk}}>> \\widehat{SH}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})\\\\\n@AAA @AAA\\\\\n\\widehat H_{\\alpha,R}(q_e) @>{\\Phi_{\\alpha,R}}>> \\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})\\\\\n@VVV @VVV\\\\\n\\widehat H_{\\alpha,K}(q_e) @>{\\Phi_{\\alpha,K}}>> \\widehat{SH}_{\\overline\\alpha,\\overline K}(q_{e+1}).\\\\\n\\end{CD}\n$$\n\\end{lem}\n\\begin{proof}[Proof]\nFirst we consider the algebras $H_{\\alpha,\\bfk}^{\\rm loc}(\\zeta_e)$, $H_{\\alpha,R}^{\\rm loc}(q_e)$ and $H_{\\alpha,K}^{\\rm loc}(q_e)$ obtained from $\\widehat H_{\\alpha,\\bfk}(\\zeta_e)$, $\\widehat H_{\\alpha,R}(q_e)$ and $\\widehat H_{\\alpha,K}(q_e)$ by inverting\n\\begin{itemize}\n\\item[\\textbullet] $(X_r-X_t)$, $(\\zeta_eX_r-X_t)$ with $r\\ne t$,\n\\item[\\textbullet] $(X_r-X_t)$, $(q_eX_r-X_t)$ with $r\\ne t$,\n\\item[\\textbullet] $(X_r-X_t)$, $(q_eX_r-X_t)$ with $r\\ne t$\n\\end{itemize}\nrespectively.\nLet $SH^{\\rm loc}_{\\overline\\alpha,\\overline\\bfk}$ and $SH^{\\rm loc}_{\\overline\\alpha,\\overline K}$ be the localizations of $\\widehat{SH}_{\\overline\\alpha,\\overline\\bfk}$ and $\\widehat{SH}_{\\overline\\alpha,\\overline K}$ such that the isomorphisms $\\Phi_{\\alpha,\\bfk}$ and $\\Phi_{\\alpha,K}$ above induce isomorphisms\n$$\n\\Phi_{\\alpha,\\bfk}\\colon H^{\\rm loc}_{\\alpha,\\bfk}(\\zeta_e)\\to{SH}^{\\rm loc}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})\\\\\n$$\n$$\n\\Phi_{\\alpha,K}\\colon H^{\\rm loc}_{\\alpha,K}(q_e)\\to{SH}^{\\rm loc}_{\\overline\\alpha,\\overline K}(q_{e+1})\\\\.\n$$\nLet $SH^{\\rm loc}_{\\overline\\alpha,\\overline R}$ be the image in $SH^{\\rm loc}_{\\overline\\alpha,\\overline K}$ of the following composition of homomorphisms\n$$\n\\mathbf{e}{H}^{\\rm loc}_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to\\mathbf{e}{H}^{\\rm loc}_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}\\to{SH}^{\\rm loc}_{\\overline\\alpha,\\overline K}(q_{e+1}).\n$$\n\n\n\n\nNext, we want to prove that there exists an algebra homomorphism $\\Phi_{\\alpha,R}\\colon H_{\\alpha,R}^{\\rm loc}(q_e)\\to SH_{\\overline\\alpha,\\overline R}^{\\rm loc}(q_{e+1})$ such that the following diagram is commutative:\n\n\n\\begin{equation}\n\\label{ch3:eq_diag-localizations}\n\\begin{CD}\nH^{\\rm loc}_{\\alpha,\\bfk}(\\zeta_e) @>{\\Phi_{\\alpha,\\bfk}}>> SH^{\\rm loc}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})\\\\\n@AAA @AAA\\\\\nH^{\\rm loc}_{\\alpha,R}(q_e) @>{\\Phi_{\\alpha,R}}>> SH^{\\rm loc}_{\\overline\\alpha,\\overline R}(q_{e+1})\\\\\n@VVV @VVV\\\\\nH^{\\rm loc}_{\\alpha,K}(q_e) @>{\\Phi_{\\alpha,K}}>> SH^{\\rm loc}_{\\overline\\alpha,\\overline K}(q_{e+1}).\\\\\n\\end{CD}\n\\end{equation}\n\n\\vspace{1cm}\n\nWe just need to check that the map $\\Phi_{\\alpha,K}$ takes an element of $H^{\\rm loc}_{\\alpha,R}(q_e)$ to an element of $SH^{\\rm loc}_{\\overline\\alpha,\\overline R}(q_{e+1})$ and that it specializes to the map $\\Phi_{\\alpha,\\bfk}\\colon H^{\\rm loc}_{\\alpha,\\bfk}(\\zeta_{e})\\to SH^{\\rm loc}_{\\overline\\alpha,\\overline\\bfk}(\\zeta_{e+1})$. We will check this on the generators $e({\\mathbf{i}})$, $X_re({\\mathbf{i}})$ and $\\Psi_re({\\mathbf{i}})$ of $H^{\\rm loc}_{\\alpha,R}(q_e)$.\n\nThis is obvious for the idempotents $e({\\mathbf{i}})$.\n\nLet us check this for $X_re({\\mathbf{i}})$.\nAssume that ${\\mathbf{i}}\\in I^\\alpha$ and ${\\mathbf{j}}\\in \\widetilde I^{|\\alpha|}$ are such that we have $\\pi_e({\\mathbf{j}})={\\mathbf{i}}$. Write ${\\mathbf{i}}'=\\phi({\\mathbf{i}})$ and ${\\mathbf{j}}'=\\widetilde\\phi({\\mathbf{j}})$. Set $r'=r'_{\\mathbf{j}}=r'_{\\mathbf{i}}$, see the notation in Section \\ref{ch3:subs_comp-pol-reps}.\nBy Theorem \\ref{ch3:thm_KLR-e-e+1} and Proposition \\ref{ch3:prop-isom_Hekce-KLR-loc} we have\n$$\n\\Phi_{\\alpha,K}(X_{r}e({\\mathbf{j}}))={\\overline p}_{j'_{r'}}^{-1}p_{j_r}X_{r'}e({\\mathbf{j}}').\n$$\nSince, $\\overline p_{j'_{r'}}^{-1}p_{j_r}$ depends only on ${\\mathbf{i}}$ and $r$ and $e({\\mathbf{i}})=\\sum_{\\pi_e({\\mathbf{j}})={\\mathbf{i}}}e({\\mathbf{j}})$, we deduce that\n$$\n\\Phi_{\\alpha,K}(X_{r}e({\\mathbf{i}}))=\\overline p_{j'_{r'}}^{-1}p_{j_r}X_{r'}e({\\mathbf{i}}').\n$$\nThus the element $\\Phi_{\\alpha,K}(X_{r}e({\\mathbf{i}}))$ is in $SH^{\\rm loc}_{\\overline\\alpha,R}$ and its image in $SH^{\\rm loc}_{\\overline\\alpha,\\bfk}$ is $\\overline p_{i'_{r'}}^{-1}p_{i_r}X_{r'}e({\\mathbf{i}}')=\\Phi_{\\alpha,\\bfk}(X_{r}e({\\mathbf{i}}))$.\n\nNext, we consider the generators $\\Psi_re({\\mathbf{i}})$.\nWe must prove that for each ${\\mathbf{j}}$ such that $\\pi_e({\\mathbf{j}})={\\mathbf{i}}$ and for each $r$ we have\n\\begin{itemize}\n \\item[\\textbullet] $\\Phi_{\\alpha,K}(\\Psi_re({\\mathbf{j}}))=\\Xi e({\\mathbf{j}}')$, for some element $\\Xi\\in H^{\\rm loc}_{\\alpha,R}(q_e)$ that depends only on $r$ and ${\\mathbf{i}}$,\n \\item[\\textbullet] the image of $\\Xi e({\\mathbf{i}}')$ in $SH^{\\rm loc}_{\\overline\\alpha,\\overline\\bfk}(q_{e+1})$ under the specialization $R\\to\\bfk$ is $\\Phi_{\\alpha,\\bfk}(\\Psi_{r}e({\\mathbf{i}}))$.\n\\end{itemize}\nThis follows from Lemma \\ref{ch3:lem_morph-Phi-intert-op}.\n\nNow we obtain the diagram from the claim of Lemma \\ref{ch3:lem_morph-Phi-over-ring} as the restriction of the diagram (\\ref{ch3:eq_diag-localizations}).\n\n\\end{proof}\n\n\n\n\\section{The category $O$}\n\\label{ch3:sec_catO}\n\\subsection{Affine Lie algebras}\n\\label{ch3:subs_aff-Lie}\nFix positive integers $N$, $l$ and $e$ such that $e>1$. Let $R$ be a deformation ring, see Section \\ref{ch3:subs_def-ring}. Set\n$$\n\\mathbf{g}_R=\\mathfrak{gl}_N(R),\\quad \\widehat\\mathbf{g}_R=\\widehat{\\mathfrak{gl}}_N(R)=\\mathfrak{gl}_N(R)[t,t^{-1}]\\oplus R\\bm{1}\\oplus R\\partial.\n$$\nFor $i,j\\in[1,N]$ let $e_{i,j}\\in\\mathbf{g}_R$ denote the matrix unit. Let $\\mathbf{h}_R\\subset\\mathbf{g}_R$ be the Cartan subalgebra generated by the $e_{i,i}$'s, and $\\epsilon_1,\\cdots,\\epsilon_N$ be the basis of $\\mathbf{h}_R^*$ dual to $e_{1,1},\\cdots,e_{N,N}$. Let $P=\\mathbb{Z}\\epsilon_1\\oplus\\cdots\\oplus\\mathbb{Z}\\epsilon_N$ be the weight lattice of $\\mathbf{g}_R$. We identify $P$ with $\\mathbb{Z}^N$.\nLet $\\Pi,~\\widehat \\Pi\\subset \\mathbf{h}_R^*$ be the sets of simple roots of $\\mathbf{g}_R$ and $\\widehat \\mathbf{g}_R$. Let $W=\\mathfrak{S}_N$ be the Weyl group of $\\mathbf{g}_R$ and $\\widetilde W=W\\ltimes \\mathbb{Z}\\Pi$, $\\widehat W=W\\ltimes P$ be the affine and the extended affine Weyl groups.\n\nRecall that $I\\in\\mathbb{Z}\/e\\mathbb{Z}$.\nWe still consider the quiver $\\Gamma=(I,H)$ as in Section \\ref{ch3:subs_not-e-e+1}. In particular, we have $X_I=X_e$, $P_I=P_e$, see Section \\ref{ch3:subs_KM-quiv} for the notation. Then we define the element $\\mathrm{\\mathbf{wt}}_e(\\lambda)\\in X_I$ given by\n$$\n\\mathrm{\\mathbf{wt}}_e(\\lambda)=\\sum_{s=1}^N\\varepsilon_{\\lambda_r},\n$$\nwhere we write $\\varepsilon_{\\lambda_r}$ for $\\varepsilon_{(\\lambda_r~\\mathrm{mod}~e)}$.\nWe will abbreviate\n\\begin{equation}\n\\label{ch3:eq_def-P[mu]}\nP[\\mu]=\\{\\lambda\\in P ; ~\\mathrm{\\mathbf{wt}}_e(\\lambda)=\\mu\\}.\n\\end{equation}\n\nSimilarly, we consider the weight\n$$\n\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda)=\\sum_{r=1}^N\\varepsilon_{\\lambda_r}+(\\sum_{r=1}^N\\lambda_r)\\delta\\in X^\\delta_e.\n$$\n\n\nFinally, let $X_I[N]\\subset X_I$ be the subset given by\n$$\nX_I[N]=\\{\\mu=\\sum_{r=1}^e\\mu_r\\varepsilon_r\\in X_I;~ \\mu_r\\geqslant 0,~ \\sum_{r=1}^e\\mu_r=N\\}.\n$$\nWe may identify $\\mu$ with the tuple $(\\mu_1,\\cdots,\\mu_e)$ if no confusion is possible.\n\n\n\n\nNow, consider the Cartan subalgebra $\\widehat\\mathbf{h}_R=\\mathbf{h}_R\\oplus R\\bm{1}\\oplus R\\partial$ of $\\widehat\\mathbf{g}_R$. Let $\\Lambda_0$ and $\\delta$ be the elements of $\\widehat\\mathbf{h}_R^*$ defined by\n$$\n\\delta(\\partial)=\\Lambda_0(\\bm{1})=1,\\quad \\delta(\\mathbf{h}_R\\oplus R\\bm{1})=\\Lambda_0(\\mathbf{h}_R\\oplus R\\partial)=0.\n$$\nLet $(\\bullet,\\bullet)\\colon \\widehat\\mathbf{h}_R^*\\times \\widehat\\mathbf{h}_R^*\\to R$ be the bilinear form such that\n$$\n\\lambda(\\alpha_r^\\vee)=(\\lambda,\\alpha_r),\\quad \\lambda(\\partial)=(\\lambda,\\Lambda_0),\\qquad \\forall\\lambda\\in \\widehat\\mathbf{h}^*_R.\n$$\n\n\nSet $P_R=P\\otimes_{\\mathbb{Z}}R$. Given a composition $\\nu=(\\nu_1,\\cdots,\\nu_l)$ of $N$, we define\n$$\n\\begin{array}{lll}\n\\rho&=&(0,-1,\\cdots,-N+1),\\\\\n\\rho_\\nu&=&(\\nu_1,\\nu_1-1\\cdots,1,\\nu_2,\\cdots,1,\\cdots,\\nu_l,\\cdots,1),\\\\\n\\tau&=&(\\tau_1^{\\nu_1},\\cdots,\\tau_l^{\\nu_l}),\n\\end{array}\n$$\nwhere $\\tau_r^{\\nu_r}$ means $\\nu_r$ copies of $\\tau_r$.\nSet also\n\\begin{equation}\n\\label{ch3:eq_lambda-tilde}\n\\widehat\\rho=\\rho+N\\Lambda_0, \\quad\n\\widetilde\\lambda=\\lambda+\\tau+z_\\lambda\\delta-(N+\\kappa)\\Lambda_0,\n\\end{equation}\nwhere $z_\\lambda=(\\lambda,2\\rho+\\lambda)\/2{\\kappa}$. Denote by $\\widehat{\\mathbf{p}}_{R,\\nu}$ the parabolic subalgebra of $\\widehat{\\mathbf{g}}_R$ of parabolic type $\\nu$.\nFor a $\\nu$-dominant weight $\\lambda\\in P$ let $\\Delta(\\lambda)_R$ be the parabolic Verma module with highest weight $\\widetilde\\lambda$ and $\\Delta_R^{\\lambda}=\\Delta(\\lambda-\\rho)_R$.\nWe will also skip the subscript $R$ when $R=\\mathbb{C}$.\n\n\\subsection{Extended affine Weyl groups}\n\\label{ch3:subs_ext-aff}\nAssume that $R=\\mathbb{C}$.\nIn this section we discuss some combinatorial aspects of the $\\widehat W$-action on $\\widehat\\mathbf{h}^*$.\n\nThe group $\\widehat W$ is generated by $\\{\\pi,s_i;~i\\in\\mathbb{Z}\/N\\mathbb{Z}\\}$ modulo the relations\n$$\n\\begin{array}{ccccc}\ns_i^2&=&1,\\\\% \\quad&\\forall i\\in\\mathbb{Z}\/N\\mathbb{Z},\\\\\ns_is_j&=&s_js_i \\quad&\\forall i\\ne j\\pm 1,\\\\\ns_is_{i+1}s_i&=&s_{i+1}s_is_{i+1},\\\\% \\quad&\\forall i\\in\\mathbb{Z}\/N\\mathbb{Z},\\\\\n\\pi s_{i+1}&=&s_{i}\\pi\n\\end{array}\n$$\n\nLet $\\widetilde W$ be the subgroup of $\\widehat W$ generated by $\\{s_i;~i\\in \\mathbb{Z}\/N\\mathbb{Z}\\}$.\nThe group $\\widehat W$ acts on $P$ in the following way:\n\\begin{itemize}\n \\item[\\textbullet] $s_r$ switches of the $r$th and $(r+1)$th components of $\\lambda$ if $r\\ne 0$,\n \\item[\\textbullet] $s_0(\\lambda_1,\\cdots,\\lambda_N)=(\\lambda_N-e,\\lambda_2,\\cdots,\\lambda_{N-1},\\lambda_1+e)$,\n \\item[\\textbullet]\n$\\pi(\\lambda_1,\\cdots,\\lambda_N)=(\\lambda_2,\\cdots,\\lambda_N,\\lambda_1+e)$.\n\\end{itemize}\nWe will call this action of $\\widehat W$ on $P$ the negative $e$-\\emph{action}. We will always consider only negative actions of $\\widehat W$ on $P$ up to Section \\ref{ch3:subs_dual-funct-in-O}. So we can skip the word \"negative\". We may write $P^{(e)}=P$ to stress that we consider the $e$-action of $\\widehat W$ on $P$. The map\n$$\nP^{(e)}\\to \\widehat\\mathbf{h}^*,\\quad \\lambda\\mapsto\\widetilde{\\lambda-\\rho}+\\widehat\\rho\n$$\nis $\\widehat W$-invariant. This means that the weights $\\lambda_1,\\lambda_2\\in P$ are in the same $\\widehat W$-orbit if and only if the highest weights of the Verma modules $\\Delta^{\\lambda_1}$ and $\\Delta^{\\lambda_2}$ are linked with respect to the Weyl group $\\widehat W$, see \\cite[Sec.~3.2]{SVV} and \\cite[Sec.~2.3]{Fie-str} for more details about linkage.\nNote that $P=\\coprod_{\\mu\\in X_I[N]} P[\\mu]$ is the decomposition of $P$ into $\\widehat W$-orbits with respect to the $e$-action. An element $\\lambda\\in P$ is $e$-\\emph{anti-dominant} if $\\lambda_1\\leqslant \\lambda_2\\leqslant\\cdots\\leqslant\\lambda_N\\leqslant \\lambda_1+e$.\n\nRecall the map $\\Upsilon\\colon\\mathbb{Z}\\to\\mathbb{Z}$ from (\\ref{ch3:eq_upsilon}).\nApplying $\\Upsilon$ coordinate by coordinate to the elements of $P$ we get a map $\\Upsilon\\colon P^{(e)}\\to P^{(e+1)}$.\n\n\\smallskip\n\\begin{lem}\nThe map $\\Upsilon\\colon P^{(e)}\\to P^{(e+1)}$ is $\\widehat W$-invariant and takes $e$-anti-dominant weights to $(e+1)$-anti-dominant weights.\n\\qed\n\\end{lem}\n\n\\subsection{The standard representation of $\\widetilde{\\mathfrak{sl}}_e$}\n\\label{ch3:subs_stand-rep-aff}\n\nLet $e_i$, $f_i$, $h_i$ the generators of the complex Lie algebra $\\widetilde{\\mathfrak{sl}}_e=\\mathfrak{sl}_e\\otimes\\mathbb{C}[t,t^{-1}]\\oplus\\mathbb{C} \\bm{1}$.\nLet $V_e$ be a $\\mathbb{C}$-vector spaces with canonical basis $\\{v_1,\\cdots,v_e\\}$ and set $U_e=V_e\\otimes \\mathbb{C}[z,z^{-1}]$. The vector space $U_e$ has a basis $\\{u_r;~r\\in\\mathbb{Z}\\}$ where $u_{a+eb}=v_a\\otimes z^{-b}$ for $a\\in[1,e]$, $b\\in\\mathbb{Z}$. It has a structure of an $\\widetilde{\\mathfrak{sl}}_e$-module such that\n$$\nf_i(u_r)=\\delta_{i\\equiv r}u_{r+1},\\quad e_i(u_r)=\\delta_{i\\equiv r-1}u_{r-1}.\n$$\nLet $\\{v'_1,\\cdots,v'_{e+1}\\}$, $\\{u'_r;r\\in\\mathbb{Z}\\}$ denote the bases of $V_{e+1}$ and $U_{e+1}$.\n\nFix an integer $0\\leqslant kk.\n\\end{array}\\right.\n$$\nIt yields an inclusion $\\mathfrak{sl}_e\\subset\\mathfrak{sl}_{e+1}$ such that\n$$\ne_r\\mapsto\n\\left\\{\\begin{array}{rl}\ne_r &\\mbox{ if }r\\in[1,k-1],\\\\\n{[e_k,e_{k+1}]} &\\mbox{ if }r=k,\\\\\ne_{r+1} &\\mbox{ if }r\\in[k+1,e-1],\n\\end{array}\\right.\n$$\n$$\nf_r\\mapsto\n\\left\\{\\begin{array}{rl}\nf_r &\\mbox{ if }r\\in[1,k-1],\\\\\n{[f_{k+1},f_k]} &\\mbox{ if }r=k,\\\\\nf_{r+1} &\\mbox{ if }r\\in[k+1,e-1],\n\\end{array}\\right.\n$$\n$$\nh_r\\mapsto\n\\left\\{\\begin{array}{rl}\nh_r &\\mbox{ if }r\\in[1,k-1],\\\\\nh_k+h_{k+1} &\\mbox{ if }r=k,\\\\\nh_{r+1} &\\mbox{ if }r\\in[k+1,e-1].\n\\end{array}\\right.\n$$\n\n\nThis inclusion lifts uniquely to an inclusion $\\widetilde{\\mathfrak{sl}}_e\\subset\\widetilde{\\mathfrak{sl}}_{e+1}$ such that\n$$\ne_0\\mapsto\n\\left\\{\\begin{array}{rl}\ne_0 &\\mbox{ if }k\\ne 0,\\\\\n{[e_0,e_1]} &\\mbox{ else},\\\\\n\\end{array}\\right.\n$$\n$$\nf_0\\mapsto\n\\left\\{\\begin{array}{rl}\nf_0 &\\mbox{ if }k\\ne 0,\\\\\n{[f_1,f_0]} &\\mbox{ else},\\\\\n\\end{array}\\right.\n$$\n$$\nh_0\\mapsto\n\\left\\{\\begin{array}{rl}\nh_0 &\\mbox{ if }k\\ne 0,\\\\\n{h_0+h_1} &\\mbox{ else}.\\\\\n\\end{array}\\right.\n$$\n\nConsider the inclusion $U_e\\subset U_{e+1}$ such that $u_{r}\\mapsto u'_{\\Upsilon(r)}$.\n\n\\smallskip\n\\begin{lem}\nThe embeddings $V_e\\subset V_{e+1}$ and $U_e\\subset U_{e+1}$ are compatible with the actions of $\\mathfrak{sl}_e\\subset \\mathfrak{sl}_{e+1}$ and $\\widetilde{\\mathfrak{sl}}_e\\subset \\widetilde{\\mathfrak{sl}}_{e+1}$ respectively.\n\\qed\n\\end{lem}\n\n\n\n\n\n\n\\subsection{Categorical representations}\n\\label{ch3:subs_categ-action}\nLet $R$ be a $\\mathbb{C}$-algebra. Fix an invertible element $q\\in R$, $q\\ne 1$. Let $\\mathcal{C}$ be an exact $R$-linear category.\n\n\\smallskip\n\\begin{df}\nA \\emph{representation datum} in $\\mathcal{C}$ is a tuple $(E,F,X,T)$ where $(E,F)$ is a pair of exact functors $\\mathcal{C}\\to\\mathcal{C}$ and $X\\in\\mathrm{End}(F)^{\\rm op}$, $T\\in\\mathrm{End}(F^2)^{\\rm op}$ are endomorphisms of functors such that\nfor each $d\\in\\mathbb{N}$, there is an $R$-algebra homomorphism $\\psi_d\\colon H_{d,R}(q)\\to \\mathrm{End}(F^d)^{\\rm op}$ given by\n$$\n\\begin{array}{ll}\nX_r\\mapsto F^{d-r}XF^{r-1} &\\forall r\\in[1,d],\\\\\nT_r\\mapsto F^{d-r-1}TF^{r-1} &\\forall r\\in[1,d-1].\n\\end{array}\n$$\n \\end{df}\n\n\\smallskip\nNow, assume that $R=\\bfk$ is a field. Assume that $\\mathcal{C}$ is a $\\mathrm{Hom}$-finite abelian category. Let $\\mathscr{F}$ be a subset of $\\bfk$ such that $q^\\mathbb{Z}\\mathscr{F}\/q^\\mathbb{Z}$ is finite. We view $\\mathscr{F}$ as the vertex set of a quiver with an arrow $i\\to j$ if and only if $j=qi$.\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_df-repl-F-by-E}\nAssume that we have a representation datum in a $\\bfk$-linear category $\\mathcal{C}$ such that the functors $E$ and $F$ are biadjoint. Then by adjointness we have an algebra isomorphism $\\mathrm{End}(E^d)\\simeq \\mathrm{End}(F^d)^{\\rm op}$. In particular we get an algebra homomorphism $H_{d,\\bfk}\\to \\mathrm{End}(E^d)$.\n\\end{rk}\n\n\\smallskip\n\\begin{df}\n\\label{ch3:def-categ_action-Hecke}\nAn $\\mathfrak{sl}_{\\mathscr{F}}$-categorical representation in $\\mathcal{C}$ is the datum of a representation datum $(E,F,X,T)$ and a decomposition $\\mathcal{C}=\\bigoplus_{\\mu\\in X_\\mathscr{F}}\\mathcal{C}_\\mu$ satisfying the conditions $(a)$ and $(b)$ below. For $i\\in\\mathscr{F}$ let $E_i$, $F_i$ be endofunctors of $\\mathcal{C}$ such that for each $M\\in\\mathcal{C}$ the objects $E_i(M)$, $F_i(M)$ are the generalized $i$-eigenspaces of $X$ acting on $E(M)$ and $F(M)$ respectively, see also Remark \\ref{ch3:rk_df-repl-F-by-E}.\nWe assume that\n\\begin{itemize}\n \\item[$(a)$] $F=\\bigoplus_{i\\in\\mathscr{F}}F_i$ and $E=\\bigoplus_{i\\in\\mathscr{F}}E_i$,\n \\item[$(b)$] $E_i(\\mathcal{C}_\\mu)\\subset \\mathcal{C}_{\\mu+\\alpha_i}$ and $F_i(\\mathcal{C}_\\mu)\\subset \\mathcal{C}_{\\mu-\\alpha_i}$.\n\\end{itemize}\n\\end{df}\n\n\\begin{rk}\n\\label{ch3:rk_cat-res-loc-H}\nIn this case the homomorphism $\\psi_d\\colon H_{d,\\bfk}\\to \\mathrm{End}(F^d)^{\\rm op}$ extends to a homomorphism $\\widehat H_{d,\\bfk}\\to \\mathrm{End}(F^d)^{\\rm op}$, where $\\widehat H_{d,\\bfk}$ is as in Section \\ref{ch3:subs_isom-KLR-Hecke-gen}.\n\\end{rk}\n\n\n\n\n\n\\medskip\nThere is an alternative definition of a categorical representation, where the affine Hecke algebra $H_{d,\\bfk}(q)$ is replaced by a KLR algebra. In this section we allow $\\Gamma=(I,H)$ be an arbitrary quiver without $1$-loops.\n\n\\smallskip\n\\begin{df}\n\\label{ch3:def-categ_action-KLR} A $\\mathfrak{g}_I$-categorical representation $(E,F,x,\\tau)$ in $\\mathcal{C}$ is the following data:\n\\begin{itemize}\n \\item[(1)] a decomposition $\\mathcal{C}=\\bigoplus_{\\mu\\in X_I}\\mathcal{C}_\\mu$,\n \\item[(2)] a pair of biadjoint exact endofunctors $(E,F)$ of $\\mathcal{C}$,\n \\item[(3)] morphisms of functors $x\\colon F\\to F$, $\\tau\\colon F^2\\to F^2$,\n \\item[(4)] decompositions $E=\\bigoplus_{i\\in I}E_i$, $F=\\bigoplus_{i\\in I}F_i$,\n\\end{itemize}\nsatisfying the following conditions.\n\\begin{itemize}\n \\item[(a)] We have $E_i(\\mathcal{C}_\\mu)\\subset\\mathcal{C}_{\\mu+\\alpha_i}$, $F_i(\\mathcal{C}_\\mu)\\subset\\mathcal{C}_{\\mu-\\alpha_i}$.\n\n \\item[(b)]\n\nFor each $d\\in\\mathbb{N}$ there is an algebra homomorphism $\\psi_d\\colon R_{d,\\bfk}\\to \\mathrm{End}(F^d)^{\\rm op}$ such that\n$\\psi_d(e({\\mathbf{i}}))$ is the projector to $F_{i_d}\\cdots F_{i_1}$, where ${\\mathbf{i}}=(i_1,\\cdots,i_d)$ and\n$$\n\\psi_d(x_r)=F^{d-r}x F^{r-1},\n\\qquad\n\\psi_d(\\tau_r)=F^{d-r-1}\\tau F^{r-1}.\n$$\n \\item[(c)] For each $M\\in\\mathcal{C}$ the endomorphism of $F(M)$ induced by $x$ is nilpotent.\n\\end{itemize}\n\n\\end{df}\n\nAssume that there is an isomorphism of quivers $I\\simeq\\mathscr{F}$. Then Definitions \\ref{ch3:def-categ_action-Hecke}, \\ref{ch3:def-categ_action-KLR} are equivalent by Proposition \\ref{ch3:prop-isom_Hekce-KLR-widetilde}.\n\n\\subsection{From $\\widetilde{\\mathfrak{sl}}_{e+1}$-categorical representations to $\\widetilde{\\mathfrak{sl}}_{e}$-categorical representations}\n\\label{ch3:subs_cat-lem}\n\n\n\n\n\n\n\n\n\nRecall that we fix $\\Gamma=(I,H)$ and $\\overline\\Gamma=(\\overline I,\\overline H)$ as in Section \\ref{ch3:subs_not-e-e+1}. In particular, we have $X_I=X_e$, $X_{\\overline I}=X_{e+1}$.\n\nAssume that $R=\\bfk$ is a field.\nLet $\\overline\\mathcal{C}$ be a $\\mathrm{Hom}$-finite abelian $\\bfk$-linear category. Let\n$$\n\\overline E=\\overline E_0\\oplus\\overline E_1\\oplus\\cdots\\oplus\\overline E_e,\\qquad \\overline F=\\overline F_0\\oplus\\overline F_1\\oplus\\cdots\\oplus\\overline F_e\n$$\nbe endofunctors defining a categorical $\\widetilde{\\mathfrak{sl}}_{e+1}$-representation in $\\overline\\mathcal{C}$.\nLet $\\overline\\psi_d\\colon R_{d,\\bfk}\\to \\mathrm{End}(\\overline F^d)^{\\rm op}$ be the corresponding algebra homomorphism.\nWe set $\\overline F_{\\mathbf{i}}=\\overline F_{i_d}\\cdots \\overline F_{i_1}$ for any tuple ${\\mathbf{i}}=(i_1,\\cdots,i_d)\\in \\overline I^d$ and $\\overline F_{\\overline\\alpha}=\\bigoplus_{{\\mathbf{i}}\\in \\overline I^{\\overline\\alpha}}\\overline F_{\\mathbf{i}}$ for any element $\\overline\\alpha\\in Q_{\\overline I}^+$.\nIf $|\\overline\\alpha|=d$ let $\\overline \\psi_{\\overline\\alpha}\\colon R_{\\overline\\alpha,\\bfk}\\to \\mathrm{End}(\\overline F_{\\overline\\alpha})^{\\rm op}$ be the $\\overline\\alpha$-component of $\\overline \\psi_d$.\n\nNow, recall the notation $X_{\\overline I}^+$ from (\\ref{ch3:eq_X+}). Assume that\n\\begin{equation}\n\\label{ch3:eq_ass-Cmu-0}\n\\overline\\mathcal{C}_\\mu=0, \\quad\\forall\\mu\\in X_{\\overline I}\\backslash X^+_{\\overline I}.\n\\end{equation}\nFor $\\mu\\in X^+_I$ set $\\mathcal{C}_\\mu=\\overline\\mathcal{C}_{\\phi(\\mu)}$, where the map $\\phi$ is as in (\\ref{ch3:eq_phi(mu)}). Let $\\mathcal{C}=\\bigoplus_{\\mu\\in X^+_I}\\mathcal{C}_\\mu$.\n\n\\smallskip\n\\begin{rk}\n$(a)$ $\\mathcal{C}$ is stable by $\\overline F_i$, $\\overline E_i$ for each $i\\ne k,k+1$,\n\n$(b)$ $\\mathcal{C}$ is stable by $\\overline F_{k+1}\\overline F_k$, $\\overline E_k\\overline E_{k+1}$,\n\n$(c)$ $\\overline F_{i_d}\\overline F_{i_{d-1}}\\cdots\\overline F_{i_1}(M)=0$ for each $M\\in \\mathcal{C}$ whenever the sequence $(i_1,\\cdots,i_d)$ is unordered (see Sections \\ref{ch3:subs_bal-quot} and \\ref{ch3:subs_not-e-e+1}).\n\\end{rk}\n\n\\smallskip\nConsider the following endofunctors of $\\mathcal{C}$:\n$$\nE_i=\n\\left\\{\n\\begin{array}{lll}\n\\restr{\\overline E_i}{\\mathcal{C}} &\\mbox{ if } 0\\leqslant i\\lambda_{r+1}$ for each $r\\in[1,N-1]\\backslash\\{\\nu_1,\\nu_1+\\nu_2,\\cdots,\\nu_1+\\cdots+\\nu_l\\}$.\nLet $P^\\nu$ be the set of $\\nu$-dominant weights of $P$. Set also $P^\\nu[\\mu]=P^\\nu\\cap P[\\mu]$, where $P[\\mu]$ is as in (\\ref{ch3:eq_def-P[mu]}). Let $O^\\nu_R$ be the $R$-linear abelian category of finitely generated $\\widehat\\mathbf{g}_R$-modules $M$ which are weight $\\widehat\\mathbf{h}_R$-modules, and such that the $\\widehat\\mathbf{p}_{R,\\nu}$-action on $M$ is locally finite over $R$, and the highest weight of any subquotient of $M$ is of the form $\\widetilde\\lambda$ with $\\lambda\\in P^\\nu$, where $\\widetilde\\lambda$ is defined in (\\ref{ch3:eq_lambda-tilde}).\nLet $O_{\\mu,R}^\\nu$ be the Serre subcategory of $O^\\nu_R$ generated by the modules $\\Delta_R^\\lambda$ for all $\\lambda\\in P^\\nu[\\mu]$. Let $O^{\\nu,\\Delta}_{\\mu,R}\\subset O^\\nu_{\\mu,R}$ be the full subcategory of $\\Delta$-filtered modules. \n\nWe will omit the upper index $\\nu$ if $\\nu=(1,1,\\cdots,1)$. Assume $\\lambda\\in P$. In the case if $R=\\bfk$ is a field we denote by $L(\\lambda)_\\bfk$ the simple quotient of $\\Delta(\\lambda)_\\bfk$.\nIn the case if $R$ is local with residue fields $\\bfk$, the simple module $L(\\lambda)_\\bfk\\in O_\\bfk$ has a simple lift $L(\\lambda)_R\\in O_R$ such that $L(\\lambda)_\\bfk=\\bfk\\otimes_RL(\\lambda)_R$ (see \\cite[Sec.~2.2]{Fie-cen}). Set also $L^\\lambda_R=L(\\lambda-\\rho)_R$.\n\nLet $U_e$, $V_e$ be as in Section \\ref{ch3:subs_stand-rep-aff}.\nSet $\\wedge^\\nu U_e=\\wedge^{\\nu_1}U_e\\otimes\\cdots\\otimes \\wedge^{\\nu_l} U_e$.\nFor each\n$\\lambda\\in P^\\nu$ define the following element in $\\wedge^\\nu U_e$:\n$$\n\\wedge_\\lambda^\\nu=(u_{\\lambda_1}\\wedge\\cdots\\wedge u_{\\lambda_{\\nu_1}})\\otimes\\cdots\\otimes(u_{\\lambda_1+\\cdots+\\lambda_{l-1}+1}\\wedge\\cdots\\wedge u_{\\lambda_{\\nu_1}+\\cdots+\\lambda_{\\nu_l}}).\n$$\nThe obvious $\\widetilde{\\mathfrak{sl}}_e$-action on $U_e$ yields an $\\widetilde{\\mathfrak{sl}}_e$-action on $\\wedge^\\nu U_e$. We identify the abelian group $X_e\/\\mathbb{Z}(\\varepsilon_1+\\cdots+\\varepsilon_e)$ with the weight lattice of $\\mathfrak{sl}_e$. In particular each element $\\mu\\in X_e$ yields a weight of $\\mathfrak{sl}_e$. For each $\\mu\\in X_e[N]$ let $(\\wedge^\\nu U_e)_\\mu$ be the weight space in $\\wedge^\\nu U_e$ corresponding to $\\mu$.\n\n\n\nSet $O^\\nu_{-e,R}=\\bigoplus_{\\mu\\in X_e[N]}O^\\nu_{\\mu,R}$ and similarly for $O^{\\nu,\\Delta}_{-e,R}$. Now we define a representation datum in the category $O^{\\nu,\\Delta}_{-e,R}$. See \\cite[Sec.~5.4]{RSVV} for more details. From now on, we assume that $R$ is either a local analytic deformation ring in general position of dimension $\\leqslant 2$ or a field. Let $\\bfk$ and $K$ be its residue field and field of fractions respectively. For an exact category $\\mathcal{C}$ denote by $[\\mathcal{C}]$ its complexified Grothendieck group. The following proposition holds, see \\cite{RSVV}.\n\\begin{prop}\n\\label{ch3:prop_functors-on-O-gen}\nThere is a pair of exact endofunctors $E$, $F$ of $O^{\\nu,\\Delta}_{-e,R}$ such that the following properties hold.\n\\begin{itemize}\n \\item[$(a)$] The functors $E$, $F$\ncommute with the base changes $K\\otimes_R\\bullet$, $\\bfk\\otimes_R\\bullet$.\n \\item[$(b)$] $(O^{\\nu,\\Delta}_{-e,R},E,F)$ admits a representation datum structure.\n \\item[$(c)$] The pair of functors $(E,F)$ is biadjoint. It extends to a pair of biadjoint functors $O^{\\nu}_{-e,R}\\to O^{\\nu}_{-e,R}$ if $R$ is a field.\n \\item[$(d)$] There are decompositions $E=\\bigoplus_{i\\in I}E_i$, $F=\\bigoplus_{i\\in I}F_i$ such that\n$$\nE_i(O_{\\mu,R}^{\\nu,\\Delta})\\subset O_{\\mu+\\alpha_i,R}^{\\nu,\\Delta},\\qquad F_i(O_{\\mu,R}^{\\nu,\\Delta})\\subset O_{\\mu-\\alpha_i,R}^{\\nu,\\Delta}.\n$$\n \\item[$(e)$] There is a vector space isomorphism $[O^{\\nu,\\Delta}_{\\mu,R}]\\simeq (\\wedge^\\nu U)_\\mu$ such that the functors $F_i$, $E_i$ act on $[O^{\\nu,\\Delta}_{-e,R}]=\\bigoplus_{\\mu\\in X_e[N]}[O^{\\nu,\\Delta}_{\\mu,R}]$ as the standard generators $e_i$, $f_i$ of $\\widetilde{\\mathfrak{sl}}_e$.\n \\item[$(f)$] If $R=\\bfk$ with the trivial deformation ring structure, then $E_i$, $F_i$ yield a categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ in $O^\\nu_{-e,\\bfk}$.\n\\qed\n\\end{itemize}\n\n\\end{prop}\n\n\n\n\n\n\n\\smallskip\nFix $k\\in [0,e-1]$. Recall the map $\\Upsilon\\colon P\\to P$ from Section \\ref{ch3:subs_ext-aff} and the map $\\phi\\colon X_I\\to X_{\\overline I}$ from (\\ref{ch3:eq_phi(mu)}).\nSet $\\mu'=\\mu-\\alpha_k$ and $\\overline\\mu=\\phi(\\mu)$.\nSet, $\\overline\\mu^0=\\overline\\mu-\\overline\\alpha_{k}$ and $\\overline\\mu'=\\overline\\mu-\\overline\\alpha_{k}-\\overline\\alpha_{k+1}$. Note that $\\Upsilon(P[\\mu])\\subset P[\\overline\\mu]$.\nFor $k\\ne 0$ we have\n$$\n\\begin{array}{llllll}\n\\mu&=&(\\mu_1,\\cdots,\\mu_k,&&\\mu_{k+1},&\\cdots,\\mu_e),\\\\\n\\mu '&=&(\\mu_1,\\cdots,\\mu_k-1,&&\\mu_{k+1}+1,&\\cdots,\\mu_e),\\\\\n\\overline\\mu&=&(\\mu_1,\\cdots,\\mu_k,&0,&\\mu_{k+1},&\\cdots,\\mu_e),\\\\\n\\overline\\mu^0&=&(\\mu_1,\\cdots,\\mu_k-1,&1,&\\mu_{k+1},&\\cdots,\\mu_e),\\\\\n\\overline\\mu '&=&(\\mu_1,\\cdots,\\mu_k-1,&0,&\\mu_{k+1}+1,&\\cdots,\\mu_e).\\\\\n\\end{array}\n$$\nFor an $e$-tuple $\\mathbf{a}=(a_1,\\cdots,a_e)$ of non-negative integers we set $1_\\mathbf{a}=(1^{a_1},\\cdots,e^{a_e})$. Note that we have\n\\begin{equation}\n\\label{ch3:eq_Upsilon(1)}\n\\Upsilon(1_{\\mu})=1_{\\overline\\mu},\\qquad \\Upsilon(1_{\\mu'})=1_{\\overline\\mu'}.\n\\end{equation}\n\n\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_choice-of-1_mu}\nThe set $P[\\mathbf{a}]$ is a $\\widehat W$-orbit in $P^{(e)}$. It is a union of $\\widetilde W$-orbits and each of them contains a unique $e$-anti-dominant weight. By definition, the weight $1_\\mathbf{a}\\in P[\\mathbf{a}]$ is $e$-anti-dominant. However, there is no canonical way to choose $1_\\mathbf{a}$.\nIn the case $k=0$ we need to change our convention and to set $1_\\mathbf{a}=(0^{a_e},1^{a_e},\\cdots,(e-1)^{a_{e-1}})$. This change is necessary to have (\\ref{ch3:eq_Upsilon(1)}).\n\\end{rk}\n\n\n\nFirst, assume that $l=N$ and $\\nu=(1,1,\\cdots,1)$.\n\n\\begin{lem}\nThere is an equivalence of categories $\\theta_{\\mu}^{\\overline\\mu}\\colon O_{\\mu,R}\\to O_{\\overline\\mu,\\overline R}$ such that $\\theta_\\mu^{\\overline\\mu}(\\Delta_{R}^{\\lambda})\\simeq\\Delta_{\\overline R}^{\\Upsilon(\\lambda)}$. It restricts to an equivalence of categories $\\theta_{\\mu}^{\\overline\\mu}\\colon O^{\\Delta}_{\\mu,R}\\to O^{\\Delta}_{\\overline\\mu,\\overline R}$.\n\\end{lem}\n\\begin{proof}[Proof]\nFor each $n\\in \\mathbb{Z}$ the weight $\\pi^n(1_\\mu)$ is $e$-anti-dominant.\nLet $\\mathcal{O}_{\\pi^n(1_\\mu),R}\\subset O_{\\mu,R}$ be the Serre subcategory generated by the Verma modules of the form $\\Delta_{R}^{w\\pi^n(1_\\mu)}$ with $w\\in\\widetilde W$.\nWe have\n\\begin{equation}\n\\label{ch3:eq_dec-O-What-Wtilde}\nO_{\\mu,R}=\\bigoplus_{n\\in\\mathbb{Z}}\\mathcal{O}_{\\pi^n(1_\\mu),R}.\n\\end{equation}\nThe weights $\\pi^n(1_\\mu)\\in P^{(e)}$ and $\\pi^n(1_{\\overline\\mu})\\in P^{(e+1)}$ have the same stabilizers in $\\widetilde W$. Thus by \\cite[Thm.~11]{Fie-str} (see also \\cite[Prop.~5.24]{RSVV}) we have an equivalence of categories\n$$\n\\mathcal{O}_{\\pi^n(1_\\mu),R}\\simeq \\mathcal{O}_{\\pi^n(1_{\\overline\\mu}),\\overline R},\\qquad \\Delta_{R}^{w\\pi^n(1_\\mu)}\\mapsto \\Delta_{\\overline R}^{w\\pi^n(1_{\\overline\\mu})} ~~\\forall w\\in\\widetilde W.\n$$\nTaking the sum by all $n\\in\\mathbb{Z}$ we get an equivalence of categories\n$$\n\\theta_{\\mu}^{\\overline\\mu}\\colon O_{\\mu,R}\\simeq O_{\\overline\\mu,\\overline R},\\qquad \\Delta_{R}^{w(1_\\mu)}\\mapsto \\Delta_{\\overline R}^{w(1_{\\overline\\mu})} ~~\\forall w\\in\\widehat W.\n$$\nRecall that we have $\\Upsilon(1_\\mu)=1_{\\overline\\mu}$.\nThus by $\\widehat W$-invariance of $\\Upsilon$ we get\n$$\n\\theta_\\mu^{\\overline\\mu}(\\Delta_{R}^{\\lambda})\\simeq\\Delta_{\\overline R}^{\\Upsilon(\\lambda)}~~\\forall \\lambda\\in P[\\mu].\n$$\n\\end{proof}\n\n\n\n\n\n\\begin{rk}\nNotice that \\cite{Fie-str} yields an equivalence of categories over a field. It is explained in \\cite{RSVV} how to get from it an equivalence of categories $O^\\Delta_R$. First, comparing the endomorphisms of projective generators one gets an equivalence of the abelian categories $O_R$. Then, comparing the highest weight structure in both sides, we deduce an equivalence of additive categories $O^\\Delta_R$.\n\n\\end{rk}\n\n\n\\smallskip\nThe equivalence $\\theta_{\\mu}^{\\overline\\mu}$ restricts to equivalences $O^\\nu_{\\mu,R}\\simeq O^\\nu_{\\overline\\mu,\\overline R}$ and $O^{\\nu,\\Delta}_{\\mu,R}\\simeq O^{\\nu,\\Delta}_{\\overline\\mu,\\overline R}$ for each parabolic type $\\nu$, see \\cite[Sec.~5.7.2]{RSVV}. We will also call this equivalence $\\theta_{\\mu}^{\\overline\\mu}$. We obtain equivalences of categories $\\theta_{\\overline\\mu'}^{\\mu'}\\colon O^\\nu_{\\overline\\mu',\\overline R}\\simeq O^\\nu_{\\mu',R}$ and $\\theta_{\\overline\\mu'}^{\\mu'}\\colon O^{\\nu,\\Delta}_{\\overline\\mu',\\overline R}\\simeq O^{\\nu,\\Delta}_{\\mu',R}$ in a similar way.\n\n\\smallskip\n\\begin{conj}\n\\label{ch3:conj_F_k-decomp}\nThere are the following commutative diagrams\n$$\n\\begin{diagram}\n\\node{O^{\\nu,\\Delta}_{\\overline\\mu,\\overline R}} \\arrow{e,t}{\\overline F_k}\n\\node{O^{\\nu,\\Delta}_{\\overline\\mu^0,\\overline R}} \\arrow{e,t}{\\overline F_{k+1}}\n\\node{\\overline O^{\\nu,\\Delta}_{\\mu',\\overline R}} \\arrow{s,r}{\\theta_{\\overline\\mu'}^{\\mu'}} \\\\\n\\node{O^{\\nu,\\Delta}_{\\mu,R}} \\arrow{n,l}{\\theta_\\mu^{\\overline\\mu}}\n\\arrow[2]{e,b}{F_k} \\node[2]{O^{\\nu,\\Delta}_{\\mu',R}}\n\\end{diagram}\n$$\nand\n$$\n\\begin{diagram}\n\\node{O^{\\nu,\\Delta}_{\\overline\\mu,\\overline R}} \\arrow{s,l}{\\theta^\\mu_{\\overline\\mu}}\n\\node{O^{\\nu,\\Delta}_{\\overline\\mu^0,\\overline R}} \\arrow{w,t}{\\overline E_k} \n\\node{O^{\\nu,\\Delta}_{\\overline\\mu',\\overline R}} \\arrow{w,t}{\\overline E_{k+1}} \\\\\n\\node{O^{\\nu,\\Delta}_{\\mu,R}} \\node[2]{O^{\\nu,\\Delta}_{\\mu',R}} \\arrow{n,r}{\\theta^{\\overline\\mu'}_{\\mu'}} \\arrow[2]{w,b}{E_k}\n\\end{diagram}.\n$$\n\\end{conj}\n\n\\subsection{The commutativity in the Grothendieck groups}\n\\label{ch3:subs_comm-Groth}\n\nWe have the following commutative diagram of vector spaces\n$$\n\\begin{CD}\n[O^{\\nu,\\Delta}_{-e-1,\\overline R}] @>>> \\wedge^\\nu U_{e+1}\\\\\n@A{\\oplus_\\mu\\theta_\\mu^{\\overline\\mu}}AA @AAA\\\\\n[O^{\\nu,\\Delta}_{-e,R}] @>>> \\wedge^\\nu U_e,\n\\end{CD}\n$$\nwhere the horizontal maps are respectively the isomorphisms of $\\widetilde{\\mathfrak{sl}}_e$-modules and $\\widetilde{\\mathfrak{sl}}_{e+1}$-modules from Proposition \\ref{ch3:prop_functors-on-O-gen} $(e)$, the right vertical map is given by the injection $U_{e}\\to U_{e+1}$ in Section \\ref{ch3:subs_stand-rep-aff}. Moreover, the right vertical map is a morphism of $\\widetilde{\\mathfrak{sl}}_e$-modules where $\\wedge^\\nu U_{e+1}$ is viewed as an $\\widetilde{\\mathfrak{sl}}_e$-module via the inclusion $\\widetilde{\\mathfrak{sl}}_e\\subset \\widetilde{\\mathfrak{sl}}_{e+1}$ introduced in Section \\ref{ch3:subs_stand-rep-aff}. Thus $\\oplus_\\mu\\theta_\\mu^{\\overline\\mu}\\colon [O^{\\nu,\\Delta}_{-e,R}] \\to [O^{\\nu,\\Delta}_{-e-1,\\overline R}]$ is a morphism of $\\widetilde{\\mathfrak{sl}}_e$-modules which intertwines\n\n\\smallskip\n\\begin{itemize}\n\\item[\\textbullet] $[E_r]$ with $[\\overline E_r]$,\\quad $[F_r]$ with $[\\overline F_r]$ if $r\\in[1,k-1]$,\n\\item[\\textbullet] $[E_k]$ with $[\\overline E_k\\overline E_{k+1}]-[\\overline E_{k+1}\\overline E_{k}]$,\\quad $[F_k]$ with $[\\overline F_{k+1}\\overline F_k]-[\\overline F_{k}\\overline F_{k+1}]$,\n\\item[\\textbullet] $[E_r]$ with $[\\overline E_{r+1}]$,\\quad $[F_r]$ with $[\\overline F_{r+1}]$ if $r\\in[k+1,e-1]$.\n\n\\smallskip\nIn particular, we see that the diagrams from Conjecture \\ref{ch3:conj_F_k-decomp} commute at the level of Grothendieck groups. Since there is no good notion of projective functors in the affine category $\\mathcal{O}$, this is not enough to prove our conjecture.\n\n\\end{itemize}\n\n\n\\subsection{Partitions}\nA \\emph{partition} of an integer $n\\geqslant 0$ is a tuple of positive integers $(\\lambda_1,\\cdots,\\lambda_s)$ such that $\\lambda_1\\geqslant \\lambda_2\\geqslant\\cdots\\geqslant \\lambda_s$ and $\\sum_{t=1}^s\\lambda_t=n$. Denote by $\\mathcal{P}_n$ the set of all partitions of $n$ and set $\\mathcal{P}=\\coprod_{n\\in\\mathbb{N}}\\mathcal{P}_n$. For a partition $\\lambda=(\\lambda_1,\\cdots,\\lambda_s)$ of $n$, we set $|\\lambda|=n$ and $\\ell(\\lambda)=s$.\nAn $l$-\\emph{partition} of an integer $n\\geqslant 0$ is an $l$-tuple $\\lambda=(\\lambda^{1},\\cdots,\\lambda^{l})$ of partitions of integers $n_1,\\cdots,n_l\\geqslant 0$ such that $\\sum_{t=1}^ln_t=n$. Let $\\mathcal{P}^l_n$ be the set of all $l$-partitions of $n$ and set $\\mathcal{P}^l=\\coprod_{n\\in\\mathbb{N}}\\mathcal{P}^l_n$.\n\n\nA partition $\\lambda$ can be represented by a Young diagram $Y(\\lambda)$ and an $l$-partition $\\lambda=(\\lambda^1,\\cdots,\\lambda^l)$ by an $l$-tuple of Young diagrams $Y(\\lambda)=(Y(\\lambda^1),\\cdots,Y(\\lambda^l))$.\n\nLet $\\lambda\\in\\mathcal{P}^l$ be an $l$-partition. For a box $b\\in Y(\\lambda)$ situated in the $i$th row, $j$th column of the $r$th component we define its \\emph{residue} $\\mathrm{Res}_\\nu(b)\\in I$ as $\\nu_r+j-i$ $\\mathrm{mod}~e$ and its \\emph{deformed residue} $\\widetilde\\mathrm{Res}_\\nu(b)\\in \\widetilde I$ as $(\\nu_r+j-i,r)$.\nSet\n$$\n\\mathrm{Res}_\\nu(\\lambda)=\\sum_{b\\in Y(\\lambda)}\\alpha_{\\mathrm{Res}_\\nu(b)}\\in Q^+_I,\\quad\\widetilde\\mathrm{Res}_\\nu(\\lambda)=\\sum_{b\\in Y(\\lambda)}\\widetilde\\alpha_{\\widetilde\\mathrm{Res}_\\nu(b)}\\in Q^+_{\\widetilde I}.\n$$\nNow for $\\alpha\\in Q^+_I$ and $\\widetilde\\alpha\\in Q^+_{\\widetilde I}$ set\n$$\n\\mathcal{P}^l_\\alpha=\\{\\lambda\\in\\mathcal{P}^l;~\\mathrm{Res}_\\nu(\\lambda)=\\alpha\\},\\qquad \\mathcal{P}^l_{\\widetilde\\alpha}=\\{\\lambda\\in\\mathcal{P}^l;~\\widetilde\\mathrm{Res}_\\nu(\\lambda)=\\widetilde\\alpha\\}.\n$$\nThis notation depends on $\\nu$. We may write $\\mathcal{P}^l_{\\alpha,\\nu}$ and $\\mathcal{P}^l_{\\widetilde\\alpha,\\nu}$ to specify $\\nu$.\nWe have decompositions\n$$\n\\mathcal{P}^l_d=\\bigoplus_{\\alpha\\in Q^+_I, |\\alpha|=d}\\mathcal{P}^l_\\alpha,\\qquad \\mathcal{P}^l_\\alpha=\\bigoplus_{\\widetilde\\alpha\\in Q^+_{\\widetilde I}, \\pi_e(\\widetilde\\alpha)=\\alpha}\\mathcal{P}^l_{\\widetilde\\alpha}.\n$$\n\n\\subsection{The category $\\mathbf{A}$}\n\\label{ch3:subs_cat-bfA}\nLet $\\mathcal{P}^\\nu_d\\subset \\mathcal{P}^l_d$ be the subset of the elements $\\lambda=(\\lambda^1,\\cdots,\\lambda^l)$ such that $\\ell(\\lambda^r)\\leqslant \\nu_r$ for each $r\\in[1,l]$.\nWe can view $\\lambda$ as the weight in $P$ given by\n$$\n(\\lambda^1_1,\\cdots,\\lambda^1_{\\ell(\\lambda^1)},0^{m_1-\\ell(\\lambda^1)},\\lambda^2_1,\\cdots,\\lambda^2_{\\ell(\\lambda^2)},0^{m_2-\\ell(\\lambda^2)},\\cdots,\\lambda^l_1,\\cdots,\\lambda^l_{\\ell(\\lambda^l)},0^{m_l-\\ell(\\lambda^l)}).\n$$\nThen, we set $\\omega(\\lambda)=\\lambda-\\rho+\\rho_\\nu$ in $P$.\n\n\\smallskip\n\\begin{df}\nLet $\\mathbf{A}^\\nu_R[d] \\subset O^\\nu_{-e,R}$ be the Serre subcategory generated by the modules $\\Delta(\\omega(\\lambda))_R$ with $\\lambda\\in \\mathcal{P}^\\nu_d$, see Section \\ref{ch3:subs_aff-Lie}. Denote by $\\mathbf{A}^{\\nu,\\Delta}_R[d]$ the full subcategory of $\\Delta$-filtered modules in $\\mathbf{A}^\\nu_R[d]$.\n\\end{df}\n\n\\smallskip\nWe abbreviate $\\Delta[\\lambda]_R=\\Delta(\\omega(\\lambda))_R$.\nThe restriction of the functor $F$ to the subcategory $\\mathbf{A}^{\\nu,\\Delta}_R[d]$ yields a functor $F\\colon\\mathbf{A}^{\\nu,\\Delta}_R[d]\\to\\mathbf{A}^{\\nu,\\Delta}_R[d+1]$. However, it is not true that $E(\\mathbf{A}^{\\nu,\\Delta}_R[d+1])\\subset \\mathbf{A}^{\\nu,\\Delta}_R[d]$. Nevertheless, we can define a functor $E\\colon\\mathbf{A}^{\\nu,\\Delta}_R[d+1]\\to \\mathbf{A}^{\\nu,\\Delta}_R[d]$ that is left adjoint to $F\\colon \\mathbf{A}^{\\nu,\\Delta}_R[d]\\to\\mathbf{A}^{\\nu,\\Delta}_R[d+1]$, see \\cite[Sec.~5.9]{RSVV}. This can be done in the following way. Let $h$ be the inclusion functor from $\\mathbf{A}^{\\nu,\\Delta}_R[d]$ to ${O^{\\nu,\\Delta}_{-e,R}}$. Abusing the notation, we will use the same symbol for the inclusion functor from $\\mathbf{A}^{\\nu,\\Delta}[d+1]$ to ${O^{\\nu,\\Delta}_{-e,R}}$. Let $h^*$ be the left adjoint functor to $h$. We define the functor $E$ for the category $\\mathbf{A}^{\\nu,\\Delta}_R$ as $h^*Eh$.\n\nThere is a decomposition $\\mathbf{A}^\\nu_R[d]=\\bigoplus_{\\alpha\\in Q^+_I,|\\alpha|=d}\\mathbf{A}^\\nu_R[\\alpha]$, where $\\mathbf{A}^\\nu_R[\\alpha]$ is the Serre subcategory of $\\mathbf{A}^\\nu_R[d]$ generated by the Verma modules $\\Delta[\\lambda]_R$ such that $\\lambda\\in\\mathcal{P}^l_\\alpha$. The functors $E$, $F$ admit decompositions\n$$\nE=\\bigoplus_{i\\in I}E_i,\\qquad F=\\bigoplus_{i\\in I}F_i\n$$\nsuch that for each $\\alpha\\in Q^+_I$ and $i\\in I$ we have\n$$\nE_i(\\mathbf{A}^{\\nu,\\Delta}_R[\\alpha])\\subset \\mathbf{A}^{\\nu,\\Delta}_R[\\alpha-\\alpha_i], \\qquad F_i(\\mathbf{A}^{\\nu,\\Delta}_R[\\alpha])\\subset \\mathbf{A}^{\\nu,\\Delta}_R[\\alpha+\\alpha_i].\n$$\n\nNote that the functor $E_i$ for the category $\\mathbf{A}^{\\nu,\\Delta}_R$ is the restriction of the functor $E_i$ for the category $O^{\\nu,\\Delta}_{-e,R}$ if $i\\ne 0$. Thus for $i\\ne 0$ the pair of functors $(E_i,F_i)$ for the category $\\mathbf{A}^{\\nu,\\Delta}_R$ is biadjoint. But we have only a one-side adjunction $(E_0,F_0)$. Note also that if $R$ is a field, then we can define the functors as above (with the same adjunction properties) for the category $\\mathbf{A}^{\\nu}_R$ instead of $\\mathbf{A}^{\\nu,\\Delta}_R$.\n\n\n\n\n\n\n\n\n\n\n\\subsection{The change of level for $\\mathbf{A}$}\n\\label{ch3:subs_rank-ch-A}\n\n\nFor $\\lambda_1,\\lambda_2\\in P$ we write $\\lambda_1\\geqslant\\lambda_2$ if $(\\lambda_1)_r\\geqslant (\\lambda_2)_r$ for each $r\\in[1,N]$. Here, $(\\lambda_i)_r$ is the $r$th entry of $\\lambda_i$ for each $r$. We identify $Q_I$ with a sublattice of $X^\\delta_I$ via the map $\\iota^\\delta$ defined in Section \\ref{ch3:subs_KM-quiv}.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_lam1>=lam2}\n$(a)$ For each $\\lambda_1,\\lambda_2\\in P$ we have $\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_1)-\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_2)\\in Q_I$.\n\n$(b)$ If we also have $\\lambda_1\\leqslant\\lambda_2$, then $\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_1)-\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_2)\\in Q^+_I$.\n\\end{lem}\n\\begin{proof}[Proof]\nIt is enough to assume that we have $\\lambda_1=\\lambda_2-\\epsilon_r$ for some $r\\in[1,N]$. In this case we have $\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_1)-\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda_2)=\\alpha_i$, where $i\\in I$ is the residue of the integer $(\\lambda_1)_r$ modulo $e$.\n\\end{proof}\n\n\\smallskip\nLet us write $\\emptyset$ for the empty $l$-partition. Note that $\\Delta[\\emptyset]_R=\\Delta^{\\rho_\\nu}_R$ is the Verma module of highest weight $\\widetilde{\\rho_\\nu-\\rho}$. Since, $\\rho_\\nu$ lies in $P[\\mathrm{\\mathbf{wt}}_e(\\rho_\\nu)]$, we have $\\Delta[\\emptyset]_R\\in O^\\nu_{\\mathrm{\\mathbf{wt}}_e(\\rho_\\nu),R}$. More generally, fix an element $\\alpha=\\sum_{i\\in I}d_i\\alpha_i$ in $Q^+_I$. Put $\\mu=\\mathrm{\\mathbf{wt}}_e(\\rho_\\nu)-\\alpha\\in X_I$. See \\cite[Sec.~2.3]{RSVV} for the definition of a highest weight category over a local ring. The following proposition holds, see \\cite[Sec.~5.5]{RSVV}.\n\\begin{prop}\nThe category $\\mathbf{A}^\\nu_R[\\alpha]$ is a full subcategory of $O^\\nu_{\\mu,R}$ that is a highest weight category.\n\\qed\n\\end{prop}\n\nFor $\\lambda\\in\\mathcal{P}^l_d$ let $P[\\lambda]_{R}$, $\\nabla[\\lambda]_{R}$ and $T[\\lambda]_{R}$ be the projective, costandard and the tilting objects in $\\mathbf{A}_R^\\nu[d]$ with parameter $\\lambda$, see \\cite[Prop.~2.1]{RSVV}.\n\n\n\nSet $\\overline\\alpha=\\phi(\\alpha)\\in Q_{\\overline I}$, $\\overline\\mu=\\phi(\\mu)\\in X_{\\overline I}$ and $\\beta=\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\rho_\\nu)-\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\Upsilon(\\rho_\\nu))$. By Lemma \\ref{ch3:lem_lam1>=lam2}, we have $\\beta\\in Q^+_{\\overline I}$.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_equiv-A-e-e+1}\nThe equivalence of categories $\\theta_{\\mu}^{\\overline\\mu}$ takes the subcategory $\\mathbf{A}_R^\\nu[\\alpha]$ of $O^\\nu_{\\mu,R}$ to the subcategory $\\mathbf{A}^\\nu_{\\overline R}[\\beta+\\overline\\alpha]$ of $O^\\nu_{\\overline\\mu,\\overline R}$.\n\\qed\n\\end{prop}\n\n\\smallskip\nLet the map $\\phi\\colon Q_I\\to Q_{\\overline I}$ be as in Section \\ref{ch3:subs_not-quiv-I-Ibar} (see also Section \\ref{ch3:subs_not-e-e+1}) and $\\Upsilon$ be as in (\\ref{ch3:eq_upsilon}). First, we prove the following lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_difference-weights}\nIf $\\lambda_1,\\lambda_2\\in P$, then\n$$\n\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_1))-\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_2))=\\phi(\\mathrm{\\mathbf{wt}}_{e}^\\delta(\\lambda_1)-\\mathrm{\\mathbf{wt}}_{e}^\\delta(\\lambda_2)).\n$$\n\n\\end{lem}\n\\begin{proof}[Proof of Lemma \\ref{ch3:lem_difference-weights}]\nIt is enough to prove the statement in the case where we have $\\lambda_1=\\lambda_2-\\epsilon_r$ for some $r\\in[1,N]$. In this case we have $\\mathrm{\\mathbf{wt}}_{e}^\\delta(\\lambda_1)-\\mathrm{\\mathbf{wt}}_{e}^\\delta(\\lambda_2)=\\alpha_i$, where $i$ is the residue of $(\\lambda_1)_r$ modulo $e$. If $i\\ne k$ then we have $\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_1))-\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_2))=\\overline\\alpha_{i'}=\\phi(\\alpha_{i})$, where $i'$ is as in (\\ref{ch3:eq_i'}).\nIf $i=k$ then we have $\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_1))-\\mathrm{\\mathbf{wt}}_{e+1}^\\delta(\\Upsilon(\\lambda_2))=\\overline\\alpha_{k}+\\overline\\alpha_{k+1}=\\phi(\\alpha_k)$.\n\\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{ch3:prop_equiv-A-e-e+1}]\nBy definition, $\\mathbf{A}^\\nu_R[\\alpha]\\subset O^\\nu_{\\mu,R}$ is the Serre subcategory of $O_{\\mu,R}^\\nu$ generated by $\\Delta_{R}^\\lambda$ such that the weight $\\lambda\\in P^\\nu$ satisfies $\\lambda\\geqslant\\rho_\\nu$ and $\\mathrm{\\mathbf{wt}}^\\delta_e(\\rho_\\nu)-\\mathrm{\\mathbf{wt}}^\\delta_e(\\lambda)=\\alpha$. Here $\\geqslant$ is the order defined before Lemma \\ref{ch3:lem_lam1>=lam2}.\n\nAs $\\theta_{\\mu}^{\\overline\\mu}(\\Delta_R^\\lambda)$ is isomorphic to $\\Delta^{\\Upsilon(\\lambda)}_{\\overline R}$, Lemma \\ref{ch3:lem_difference-weights} implies that $\\theta_{\\mu}^{\\overline\\mu}(\\mathbf{A}^\\nu_R[\\alpha])$ is the Serre subcategory of $O^\\nu_{\\overline\\mu,R}$ generated by $\\Delta^{\\overline\\lambda}_{\\overline R}$ for $\\overline\\lambda\\in P^\\nu$ such that $\\overline\\lambda\\geqslant\\Upsilon(\\rho_\\nu)$ and $\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\Upsilon(\\rho_\\nu))-\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\overline\\lambda)=\\overline\\alpha$.\n\nMoreover, for each module $\\Delta^{\\overline\\lambda}_{\\overline R}\\in O^\\nu_{\\overline\\mu,\\overline R}$, the weight $\\overline\\lambda$ has no coordinates that are congruent to $k+1$ modulo $e+1$.\nThen $\\overline\\lambda$ satisfies $\\overline\\lambda\\geqslant \\rho_\\nu$ if and only if it satisfies $\\overline\\lambda\\geqslant \\Upsilon(\\rho_\\nu)$. We have $\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\rho_\\nu)-\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\Upsilon(\\rho_\\nu))=\\beta$. Thus $\\theta_{\\mu}^{\\overline\\mu}(\\mathbf{A}^\\nu_R[\\alpha])$ is the Serre subcategory of $O^\\nu_{\\overline\\mu,\\overline R}$ generated by the modules $\\Delta^{\\overline\\lambda}_{\\overline R}$ where $\\overline\\lambda$ runs over the set of all $\\overline\\lambda\\in P^\\nu$ such that $\\overline\\lambda\\geqslant\\rho_\\nu$ and $\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\rho_\\nu)-\\mathrm{\\mathbf{wt}}^\\delta_{e+1}(\\overline\\lambda)=\\overline\\alpha+\\beta$. This implies $\\theta_{\\mu}^{\\overline\\mu}(\\mathbf{A}_R^\\nu[\\alpha])=\\mathbf{A}_{\\overline R}^\\nu[\\overline\\alpha+\\beta]$.\n\\end{proof}\n\n\n\\subsection{The category $\\mathcal{A}$}\n\nFrom now on, to avoid cumbersome notation we will use the following abbreviations.\nFirst, for each $\\alpha\\in Q^+_I$ we set\n$$\n\\mathcal{A}^\\nu_{R}[\\alpha]=\\mathbf{A}^\\nu_{\\overline R}[\\beta+\\overline\\alpha],\\qquad \\mathcal{A}^\\nu_R[d]=\\bigoplus_{|\\alpha|=d}\\mathcal{A}^\\nu_R[\\alpha], \\qquad \\mathcal{A}^\\nu_R=\\bigoplus_{d\\in\\mathbb{N}}\\mathcal{A}^\\nu_R[d].\n$$\nNext, we define the endofunctors $E_0,\\cdots,E_{e-1}$, $F_0,\\cdots, F_{e-1}$ of $\\mathcal{A}^{\\nu,\\Delta}_R$ (or of $\\mathcal{A}^{\\nu}_R$ is $R$ is a field) by\n$$\nF_0=\\restr{\\overline F_0}{\\mathcal{A}^{\\nu,\\Delta}_R},\\cdots, F_{k-1}=\\restr{\\overline F_{k-1}}{\\mathcal{A}^{\\nu,\\Delta}_R},\\quad F_k=\\restr{\\overline F_{k+1}\\overline F_k}{\\mathcal{A}^{\\nu,\\Delta}_R},\n$$\n\\begin{equation}\n\\label{ch3:eq_E-from-E-bar-A}\nF_{k+1}=\\restr{\\overline F_{k+2}}{\\mathcal{A}^{\\nu,\\Delta}_R}, \\cdots , F_{e-1}=\\restr{\\overline F_e}{\\mathcal{A}^{\\nu,\\Delta}_R},\n\\end{equation}\n$$\nE_0=\\restr{\\overline E_0}{\\mathcal{A}^{\\nu,\\Delta}_R},\\cdots, E_{k-1}=\\restr{\\overline E_{k-1}}{\\mathcal{A}^{\\nu,\\Delta}_R}, \\quad E_k=\\restr{\\overline E_{k}\\overline F_{k+1}}{\\mathcal{A}^{\\nu,\\Delta}_R},\n$$\n$$\nE_{k+1}=\\restr{\\overline E_{k+2}}{\\mathcal{A}^{\\nu,\\Delta}_R}, \\cdots , E_{e-1}=\\restr{\\overline E_e}{\\mathcal{A}^{\\nu,\\Delta}_R}.\n$$\n\nBy definition, we have $E_i(\\mathcal{A}^{\\nu,\\Delta}_R[\\alpha])\\subset \\mathcal{A}^{\\nu,\\Delta}_R[\\alpha-\\alpha_i]$ and $F_i(\\mathcal{A}^{\\nu,\\Delta}_R[\\alpha])\\subset \\mathcal{A}^{\\nu,\\Delta}_R[\\alpha+\\alpha_i]$. Consider the endofunctors $E=\\bigoplus_{i\\in I}E_i$ and $F=\\bigoplus_{i\\in I}F_i$ of $\\mathcal{A}^{\\nu,\\Delta}_R$. We have $E(\\mathcal{A}^{\\nu,\\Delta}_R[d])\\subset \\mathcal{A}^{\\nu,\\Delta}_R[d-1]$ and $F(\\mathcal{A}^{\\nu,\\Delta}_R[d])\\subset \\mathcal{A}^{\\nu,\\Delta}_R[d+1]$.\n\nLet $\\theta_\\alpha\\colon\\mathbf{A}^{\\nu}_R[\\alpha]\\to\\mathcal{A}^{\\nu}_R[\\alpha]$ be the equivalence of categories in Proposition \\ref{ch3:prop_equiv-A-e-e+1}. Taking the sum over $\\alpha$'s, we get an equivalence $\\theta\\colon\\mathbf{A}^{\\nu}_R\\to \\mathcal{A}_R^{\\nu}$ and an equivalence $\\theta_d\\colon\\mathbf{A}^{\\nu}_R[d]\\to \\mathcal{A}_R^{\\nu}[d]$. Moreover, we have the following commutative diagram of Grothendieck groups\n\\begin{equation}\n\\label{ch3:eq_comm-diag-Groth-A}\n\\begin{CD}\n[\\mathbf{A}^{\\nu,\\Delta}_R] @>{F_i}>> [\\mathbf{A}^{\\nu,\\Delta}_R]\\\\\n@V{\\theta}VV @V{\\theta}VV\\\\\n[\\mathcal{A}^{\\nu,\\Delta}_R] @>{F_i}>> [\\mathcal{A}^{\\nu,\\Delta}_R],\\\\\n\\end{CD}\n\\end{equation}\nsee Section \\ref{ch3:subs_comm-Groth}.\n\nFor $\\lambda\\in\\mathcal{P}^l_d$ we set $\\overline\\Delta[\\lambda]_R=\\Delta_{\\overline R}^{\\Upsilon(\\rho_\\nu+\\lambda)}\\in\\mathcal{A}_R^\\nu[d]$. By construction we have $\\theta_d(\\Delta[\\lambda]_R)\\simeq \\overline\\Delta[\\lambda]_R$.\nLet $\\overline P_R[\\lambda]$, $\\overline\\nabla_R[\\lambda]$ and $\\overline T_R[\\lambda]$ be the projective, the costandard and the tilting object with parameter $\\lambda$ in $\\mathcal{A}^\\nu_R[d]$.\n\n\n\\subsection{The categorical representation in the category $O$ over the field $K$}\nIn this section we compare the categorical representation in $O^\\nu_{-e,K}$ with the representation datum in $O^{\\nu,\\Delta}_{-e,R}$ introduced above. First, for each $\\lambda\\in P$ we define the following weight in $X^+_{\\widetilde I}$\n$$\n\\widetilde\\mathrm{\\mathbf{wt}}_{e}(\\lambda)=\\sum_{r=1}^l\\left(\\sum_{t=\\nu_1+\\cdots+\\nu_{r-1}+1}^{\\nu_1+\\cdots+\\nu_{r}}\\widetilde\\varepsilon_{(\\lambda_t,r)}\\right).\n$$\n\nFor $\\widetilde\\mu\\in X^+_{\\widetilde I}$ let $O^\\nu_{\\widetilde\\mu,K}$ be the Serre subcategory of $O^\\nu_{-e,K}$ generated by the Verma modules $\\Delta^\\lambda_K$ such that $\\widetilde\\mathrm{\\mathbf{wt}}_{e}(\\lambda)=\\widetilde\\mu$. This decomposition is a refinement of the decomposition $O_{-e,K}^\\nu=\\bigoplus_{\\mu\\in X_I}O^\\nu_{\\mu,K}$ introduced in Section \\ref{ch3:subs_cat-O}. More precisely, we have\n$$\nO^\\nu_{\\mu,K}=\\bigoplus_{\\widetilde\\mu\\in X^+_{\\widetilde I},\\pi_e(\\widetilde\\mu)=\\mu}O^\\nu_{\\widetilde\\mu,K}.\n$$\nSimilarly, there are decompositions\n\n\\begin{equation}\n\\label{ch3:eq_Fj-Ej-K}\nE=\\bigoplus_{j\\in\\widetilde I}E_j, \\qquad F=\\bigoplus_{j\\in\\widetilde I}F_j\n\\end{equation}\nsuch that $E_j$ and $F_j$ map $O_{\\widetilde\\mu,K}$\n to $O_{\\widetilde\\mu+\\widetilde\\alpha_j,K}$ and $O_{\\widetilde\\mu-\\widetilde\\alpha_j,K}$ respectively. We set\n$$\nE_i=\\bigoplus_{j\\in \\widetilde I,\\pi_e(j)=i}E_j,\\qquad F_i=\\bigoplus_{j\\in\n\\widetilde I,\\pi_e(j)=i}F_j.\n$$\nWe have commutative diagrams\n$$\n\\begin{CD}\nO^{\\nu,\\Delta}_{-e,R} @>{E_i}>> O^{\\nu,\\Delta}_{-e,R}\\\\\n@V{K\\otimes_R\\bullet}VV @V{K\\otimes_R\\bullet}VV\\\\\nO^{\\nu,\\Delta}_{-e,K} @>{E_i}>> O^{\\nu,\\Delta}_{-e,K},\\\\\n\\end{CD}\n\\qquad\n\\begin{CD}\nO^{\\nu,\\Delta}_{-e,R} @>{F_i}>> O^{\\nu,\\Delta}_{-e,R}\\\\\n@V{K\\otimes_R\\bullet}VV @V{K\\otimes_R\\bullet}VV\\\\\nO^{\\nu,\\Delta}_{-e,K} @>{F_i}>> O^{\\nu,\\Delta}_{-e,K}.\\\\\n\\end{CD}\n$$\n\nFor each element $\\widetilde\\alpha\\in Q^+_{\\widetilde I}$ let $\\mathbf{A}^\\nu_K[\\widetilde\\alpha]$ be the Serre subcategory of $\\mathbf{A}^\\nu_K$ generated by the Verma modules $\\Delta[\\lambda]_K$ such that $\\lambda\\in \\mathcal{P}^l_{\\widetilde\\alpha,\\nu}$. Similarly to Section \\ref{ch3:subs_cat-bfA}, we have\n$$\nE_j(\\mathbf{A}^\\nu_K[\\widetilde\\alpha])\\subset \\mathbf{A}^\\nu_K[\\widetilde\\alpha-\\widetilde\\alpha_j],\\quad F_j(\\mathbf{A}^\\nu_K[\\widetilde\\alpha])\\subset \\mathbf{A}^\\nu_K[\\widetilde\\alpha+\\widetilde\\alpha_j].\n$$\nSee \\cite[Sec.~7.4]{RSVV} for details.\n\nSimilarly, for $j\\in\\widetilde I$ we can define the endofunctor $\\overline E_j$, $\\overline F_j$ of the categories $O^\\nu_{-e-1,\\overline K}$ and $\\overline \\mathbf{A}^\\nu_{\\overline K}$.\n\nThe decomposition $\\mathbf{A}^\\nu_K[\\alpha]=\\bigoplus_{\\pi_e(\\widetilde\\alpha)=\\alpha}\\mathbf{A}^\\nu_K[\\widetilde\\alpha]$ yields a decomposition $\\mathcal{A}^\\nu_K[\\alpha]=\\bigoplus_{\\pi_e(\\widetilde\\alpha)=\\alpha}\\mathcal{A}^\\nu_K[\\widetilde\\alpha]$.\nWe also consider the endofunctors $E_j$, $F_j$ of $\\mathcal{A}^\\nu_{K}$ such that for $j=(a,b)\\in \\widetilde I$ we have the following analogue of (\\ref{ch3:eq_E-from-E-bar-A}):\n$$\nE_j=\n\\left\\{\\begin{array}{lll}\n\\restr{\\overline E_{(\\Upsilon(a),b)}}{\\mathcal{A}^\\nu_K} & \\mbox{ if }\\pi_{e}(j)\\ne k,\\\\\n \\restr{\\overline E_{(\\Upsilon(a),b)}\\overline E_{(\\Upsilon(a)+1,b)}}{{\\mathcal{A}^\\nu_K}}& \\mbox{ if }\\pi_{e}(j)= k,\n\\end{array}\\right.\n$$\n$$\nF_j=\n\\left\\{\\begin{array}{lll}\n\\restr{\\overline F_{(\\Upsilon(a),b)}}{\\mathcal{A}^\\nu_K} & \\mbox{ if }\\pi_{e}(j)\\ne k,\\\\\n\\restr{\\overline F_{(\\Upsilon(a)+1,b)}\\overline F_{(\\Upsilon(a),b)}}{\\mathcal{A}^\\nu_K}& \\mbox{ if }\\pi_{e}(j)= k.\n\\end{array}\\right.\n$$\nWe have $E_j(\\mathcal{A}^\\nu_K[\\widetilde\\alpha])\\subset \\mathcal{A}^\\nu_K[\\widetilde\\alpha-\\widetilde\\alpha_j]$ and $F_j(\\mathcal{A}^\\nu_K[\\widetilde\\alpha])\\subset \\mathcal{A}^\\nu_K[\\widetilde\\alpha+\\widetilde\\alpha_j]$.\n\n\n\n\\subsection{The modules $T_{\\alpha,R}$, $\\overline T_{\\alpha,R}$}\n\nConsider the module $T_{\\alpha,R}=F_\\alpha(\\Delta[\\emptyset]_R)$ in $\\mathbf{A}^\\nu_R[\\alpha]$ and the module $\\overline T_{\\alpha,R}=F_{\\alpha}(\\overline\\Delta[\\emptyset]_R)$ in $\\mathcal{A}^\\nu_R[\\alpha]$. The commutativity of the diagram (\\ref{ch3:eq_comm-diag-Groth-A}) implies that we have the following equality of classes $[\\theta_\\alpha(T_{\\alpha,R})]=[\\overline T_{\\alpha,R}]$ in $[\\mathcal{A}^\\nu_R]$. The modules $T_{\\alpha,R}\\in\\mathbf{A}^\\nu_R[\\alpha]$ and $\\overline T_{\\alpha,R}\\in\\mathcal{A}^\\nu_R[\\alpha]$ are tilting because $\\Delta[\\emptyset]_R\\in \\mathbf{A}^\\nu_R[0]$ and $\\overline\\Delta[\\emptyset]_R\\in\\mathcal{A}_R^\\nu[0]$ are tilting and the functors $F$ and $\\overline F$ preserve tilting modules, see \\cite[Lem.~8.33,~Lem.~5.16~(b)]{RSVV}. Since a tilting module is characterized by its class in the Grothendieck group, we deduce that there is an isomorphism of modules\n\\begin{equation}\n\\label{ch3:eq_theta-T-Tbar}\n\\theta_\\alpha(T_{\\alpha,R})\\simeq\\overline T_{\\alpha,R}.\n\\end{equation}\n\nFrom now on, we assume that $\\nu_r\\geqslant |\\alpha|$ for each\n$r\\in[1,l]$.\nConsider the weight\n\\begin{equation}\n\\label{ch3:eq_Lambda}\n\\Lambda_\\nu:=\\sum_{r=1}^l\\Lambda_{\\nu_r}\\in P_I.\n\\end{equation}\nWe may abbreviate $\\Lambda=\\Lambda_\\nu$.\nAssume that $R=\\bfk$.\nThe following result is proved in \\cite[Theorem~5.37]{RSVV}.\n\n\\smallskip\n\\begin{prop}\nThe homomorphism $R_{\\alpha,\\bfk}\\to \\mathrm{End}(F_\\alpha(\\Delta[\\emptyset]_\\bfk))^{\\rm op}$ induced by the categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ in $O^{\\nu}_{-e,\\bfk}$ yields an isomorphism $\\psi_{\\alpha,\\bfk}\\colon R_{\\alpha,\\bfk}^\\Lambda\\simeq \\mathrm{End}(T_{\\alpha,\\bfk})^{\\rm op}$.\n\\qed\n\\end{prop}\n\n\\smallskip\nConsider the category $\\overline\\mathcal{C}=O^\\nu_{-e-1,\\overline\\bfk}$. Now we must construct a similar isomorphism $\\overline\\psi_{\\alpha,\\bfk}\\colon R_{\\alpha,\\bfk}^\\Lambda\\simeq \\mathrm{End}(\\overline T_{\\alpha,\\bfk})^{\\rm op}$ coming from the $\\widetilde{\\mathfrak{sl}}_{e+1}$-categorical representation in $\\overline\\mathcal{C}$.\n\n\n\n\nLemma \\ref{ch3:lem_categ-e-e+1} yields a categorical representation of $\\widetilde{\\mathfrak{sl}}_e$ in a subcategory $\\mathcal{C}$ of $\\overline\\mathcal{C}$. As we have $\\overline\\Delta[\\emptyset]_R\\in\\mathcal{C}$, there is an algebra homomorphism\n\\begin{equation}\n\\label{ch3:eq_R-al-to-Tbar}\nR_{\\alpha,\\bfk}\\to \\mathrm{End}(\\overline T_{\\alpha,\\bfk})^{\\rm op}.\n\\end{equation}\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_cycl-quot-Tbar-k}\nThe homomorphism\n(\\ref{ch3:eq_R-al-to-Tbar}) factors through a homomorphism\n$\\overline\\psi_{\\alpha,\\bfk}\\colon R^{\\Lambda}_{\\alpha,\\bfk}\\to \\mathrm{End}(\\overline T_{\\alpha,\\bfk})^{\\rm op}$.\n\\end{lem}\n\\begin{proof}[Proof]\nThe statement follows from Lemma \\ref{ch3:lem_cycl-gen-hw-KLR} below applied to $M=\\overline\\Delta[\\emptyset]_\\bfk$.\n\\end{proof}\n\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_cycl-gen-hw-KLR}\nLet $\\mathcal{C}=\\bigoplus_{\\mu\\in X_I}\\mathcal{C}_\\mu$ be a categorical representation of $\\mathfrak{g}_I$ over $\\bfk$.\nLet $M\\in\\mathcal{C}$ such that there are non-negative integers $t_i$ for $i\\in I$ such that\n\\begin{itemize}\n \\item[\\textbullet] $\\mathrm{End}(M)\\simeq \\bfk$,\n \\item[\\textbullet] $E_iF_i(M)\\simeq M^{\\oplus t_i},\\quad \\forall~i\\in I$.\n\\end{itemize}\nThen, for each $d\\in\\mathbb{N}$, the homomorphism $R_{d,\\bfk}\\to \\mathrm{End}(F^d(M))^{\\rm op}$ factors through the cyclotomic quotient with respect to the weight $\\sum_{i\\in I}t_i\\Lambda_i$.\n\\end{lem}\n\\begin{proof}[Proof]\nIt suffices to prove the statement for $d=1$.\nBy adjointness we have the following isomorphisms of vector spaces\n$$\n\\mathrm{Hom}(F_i(M),F_i(M))\\simeq \\mathrm{Hom}(E_iF_i(M),M)\\simeq \\mathrm{Hom}(M,M)^{\\oplus t_i}\\simeq \\bfk^{t_i}.\n$$\nThe image of $x$ in $\\mathrm{End}(F_i(M))$ is nilpotent. Thus it must be killed by the $t_i$th power because $\\dim \\mathrm{End}(F_i(M))=t_i$.\n\n\\end{proof}\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_cycl-gen-hw-Hecke}\nThe lemma above admits the following equivalent version.\n\nLet $\\mathcal{C}=\\bigoplus_{\\mu\\in X_\\mathscr{F}}\\mathcal{C}_\\mu$ be a categorical representation of $\\mathfrak{g}_\\mathscr{F}$ over $\\bfk$.\nLet $M\\in\\mathcal{C}$ be an object such that there are non-negative integers $t_i$ for $i\\in \\mathscr{F}$ such that $t_i$ is non-zero only for finitely many $i$ and\n\\begin{itemize}\n \\item[\\textbullet] $\\mathrm{End}(M)\\simeq \\bfk$,\n \\item[\\textbullet] $E_iF_i(M)\\simeq M^{\\oplus t_i},\\quad \\forall~i\\in \\mathscr{F}$.\n\\end{itemize}\nFor each $d\\in\\mathbb{N}$, the homomorphism $H_{d,\\bfk}(q)\\to \\mathrm{End}(F^d(M))^{\\rm op}$ factors through the cyclotomic quotient $H^{Q}_{d,\\bfk}(q)$, where $Q$ is a tuple of elements of $\\mathscr{F}$ such that $Q$ contains $t_i$ copies of the element $i$ for each $i\\in\\mathscr{F}$.\n\\end{rk}\n\n\\smallskip\nNow we are going to prove that the algebra homomorphism \n$\\overline\\psi_{\\alpha,\\bfk}$ constructed in Lemma \\ref{ch3:lem_cycl-quot-Tbar-k} is an isomorphism. To do this, we will use a deformation argument. Recall from Corollary \\ref{ch3:coro_isom-Hecke-KLR-cycl-gen} that the cyclotomic KLR algebra $R^\\Lambda_{\\alpha,\\bfk}$ is isomorphic to the cyclotomic Hecke algebra $H^\\nu_{\\alpha,\\bfk}(\\zeta_e)$. We will use the algebra $H^\\nu_{\\alpha,R}(q_e)$ as an $R$-version of $R^\\Lambda_{\\alpha,\\bfk}$.\n\n\\begin{rk}\nThe homomorphism $H_{d,\\overline R}(q_{e+1})\\to \\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_R))^{\\rm op}$ coming from the representation datum structure commutes with base changes $\\bfk\\otimes_R\\bullet$ and $K\\otimes_R\\bullet$, see \\cite[Prop.~8.30]{RSVV}.\n\\end{rk}\n\n\\medskip\n\\begin{lem}\n\\label{ch3:lem_Hd-loc-to-Tbar-ring}\nFor each $d\\in\\mathbb{N}$ the homomorphism $H_{d,\\overline R}(q_{e+1})\\to \\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_R))^{\\rm op}$ extends to the algebra $\\widehat H_{d,\\overline R}(q_{e+1})$ in such a way that the idempotent $e({\\mathbf{i}})$ goes to the projector to $\\overline F_{{\\mathbf{i}}}(\\overline\\Delta[\\emptyset]_R)$ for each ${\\mathbf{i}}$.\n\\end{lem}\n\\begin{proof}[Proof]\nWe must prove that for each ${\\mathbf{i}}\\in \\overline I^d$ we have the following.\n\\begin{itemize}\n\\item[$(a_1)$] The element $(X_r-X_t)e({\\mathbf{i}})\\in H_{d,\\overline R}(q_{e+1})$ acts on $F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_R)$ by an invertible operator for each $r\\ne t$ such that $i_r\\ne i_t$.\n\\item[$(a_2)$] The element $(q_{e+1}X_r-X_t)e({\\mathbf{i}})\\in H_{d,\\overline R}(q_{e+1})$ acts on $F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_R)$ by an invertible operator for each $r\\ne t$ such that $i_r+1\\ne i_t$.\n\\end{itemize}\n\nTo prove this we need the following standard lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Nakayama-coro}\nLet $M$ be an $R$-module. Let $\\phi\\colon M\\to M$ be an endomorphism of $M$.\n\n$(a)$ If $M$ is finitely generated over $R$ and the endomorphism $\\bfk\\phi$ of $\\bfk\\otimes_R M$ is surjective then $\\phi$ is surjective.\n\n$(b)$ If $M$ is free over $R$ and the endomorphism $K\\phi$ of $K\\otimes_R M$ is injective then $\\phi$ is injective.\n\\qed\n\\end{lem}\n\n\\smallskip\nWe have a commutative diagram\n$$\n\\begin{CD}\nH_{d,\\overline \\bfk}(\\zeta_{e+1}) @>>> \\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_\\bfk))^{\\rm op}\\\\\n@AAA @AAA\\\\\nH_{d,\\overline R}(q_{e+1})@>>>\\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_R))^{\\rm op}\\\\\n@VVV @VVV\\\\\nH_{d,\\overline K}(q_{e+1})@>>>\\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_K))^{\\rm op}\n\\end{CD}\n$$\nBy Remark \\ref{ch3:rk_cat-res-loc-H} the top and the bottom vertical homomorphisms extend to homomorphisms\n$$\n\\widehat H_{d,\\overline \\bfk}(\\zeta_{e+1})\\to\\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_\\bfk))^{\\rm op}\n$$\nand\n$$\n\\widehat H_{d,\\overline K}(q_{e+1})\\to\\mathrm{End}(\\overline F^d(\\overline\\Delta[\\emptyset]_K))^{\\rm op}.\n$$\nIn particular the elements from $(a_1)$ and $(a_2)$ go to elements of $H_{d,\\overline\\bfk}(\\zeta_{e+1})$ and $H_{d,\\overline K}(q_{e+1})$ that act on $\\overline F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_\\bfk)$ and $\\overline F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_K)$ by invertible operators. To conclude we apply Lemma \\ref{ch3:lem_Nakayama-coro} to each weight space of the $R$-module $\\overline F_{{\\mathbf{i}}}(\\overline\\Delta[\\emptyset]_R)$ (that is a free $R$-module of finite rank because $\\overline F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_R)$ is tilting by \\cite[Lem.~8.33,~Lem.~5.16~(b)]{RSVV}).\n\\end{proof}\n\nThe homomorphism $\\widehat H_{d,\\overline R}(q_{e+1})\\to \\mathrm{End}(\\overline F^d(\\Delta[\\emptyset]_R))^{\\rm op}$ from Lemma \\ref{ch3:lem_Hd-loc-to-Tbar-ring} yields a homomorphism\n$\n\\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})\\to \\mathrm{End}(\\overline F_{\\overline\\alpha}(\\Delta[\\emptyset]_R))^{\\rm op}\n$\nfor each $\\alpha\\in Q^+_{I}$. Consider the sum $\\mathbf{e}=\\sum_{{\\mathbf{i}}\\in I^{\\overline\\alpha}_{\\rm ord}}e({\\mathbf{i}})$ in $\\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})$. We get an algebra homomorphism\n\\begin{equation}\n\\label{ch3:eq_eHe-to-Tbar}\n\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}\n\\end{equation}\nbecause $\\overline T_{\\alpha,R}=\\bigoplus_{{\\mathbf{i}}\\in I^{\\overline\\alpha}_{\\rm ord}}\\overline F_{\\mathbf{i}}(\\overline\\Delta[\\emptyset]_R)$.\n\\begin{lem}\n\\label{ch3:lem_eHe-to-End-SH-R}\nThe homomorphism (\\ref{ch3:eq_eHe-to-Tbar}) factors through $\\widehat{SH}_{\\overline\\alpha,\\overline R}(q_{e+1})$.\n\\end{lem}\n\\begin{proof}[Proof]\nWe can construct a $K$-linear version $\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}\\to\\mathrm{End}(\\overline T_{\\alpha,K})^{\\rm op}$ of the homomorphism (\\ref{ch3:eq_eHe-to-Tbar}). It factors through a homomorphism\n$\\widehat {SH}_{\\overline\\alpha,\\overline K}(q_{e+1})\\to\\mathrm{End}(\\overline T_{\\alpha,K})^{\\rm op}$ because $\\overline F_{{\\mathbf{j}}}(\\overline \\Delta[\\emptyset]_K)=0$ for each ${\\mathbf{j}}\\in \\widetilde I^{\\overline\\alpha}_{\\rm un}$. Here we set\n$$\n\\widetilde I^{\\overline\\alpha}_{\\rm un}=\\coprod_{\\widetilde\\alpha\\in Q^+_{\\widetilde I},\\pi_{e+1}(\\widetilde\\alpha)=\\overline\\alpha}\\widetilde I^{\\widetilde\\alpha}_{\\rm un}.\n$$\n\nMoreover, we have a commutative diagram\n\\begin{equation}\n\\label{ch3:eq_diag-eHe-R-K}\n\\begin{CD}\n\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e} @>>>\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}\\\\\n@VVV @VVV\\\\\n\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e} @>>>\\mathrm{End}(\\overline T_{\\alpha,K})^{\\rm op}.\n\\end{CD}\n\\end{equation}\nBy definition, the kernel of the homomorphism\n$$\n\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}\\to \\widehat {SH}_{\\overline\\alpha,\\overline R}(q_{e+1})\n$$\nis the intersection of the kernel of the homomorphism\n$$\n\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline K}(q_{e+1})\\mathbf{e}\\to \\widehat {SH}_{\\overline\\alpha,\\overline K}(q_{e+1})\n$$\nwith $\\mathbf{e} \\widehat H_{\\overline\\alpha,\\overline R}(q_{e+1})\\mathbf{e}$. This proves the statement because the right vertical map in (\\ref{ch3:eq_diag-eHe-R-K}) is injective because the module $\\overline T_{\\alpha,R}$ is tilting.\n\\end{proof}\n\nComposing the homomorphism $\\widehat {SH}_{\\overline\\alpha,\\overline R}(q_{e+1})\\to\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}$ in Lemma \\ref{ch3:lem_eHe-to-End-SH-R} with the homomorphism $\\Phi_{\\alpha,R}$ in Lemma \\ref{ch3:lem_morph-Phi-over-ring} yield an algebra homomorphism\n\\begin{equation}\n\\label{ch3:eq_H-to-Tbar-R}\n\\widehat H_{\\alpha,R}(q_{e})\\to\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}\n\\end{equation}\n\nNote that there is a surjective algebra homomorphism $\\widehat H_{\\alpha,R}(q_e)\\to H^\\nu_{\\alpha,R}(q_e)$ (see also Corollary \\ref{ch3:coro_isom-Hecke-KLR-cycl-gen}).\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_H-cycl-to-Tbar-R}\nThe homomorphism (\\ref{ch3:eq_H-to-Tbar-R}) factors through a homomorphism $\\overline\\psi_{\\alpha,R}\\colon H^\\nu_{\\alpha,R}(q_e)\\to\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}$.\n\\end{lem}\n\\begin{proof}[Proof]\nThe proof is similar to the proof of Lemma \\ref{ch3:lem_eHe-to-End-SH-R}. The $K$-linear version $\\widehat H_{\\overline\\alpha,K}(q_{e})\\to\\mathrm{End}(\\overline T_{\\alpha,K})^{\\rm op}$ of this homomorphism factors through the cyclotomic quotient $H^\\nu_{d,K}(q_e)$ by Remark \\ref{ch3:rk_cycl-gen-hw-Hecke}. Then we deduce the statement from the commutativity of the following diagram\n$$\n\\begin{CD}\n\\widehat H_{\\overline\\alpha,R}(q_{e}) @>>>\\mathrm{End}(\\overline T_{\\alpha,R})^{\\rm op}\\\\\n@VVV @VVV\\\\\n\\widehat H_{\\overline\\alpha,K}(q_{e}) @>>>\\mathrm{End}(\\overline T_{\\alpha,K})^{\\rm op}.\n\\end{CD}\n$$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nThe algebra homomorphism $\\overline\\psi_{\\alpha,R}$ in Lemma \\ref{ch3:lem_H-cycl-to-Tbar-R} is an $R$-linear version of the homomorphism $\\overline\\psi_{\\alpha,\\bfk}$ in Lemma \\ref{ch3:lem_cycl-quot-Tbar-k}.\n\n\n\n\n\n\n\\subsection{The proof of invertibility}\nThe goal of this section is to prove that the homomorphism $\\overline\\psi_{\\alpha,R}$ in Lemma \\ref{ch3:lem_H-cycl-to-Tbar-R} is an isomorphism.\n\nConsider the functors\n$$\n\\begin{array}{ll}\n\\Psi^\\nu_\\alpha\\colon \\mathbf{A}^\\nu_R[\\alpha]\\to \\mathrm{mod}(H^\\nu_{\\alpha,R}(q_e)),& M\\mapsto \\mathrm{Hom}(T_{\\alpha,R},M),\\\\\n\\overline\\Psi^\\nu_\\alpha\\colon \\mathcal{A}^\\nu_R[\\alpha]\\to \\mathrm{mod}(H^\\nu_{\\alpha,R}(q_e)),& M\\mapsto \\mathrm{Hom}(\\overline T_{\\alpha,R},M),\\\\\n\\end{array}\n$$\nwhere $\\mathrm{Hom}(T_{\\alpha,R},M)$ and $\\mathrm{Hom}(\\overline T_{\\alpha,R},M)$ are considered as $H^{\\nu}_{d,R}(q_e)$-modules with respect to the homomorphisms $\\psi_{\\alpha,R}$ and $\\overline \\psi_{\\alpha,R}$.\n\nLet us abbreviate $\\Psi=\\Psi^\\nu_{\\alpha}$, $\\overline\\Psi=\\overline\\Psi^\\nu_{\\alpha}$ $T_{R}=T_{\\alpha,R}$ and $\\overline T_R=\\overline T_{\\alpha, R}$. We may write $\\Psi_R$, $\\overline\\Psi_R$ to specify the ring $R$. For $\\lambda\\in\\mathcal{P}^l_{\\alpha,\\nu}$ denote by $S[\\lambda]_R$ the Specht module of $H^\\nu_{\\alpha,R}(q_e)$. We will use similar notation for $\\bfk$ or $K$ instead of $R$. See also \\cite[Sec.~2.4.3]{RSVV}.\n\n\n\n\n\n\n\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_isom-K}\n$(a)$ The homomorphism $\\overline\\psi_{\\alpha,K}\\colon H^{\\nu}_{\\alpha,K}(q_e)\\to \\mathrm{End}(\\overline T_K)^{\\rm op}$ is an isomorphism.\n\n$(b)$ For each $\\lambda\\in\\mathcal{P}^l_{\\alpha,\\nu}$ we have $\\overline\\Psi(\\overline\\Delta[\\lambda]_K)\\simeq S[\\lambda]_K$.\n\\end{lem}\n\\begin{proof}[Proof]\n\nFirst, we prove that the homomorphism $\\overline\\psi_{\\alpha,K}$ is injective.\nThe algebra $H^\\nu_{\\alpha,K}(q_e)$ is finite dimensional and semisimple. Its center is spanned by the idempotents $e(\\widetilde\\alpha)$ such that $\\widetilde\\alpha\\in Q^+_{\\widetilde I}$ and $\\pi_{e}(\\widetilde\\alpha)=\\alpha$. The idempotent $e(\\widetilde\\alpha)$ acts on $T_{K}=F_\\alpha(\\overline\\Delta[\\emptyset]_K)$ by projection onto $F_{\\widetilde\\alpha}(\\overline\\Delta[\\emptyset]_K)$. Thus, to prove the injectivity of $\\overline\\psi_{\\alpha,K}$ we need to check that $F_{\\widetilde\\alpha}(\\overline\\Delta[\\emptyset]_K)$ is nonzero whenever $e(\\widetilde\\alpha)$ is nonzero. Similarly to the argument in Section \\ref{ch3:subs_comm-Groth}, we see that the equivalence $\\theta\\colon \\mathbf{A}^\\nu_K\\simeq \\mathcal{A}^\\nu_K$ yields an isomorphism of Grothendieck groups $[\\mathbf{A}^\\nu_K]\\simeq [\\mathcal{A}^\\nu_K]$ that commutes with functors $F_j$. Thus the module $F_{\\widetilde\\alpha}(\\overline\\Delta[\\emptyset]_K)\\in\\mathcal{A}^\\nu_K$ is nonzero if and only if the module $F_{\\widetilde\\alpha}(\\Delta[\\emptyset]_K)\\in\\mathbf{A}^\\nu_K$ is nonzero. By \\cite[Prop.~5.22~(d)]{RSVV}, the module $F_{\\widetilde\\alpha}(\\Delta[\\emptyset]_K)\\in\\mathbf{A}^\\nu_K$ is nonzero whenever $e(\\widetilde\\alpha)$ is nonzero. Thus $\\overline\\psi_{\\alpha,K}$ is injective.\n\n\n\nThus it is also surjective because\n$$\n\\dim_K H^\\nu_{\\alpha,K}(q_e)=\\dim_K\\mathrm{End}(T_K)^{\\rm op}=\\dim_K\\mathrm{End}(\\overline T_K)^{\\rm op},\n$$\nwhere the first equality holds by \\cite[Prop.~5.22~(d)]{RSVV} and the second holds by (\\ref{ch3:eq_theta-T-Tbar}). This implies part $(a)$.\n\nThe discussion above implies that $\\overline T_K$ contains each $\\overline\\Delta[\\lambda]_K$, $\\lambda\\in \\mathcal{P}^l_\\alpha$ as a direct factor. In particular $\\overline T_K$ is a projective generator of $\\mathcal{A}^\\nu_K[\\alpha]$. Thus $\\overline\\Psi_K$ is an equivalence of categories. It must take $\\overline\\Delta[\\lambda]_K$ to $S[\\lambda]_K$ because $S[\\lambda]_K$ is the unique simple module in the block $\\mathrm{mod}(H^\\nu_{\\widetilde\\alpha,K}(q_e))$ of $\\mathrm{mod}(H^\\nu_{\\alpha,K}(q_e))$.\n\\end{proof}\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_isom-R}\n$(a)$ The homomorphism $\\overline\\psi_{\\alpha,K}\\colon H^{\\nu}_{\\alpha,R}(q_e)\\to \\mathrm{End}(\\overline T_R)^{\\rm op}$ is an isomorphism.\n\n$(b)$ For each $\\lambda\\in\\mathcal{P}^l_{\\alpha,\\nu}$ we have $\\overline\\Psi(\\overline\\Delta[\\lambda]_R)\\simeq S[\\lambda]_R$.\n\\end{lem}\n\\begin{proof}[Proof]\nConsider the endomorphism $u$ of $H^\\nu_{\\alpha,R}(q_e)$ obtained from the following chain of homomorphisms\n$$\nu\\colon H^\\nu_{\\alpha,R}(q_e)\\stackrel{\\overline\\psi_{\\alpha,R}}{\\to}\\mathrm{End}_{\\mathcal{A}^\\nu}(\\overline T_R)^{\\rm op}\\stackrel{\\theta_{\\alpha}^{-1}}{\\to}\\mathrm{End}_{\\mathbf{A}^\\nu}(T_R)^{\\rm op}\\stackrel{\\psi_{\\alpha,R}^{-1}}{\\to}H^\\nu_{\\alpha,R}(q_e).\n$$\nThe invertibility of $\\overline\\psi_{\\alpha,R}$ is equivalent to the invertibility of $u$. By \\cite[Prop.~2.23]{RSVV} to prove that $u$ is an isomorphism it is enough to show that its localization $Ku\\colon H^\\nu_{\\alpha,K}(q_e)\\to H^\\nu_{\\alpha,K}(q_e)$ is an isomorphism and that $Ku$ induces the identity map on Grothendieck groups $[\\mathrm{mod}(H^\\nu_{\\alpha,K}(q_e))]\\to [\\mathrm{mod}(H^\\nu_{\\alpha,K}(q_e))]$. The bijectivity of $Ku$ follows from Lemma \\ref{ch3:lem_isom-K} $(a)$.\n\nNow we check the condition on the Grothendieck group. We already know from \\cite[Prop.~5.22~(c)]{RSVV} and the proof of Lemma \\ref{ch3:lem_isom-K} that $\\Psi_K$ and $\\overline\\Psi_K$ are equivalences of categories. Thus, by semisimplicity of the categories $\\mathbf{A}^\\nu_K[\\alpha]$, $\\mathcal{A}^\\nu_K[\\alpha]$ and $\\mathrm{mod}(H^\\nu_{\\alpha,K}(q_e))$, we have an isomorphism of functors $\\Psi_K\\simeq \\overline \\Psi_K\\circ \\theta_{\\alpha}$ because $\\Psi_K(M)\\simeq \\overline\\Psi_K\\circ \\theta_\\alpha(M)$ for each $M\\in\\mathbf{A}_K^\\nu[\\alpha]$. This implies that $Ku$ is the identity on the Grothendieck group. This proves part $(a)$.\n\nPart $(b)$ follows from Lemma \\ref{ch3:lem_isom-K} and the characterization of Specht modules, see \\cite[Sec.~2.4.3]{RSVV}.\n\\end{proof}\n\n\\smallskip\n\\begin{rk}\nThere is no reason why the automorphism $u\\colon H^\\nu_{\\alpha,R}(q_e)\\to H^\\nu_{\\alpha,R}(q_e)$ in the proof of Lemma \\ref{ch3:lem_isom-R} should be identity. Because of this the functor $\\Psi$ has no reason to coincide with $\\overline\\Psi\\circ\\theta_\\alpha$. However the automorphism $u$ of $H^\\nu_{\\alpha,R}(q_e)$ induces an autoequivalence $u^*$ of $\\mathrm{mod}(H^\\nu_{\\alpha,R}(q_e))$ such that we have\n\\begin{equation}\n\\label{ch3:eq_isom-Psi-Psibar-u*}\n\\Psi= u^*\\circ\\overline\\Psi\\circ\\theta_\\alpha.\n\\end{equation}\n\\end{rk}\n\n\\smallskip\nNow, specializing to $\\bfk$, we obtain the following.\n\n\\smallskip\n\\begin{coro}\n\\label{ch3:coro_isom-C}\n$(a)$ The homomorphism $\\overline\\psi_{\\alpha,\\bfk}\\colon R^{\\Lambda}_{\\alpha,\\bfk}\\to \\mathrm{End}(\\overline T_\\bfk)^{\\rm op}$ is an isomorphism.\n\n$(b)$ For each $\\lambda\\in\\mathcal{P}^l_{\\alpha,\\nu}$ we have $\\overline\\Psi_\\bfk(\\overline\\Delta[\\lambda]_\\bfk)\\simeq S[\\lambda]_\\bfk$.\n\\qed\n\\end{coro}\n\n\n\n\n\\subsection{Rational Cherednik algebras}\nLet $R$ be a local commutative $\\mathbb{C}$-algebra with residue field $\\mathbb{C}$. Let $W$ be a complex reflection group. Denote by $S=S(W)$ and $\\mathcal{A}$ the set of pseudo-reflections in $W$ and the set of reflection hyperplanes respectively. Let $\\mathfrak{h}$ be the reflection representation of $W$ over $R$. Let $c\\colon S\\to R$ be a map which is constant on the $W$-conjugacy classes.\n\nDenote by $\\langle\\bullet,\\bullet\\rangle$ the canonical pairing between $\\mathfrak{h}^*$ and $\\mathfrak{h}$. For each $s\\in S$ fix a generator $\\alpha_s\\in \\mathfrak{h}^*$ of ${\\rm Im}(\\restr{s}{\\mathfrak{h}^*}-1)$ and a generator $\\check\\alpha_s\\in\\mathfrak{h}$ of ${\\rm Im}(\\restr{s}{\\mathfrak{h}}-1)$ such that $\\langle\\alpha_s,\\check{\\alpha}_s\\rangle=2$.\n\n\\smallskip\n\\begin{df}\nThe \\emph{rational Cherednik algebra} $H_c(W,\\mathfrak{h})_R$ is the quotient of the smash product $RW\\ltimes T(\\mathfrak{h}\\oplus\\mathfrak{h}^*)$ by the relations\n$$\n[x,x']=0,\\qquad [y,y']=0,\\qquad [y,x]=\\langle x,y\\rangle-\\sum_{s\\in S}c_s\\langle\\alpha_s,y\\rangle\\langle x,\\check{\\alpha}_s\\rangle s,\n$$\nfor each $x,x'\\in\\mathfrak{h}^*$, $y,y'\\in\\mathfrak{h}$. Here $T(\\bullet)$ denotes the tensor algebra.\n\\end{df}\n\nDenote by $\\mathcal{O}_c(W,\\mathfrak{h})_R$ the category $\\mathcal{O}$ of $H_c(W,\\mathfrak{h})_R$, see \\cite[Sec.~3.2]{GGOR} and \\cite[Sec.~6.1.1]{RSVV}.\nLet $E$ be an irreducible representation of $\\mathbb{C} W$.\n\n\\smallskip\n\\begin{df}\nA \\emph{Verma module} associated with $E$ is the following module in $\\mathcal{O}_c(W,\\mathfrak{h})_R$\n$$\n\\Delta_R(E):=\\mathrm{Ind}_{RW\\ltimes \\mathfrak{h}^*}^{H_c(W,\\mathfrak{h})_R}(RE).\n$$\nHere each element $x\\in\\mathfrak{h}^*$ acts on $RE$ by zero.\n\\end{df}\nThe category $\\mathcal{O}_c(W,\\mathfrak{h})_R$ is a highest weight category over $R$ with standard modules $\\Delta_R(E)$.\n\nWe call a subgroup $W'$ of $W$ \\emph{parabolic} if it is a stabilizer of some point of $b\\in\\mathfrak{h}$. In this case $W'$ is a complex reflection group with reflection representation $\\mathfrak{h}\/\\mathfrak{h}^{W'}$, where $\\mathfrak{h}^{W'}$ is the set of $W'$-stable points in $\\mathfrak{h}$. Moreover, the map $c\\colon S(W)\\to R$ restricts to a map $c\\colon S(W')\\to R$. There are induction and restriction functors\n$$\n^\\mathcal{O}\\mathrm{Ind}_{W'}^W\\colon \\mathcal{O}_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R\\to \\mathcal{O}_c(W,\\mathfrak{h})_R,\\quad ^\\mathcal{O}\\mathrm{Res}_{W'}^W\\colon \\mathcal{O}_c(W,\\mathfrak{h})_R\\to \\mathcal{O}_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R,\n$$\nsee \\cite{BE}. The definitions of these functors depend on $b$ but their isomorphism classes are independent of the choice of $b$.\n\n\nThe following lemma holds.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_proj-conj-par-same}\nAssume that $W'$ and $W''$ are conjugated parabolic subgroups in $W$. Let $P\\in\\mathcal{O}_c(W,\\mathfrak{h})_R$ be a projective module. Then the following conditions are equivalent\n \\begin{itemize}\n \\item[\\textbullet] the module $P$ is isomorphic to a direct factor of the module $^\\mathcal{O}\\mathrm{Ind}_{W'}^W(P')$ for some projective module $P'\\in \\mathcal{O}_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R$,\n \\item[\\textbullet] the module $P$ is isomorphic to a direct factor of a module $^\\mathcal{O}\\mathrm{Ind}_{W''}^W(P'')$ for some projective module $P''\\in \\mathcal{O}_c(W'',\\mathfrak{h}\/\\mathfrak{h}^{W''})_R$.\n\\end{itemize}\n\\end{lem}\n\\begin{proof}[Proof]\nLet $w$ be an element of $W$ such that $wW'w^{-1}=W''$. The conjugation by $w$ yields an isomorphism $W'\\simeq W''$. Hence, the element $w$ takes $\\mathfrak{h}^{W'}$ to $\\mathfrak{h}^{W''}$. Thus we get an algebra isomorphism $H_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R\\simeq H_c(W'',\\mathfrak{h}\/\\mathfrak{h}^{W''})_R$ and an equivalence of categories $\\mathcal{O}_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R\\simeq \\mathcal{O}_c(W'',\\mathfrak{h}\/\\mathfrak{h}^{W''})_R$. Moreover, the conjugation by $w$ yields an automorphism $t$ of $H_c(W,\\mathfrak{h})_R$ such that for each $x\\in\\mathfrak{h}^*$, $y\\in\\mathfrak{h}$, $u\\in W$ we have\n$$\nt(x)=w(x),\\quad t(y)=w(y),\\quad t(u)=wuw^{-1}.\n$$\nThe following diagram of functors is commutative up to equivalence of functors\n$$\n\\begin{CD}\n\\mathcal{O}_c(W,\\mathfrak{h})_R @<{t^*}<< \\mathcal{O}_c(W,\\mathfrak{h})_R\\\\\n@A{^\\mathcal{O}\\mathrm{Ind}_{W'}^W}AA @A{^\\mathcal{O}\\mathrm{Ind}_{W''}^W}AA\\\\\n\\mathcal{O}_c(W',\\mathfrak{h}\/\\mathfrak{h}^{W'})_R @<<< \\mathcal{O}_c(W'',\\mathfrak{h}\/\\mathfrak{h}^{W''})_R\n\\end{CD}\n$$\n\nTo conclude, we need only to prove that the pull-back $t^*$ induces the identity map on the Grothendieck group of $\\mathcal{O}_c(W,\\mathfrak{h})_R$ (and thus it maps each projective module to an isomorphic one). This is true because $t^*$ maps each Verma module $\\Delta(E)_R$ to an isomorphic one because the representation $E$ of $W$ does not change the isomorphism class when we twist the $W$-action by an inner automorphism.\n\\end{proof}\n\n\\subsection{Cyclotomic rational Cherednik algebras}\nFrom now on, we assume that $R$ is either a local analytic deformation ring in general position of dimension $\\leqslant 2$ or a field. Let $\\bfk$ and $K$ be its residue field and the field of fractions respectively.\n\nLet $\\Gamma\\simeq\\mathbb{Z}\/l\\mathbb{Z}$ be the group of complex $l$th roots of unity and set $\\Gamma_d=\\mathfrak{S}_d\\ltimes \\Gamma^d$. For $\\gamma\\in\\Gamma$, $r\\in[1,l]$ denote by $\\gamma_r$ the element of $\\Gamma^d$ having $\\gamma$ at the position $r$ and $1$ at other positions. Let $s_{r,t}$ be the transposition in $\\mathfrak{S}_d$ exchanging $r$ and $t$. For $\\gamma\\in\\Gamma$, $r,t\\in[1,l]$ set $s_{r,t}^\\gamma:=s_{r,t}\\gamma_r\\gamma^{-1}_t\\in\\Gamma_d$. From now on we suppose that the group $W$ is $\\Gamma_d$ and $\\mathfrak{h}=R^d$ is the obvious reflection representation of $\\Gamma_d$. Assume also that $h,h_1,\\cdots,h_{l-1}$ are some elements of $R$ and set $h_{-1}=h_{l-1}$. Let us chose the parameter $c$ in the following way\n$$\nc(s_{r,t}^\\gamma)=-h \\qquad \\mbox{ for each $r,t\\in [1,t]$, $r\\ne t$, $\\gamma\\in\\Gamma$},\n$$\n$$\nc(\\gamma_r)=-\\frac{1}{2}\\sum_{p=0}^{l-1}\\gamma^{-p}(h_p-h_{p-1})\\qquad \\mbox{ for each $r\\in[1,l]$, $\\gamma\\in\\Gamma$, $\\gamma\\ne 1$}.\n$$\n\nLet $\\nu_1,\\cdots,\\nu_l$ be as above. We set\n$$\nh=-1\/\\kappa,\\quad h_p=-(\\nu_{p+1}+\\tau_{p+1})\/\\kappa-p\/l, \\quad p\\in[1,l-1].\n$$\nLet us abbreviate $\\mathcal{O}^\\nu_R[d]=\\mathcal{O}_c(\\Gamma_d,R^d)_R$. Consider the $\\mathrm{KZ}$-functor $\\mathrm{KZ}^\\nu_d\\colon \\mathcal{O}^\\nu_R[d]\\to \\mathrm{mod}(H^\\nu_{d,R}(q_e))$ introduced in \\cite[Sec.~6]{RSVV}. Denote by $^*\\mathcal{O}^\\nu_R[d]$ the category defined in the same way as $\\mathcal{O}^\\nu_R[d]$ with replacement of $(\\nu_1,\\cdots,\\nu_l)$ by $(-\\nu_l,\\cdots,-\\nu_1)$ and $(\\tau_1,\\cdots,\\tau_l)$ by $(-\\tau_l,\\cdots,-\\tau_1)$. Similarly, denote by $^*H^\\nu_{d,R}(q_e)$ the affine Hecke algebra defined in the same way as $H^\\nu_{d,R}(q_e)$ with the replacement of parameters as above. There is also a $\\mathrm{KZ}$-functor $^*\\mathrm{KZ}^\\nu_d\\colon {^*\\mathcal{O}}^\\nu_R[d]\\to \\mathrm{mod}(^*H^\\nu_{d,R}(q_e))$.\n\nThe simple $\\mathbb{C}\\Gamma_d$-modules are labeled by the set $\\mathcal{P}^l_d$. We write $E(\\lambda)$ for the simple module corresponding to $\\lambda$. Set $\\Delta[\\lambda]_R=\\Delta(E(\\lambda))_R$. Similarly, write $P[\\lambda]_R$ and $T[\\lambda]_R$ for the projective and tilting object in $\\mathcal{O}^\\nu_R[d]$ with index $\\lambda$.\n\nThe category $^*\\mathcal{O}^{\\nu}_R[d]$ is the Ringel dual of the category $\\mathcal{O}^{\\nu}_R[d]$, see \\cite[Sec.~6.2.4]{RSVV}. In particular we have an equivalence between the categories of standardly filtered objects $\\mathscr{R}\\colon{^*\\mathcal{O}}^{\\nu}_R[d]^{\\Delta}\\to (\\mathcal{O}^{\\nu}_R[d]^{\\Delta})^{\\rm op}$. Let $\\mathrm{proj}(R)$ be the category of projective finitely generated $R$-modules. There is an algebra isomorphism\n$$\n\\iota\\colon H^\\nu_{d,R}(q_e)\\to(^*H^\\nu_{d,R}(q_e))^{\\rm op},\\quad T_r\\mapsto -q_eT_r^{-1},~X_r\\mapsto X_r^{-1},\n$$\nsee \\cite[Sec.~6.2.4]{RSVV}. It induces an equivalence\n$$\n\\mathscr{R}_H=\\iota^*(\\bullet^\\vee)\\colon \\mathrm{mod}(^*H^\\nu_{d,R}(q_e))\\cap\\mathrm{proj}(R)\\to(\\mathrm{mod}(H^\\nu_{d,R}(q_e))\\cap\\mathrm{proj}(R))^{\\rm op},\n$$\nwhere $\\bullet^\\vee$ is the dual as an $R$-module.\nBy \\cite[(6.3)]{RSVV},\nthe following diagram of functors is commutative\n\\begin{equation}\n\\label{ch3:eq_diag-scrR}\n\\begin{CD}\n^*\\mathcal{O}^\\nu_R[d]^\\Delta @>{\\mathscr{R}}>> (\\mathcal{O}^\\nu_R[d]^\\Delta)^{\\rm op}\\\\\n@V{^*\\mathrm{KZ}^\\nu_{d}}VV @V{\\mathrm{KZ}^\\nu_{d}}VV\\\\\n\\mathrm{mod}({^*H}^\\nu_{d,R}(q_e))\\cap\\mathrm{proj}(R) @>{\\mathscr{R}_H}>>(\\mathrm{mod}({H}^\\nu_{d,R}(q_e))\\cap\\mathrm{proj}(R))^{\\rm op}.\n\\end{CD}\n\\end{equation}\n\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_d=1-Cher}\nAssume that $d=1$.\nFor each $l$-partition of $1$ $\\lambda$ we have an isomorphism of $H^\\nu_{1,R}(q_e)$-modules $\\mathrm{KZ}^\\nu_1(P[\\lambda]_R)\\simeq \\overline \\Psi^\\nu_1(\\overline T[\\lambda]_R)$.\n\\end{lem}\n\\begin{proof}[Proof]\nThe proof is similar to the proof of \\cite[Prop.~6.7]{RSVV}. The commutativity of the diagram (\\ref{ch3:eq_diag-scrR}) implies\n$$\n\\mathrm{KZ}^\\nu_1(P[\\lambda]_R)=\\mathrm{KZ}^\\nu_1(\\mathscr{R}(T[\\lambda]_R))=\\mathscr{R}_H({^*\\mathrm{KZ}^\\nu_1}(T[\\lambda]_R)).\n$$\n\n\nTo conclude, we just need to compare the highest weight covers $\\mathscr{R}_H\\circ{^*\\mathrm{KZ}^\\nu_1}$ and $\\overline\\Psi^\\nu_1$ of $H^\\nu_{1,R}(q_e)$ using Lemma \\ref{ch3:lem_isom-K} $(b)$ and \\cite[Prop.~2.21]{RSVV}.\n\n\n\\end{proof}\n\nLet $O^+_{\\mu,R}$ be the affine parabolic category $\\mathcal{O}$ associated with the parabolic type consisting of the single block of size $\\nu_1+\\cdots+\\nu_l$.\n\nWe define the categories $\\mathbf{A}^+_{R}[d]$, $\\mathcal{A}^+_R[d]$ and $O^+_R[d]$ similarly. In this case we will also write the upper index $+$ in the notation of modules and functors (for example $\\Delta^+[\\lambda]_R$, $T^+_{d,R}$, $\\mathrm{KZ}^+_{d}$, etc.) Let also $H^+_{d,R}(q_e)$ be the cyclotomic Hecke algebra with $l=1$. It is isomorphic to the Hecke algebra of $\\mathfrak{S}_d$.\n\nWe can prove the following result.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_d=2,l=1-Cher}\nFor each $\\lambda\\in\\mathcal{P}^1_2$ we have $\\mathrm{KZ}^+_2(P^+[\\lambda]_R)\\simeq \\overline \\Psi^+_2(\\overline T^+[\\lambda]_R)$.\\qed\n\\end{lem}\n\\begin{proof}[Proof]\nSimilarly to the proof of Lemma \\ref{ch3:lem_d=1-Cher} we compare the highest weight covers $\\mathscr{R}_H\\circ{^*\\mathrm{KZ}^+_2}$ and $\\overline\\Psi^+_2$ of $H^+_{2,R}(q_e)$ using Lemma \\ref{ch3:lem_isom-K} $(b)$ and \\cite[Prop.~2.21]{RSVV}.\n\n\\end{proof}\n\nDenote by $\\mathrm{Ind}_{d,+}^{d,\\nu}$ the induction functor with respect to the inclusion $H^{+}_{d,R}(q_e)\\subset H^{\\nu}_{d,R}(q_e)$.\nWe will also need the following lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_d=1-to-d=2,l=1}\nAssume $\\nu_r\\geqslant 2$ for each $r\\in[1,l]$. Assume also that $e>2$. For each $\\lambda\\in\\mathcal{P}^1_2$ there exists a tilting module $\\overline T_{\\lambda,R}\\in\\mathcal{A}^\\nu_R[2]$ such that $\\Psi^\\nu_2(\\overline T_{\\lambda,R})\\simeq \\mathrm{Ind}_{2,+}^{2,\\nu}(\\overline\\Psi^+_2(\\overline T^+[\\lambda]_R))$.\n\\end{lem}\n\\begin{proof}[Proof]\nSet $\\lambda^+=(2)$, $\\lambda^-=(1,1)$.\nWe have $\\zeta_e\\ne -1$ because $e>2$. In this case the algebra $H^+_{2,\\bfk}(\\zeta_e)$ is semisimple. The category $\\mathcal{A}^+_\\bfk[2]$ is also semisimple. This implies\n$$\n\\overline T^+_{2,R}\\simeq \\overline\\Delta[\\lambda^+]_R\\oplus\\overline\\Delta[\\lambda^-]_R= \\overline T[\\lambda^+]_R\\oplus \\overline T[\\lambda^-]_R.\n$$\nBy definition, we have $\\overline\\Psi^+_{2}(\\overline T^+_{2,R})\\simeq H^+_{2,R}(q_e)$ and $\\overline\\Psi^\\nu_{2}(\\overline T_{2,R})\\simeq H^\\nu_{2,R}(q_e)$. This implies\n$$\n\\overline\\Psi^\\nu_{2}(\\overline T_{2,R})\\simeq \\mathrm{Ind}_{2,+}^{2,\\nu}(\\overline\\Psi^+_{2}(\\overline T^+_{2,R})).\n$$\nBy the proof of \\cite[Prop.~6.8]{RSVV},\nthe functor $\\Psi_2^\\nu$ takes indecomposable factors of $T_{2,R}$ to indecomposable modules. Thus, by (\\ref{ch3:eq_isom-Psi-Psibar-u*}), the functor $\\overline\\Psi_2^\\nu$ takes indecomposable factors of $\\overline T_{2,R}$ to indecomposable modules. Thus there is a decomposition $\\overline T_{2,R}=\\overline T_{\\lambda^+,R}\\oplus \\overline T_{\\lambda^-,R}$ such that $\\overline T_{\\lambda^+,R}$, $\\overline T_{\\lambda^-,R}$ satisfy the required properties.\n\n\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{ch3:thm_intro-main-decomp-functors}}\nIn this section we finally give a proof of over main result.\n\n\nA priori there is no reason to have the following isomorphism of functors $\\Psi^\\nu_\\alpha\\simeq \\overline \\Psi^\\nu_\\alpha\\circ \\theta_\\alpha$. However, we can modify the equivalence $\\theta_\\alpha$ to make this true.\n\nFor $d_1 d$ for each $r\\in[1,l]$. Then the following diagram of functors is commutative.\n$$\n\\begin{CD}\n\\mathcal{A}^\\nu_R[d] @>{F}>> \\mathcal{A}^\\nu_R[d+1]\\\\\n@V{\\overline\\Psi^\\nu_d}VV @V{\\overline\\Psi^\\nu_{d+1}}VV\\\\\n\\mathrm{mod}(H^\\nu_{d,R}(q_e))@>{\\mathrm{Ind}_d^{d+1}}>>\\mathrm{mod}(H^\\nu_{d+1,R}(q_e))\n\\end{CD}\n$$\n\\qed\n\\end{lem}\n\n\nFor a partition $\\lambda$ denote by $\\lambda^*$ the transposed partition. For an $l$-partition $\\lambda=(\\lambda_1,\\cdots,\\lambda_l)$ set $\\lambda^*=((\\lambda_l)^*,\\cdots,(\\lambda_1)^*)$.\nThere is an algebra isomorphism\n$$\n{\\rm IM}\\colon H^\\nu_{d,R}(q_e)\\to {^*H}^\\nu_{d,R}(q_e), \\quad T_r\\mapsto -q_eT_r^{-1},~X_r\\mapsto X_r^{-1},\n$$\nsee \\cite[Sec.~6.2.4]{RSVV}. Let ${\\rm IM}^*\\colon \\mathrm{mod}({^*H}^\\nu_{d,R}(q_e))\\to \\mathrm{mod}(H^\\nu_{d,R}(q_e))$ be the induced equivalence of categories. We have\n\\begin{equation}\n\\label{ch3:eq_IM-on-Specht}\n{\\rm IM}^*(S[\\lambda^*]_R)\\simeq S[\\lambda]_R.\n\\end{equation}\n\nThe following proposition is proved in \\cite[Thm.~6.9]{RSVV}.\n\\begin{prop}\n\\label{ch3:prop_equiv-Cher-bfA}\nAssume that $\\nu_r\\geqslant d$ for each $r\\in[1,l]$. Then there is an equivalence of categories $\\gamma_d\\colon{^*\\mathcal{O}}^{\\nu}_R[d]\\simeq \\mathbf{A}^\\nu_R[d]$ taking $\\Delta[\\lambda^*]_R$ to $\\Delta[\\lambda]_R$. Moreover, we have the following isomorphism of functors $\\Psi^\\nu_d\\circ\\gamma_d\\simeq {\\rm IM}^*\\circ{^*\\mathrm{KZ}}^{\\nu}_d$.\n\\end{prop}\n\nNow, we prove a similar statement for $\\mathcal{A}^\\nu_R[d]$. For each reflection hyperplane $H$ of the complex reflection group $\\Gamma_d$ let $W_H\\subset \\Gamma_d$ be the pointwise stabilizer of $H$.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_constr-of-equiv}\nAssume that $\\nu_r\\geqslant d$ for each $r\\in[1,l]$ and $e>2$.\nThere is an equivalence of categories $\\overline\\gamma_d\\colon {^*\\mathcal{O}}^{\\nu}_R[d]\\simeq \\mathcal{A}^\\nu_R[d]$, taking $\\Delta[\\lambda^*]_R$ to $\\overline\\Delta[\\lambda]_R$. Moreover, we have the following isomorphism of functors $\\overline\\Psi^\\nu_d\\circ\\overline\\gamma_d\\simeq {\\rm IM}^*\\circ{^*\\mathrm{KZ}}^{\\nu}_d$.\n\\end{prop}\n\\begin{proof}[Proof]\nThe proof is similar to the proof of \\cite[Thm. 6.9]{RSVV}.\nWe set $\\mathcal{C}={^*\\mathcal{O}}^{\\nu}_R[d]$, $\\mathcal{C}'=\\mathcal{A}^\\nu_R[d]$. Consider the following functors\n$$\n\\begin{array}{lll}\nY\\colon \\mathcal{C}\\to \\mathrm{mod}(H^\\nu_{d,R}(q_e)),\\quad &Y={\\rm IM}^*\\circ {^*\\mathrm{KZ}}^{\\nu}_d,\\\\\nY'\\colon \\mathcal{C}'\\to \\mathrm{mod}(H^\\nu_{d,R}(q_e)),\\quad &Y'=\\overline\\Psi^\\nu_d.\n\\end{array}\n$$\nBy \\cite[Prop.~2.20]{RSVV} it is enough to check the following four conditions.\n\\begin{itemize}\n\\item[(1)] We have $Y(\\Delta[\\lambda^*]_R)\\simeq Y'(\\Delta[\\lambda]_R)$ and the bijection $\\Delta[\\lambda^*]_R\\mapsto \\Delta[\\lambda]_R$ between the sets of standard objects in $\\mathcal{C}$ and $\\mathcal{C}'$ respects the highest weight orders.\n\\item[(2)] The functor $Y$ is fully faithful on $\\mathcal{C}^\\Delta$ and $\\mathcal{C}^\\nabla$.\n\\item[(3)] The functor $Y'$ is fully faithful on $\\mathcal{C}'^\\Delta$ and $\\mathcal{C}'^\\nabla$.\n\\item[(4)] For each reflection hyperplane $H$ of $\\Gamma_d$ and each projective module $P\\in\\mathcal{O}(W_H)_R$ we have\n$$\n\\mathrm{KZ}^\\nu_d(^\\mathcal{O}\\mathrm{Ind}^{\\Gamma_d}_{W_H}P)\\in F'(\\mathcal{C}'^{\\rm tilt}).\n$$\n\\end{itemize}\n\nIt is explained in the proof of \\cite[Thm.~6.9]{RSVV} that condition $(4)$ announced here implies the fourth condition in \\cite[Prop.~2.20]{RSVV}.\n\nWe have $Y(\\Delta[\\lambda^*]_R)\\simeq Y'(\\Delta[\\lambda]_R)$ by Lemma \\ref{ch3:lem_isom-R} $(b)$, \\cite[Lem.~6.6]{RSVV} and (\\ref{ch3:eq_IM-on-Specht}). The composition of the equivalence $\\theta_d\\colon\\mathbf{A}^\\nu_R[d]\\to\\mathcal{A}^\\nu_R[d]$ with the equivalence $\\gamma_d\\colon{^*\\mathcal{O}}^\\nu_R[d]\\simeq\\mathbf{A}^\\nu_R[d]$ yields an equivalence of highest weight categories $\\mathcal{C}\\simeq \\mathcal{C}'$ that takes $\\Delta[\\lambda^*]_R$ to $\\Delta[\\lambda]_R$. This implies $(1)$.\n\nCondition $(2)$ is already checked in \\cite[Sec.~6.3.2]{RSVV}.\n\n\nThe functor $\\Psi^\\nu_d$ is fully faithful on $\\mathbf{A}^{\\nu,\\Delta}_R[d]$ and $\\mathbf{A}^{\\nu,\\nabla}_R[d]$ by \\cite[Thm.~5.37~(c)]{RSVV}. Thus the functor $\\overline\\Psi^\\nu_d$ is fully faithful on $\\mathcal{A}^{\\nu,\\Delta}_R[d]$ and $\\mathcal{A}^{\\nu,\\nabla}_R[d]$ by (\\ref{ch3:eq_isom-Psi-Psibar-u*}). This implies $(3)$.\n\n\n\nLet us check condition $(4)$.\nThere are two possibilities for the hyperplane $H$.\n\\begin{itemize}\n \\item[\\textbullet] The hyperplane is $\\mathrm{Ker}(\\gamma_r-1)$ for $r\\in[1,d]$.\nBy Lemma \\ref{ch3:lem_proj-conj-par-same}, we can assume that $H=\\mathrm{Ker}(\\gamma_1-1)$. By Lemma \\ref{ch3:lem_d=1-Cher} there exists a tilting module $\\overline T\\in \\mathcal{A}^\\nu_R[1]$ such that $\\mathrm{KZ}^\\nu_1(P)\\simeq \\overline\\Psi^\\nu_1(\\overline T)$.\nWe get\n$$\n\\mathrm{KZ}^\\nu_d(^\\mathcal{O}\\mathrm{Ind}^{\\Gamma_d}_{W_H}P)\\simeq\\mathrm{Ind}_1^d(\\mathrm{KZ}^\\nu_1(P))\\simeq \\mathrm{Ind}_1^d(\\overline \\Psi^\\nu_1(\\overline T))\\simeq \\overline\\Psi^\\nu_d(F^{d-1}(\\overline T)).\n$$\nHere the first isomorphism follows from \\cite[(6.1)]{RSVV}, the third isomorphism follows from Lemma \\ref{ch3:lem_com-F-calA-Hecke}.\n \\item[\\textbullet] The hyperplane is $\\mathrm{Ker}(s_{r,t}^\\gamma-1)$ for $r,t\\in[1,d]$, $\\gamma\\in\\Gamma$.\nBy Lemma \\ref{ch3:lem_proj-conj-par-same}, we can assume that $H=\\mathrm{Ker}(s_{1,2})$. By Lemma \\ref{ch3:lem_d=2,l=1-Cher} there is a tilting module $\\overline T^+\\in\\mathcal{A}_R^+[2]$ such that $\\mathrm{KZ}^+_2(P)\\simeq \\overline \\Psi^+_2(\\overline T^+)$. By Lemma \\ref{ch3:lem_d=1-to-d=2,l=1} there is a tilting module $\\overline T\\in\\mathcal{A}^\\nu_R[2]$ such that $\\mathrm{Ind}_{2,+}^{2,\\nu}(\\overline\\Psi^+_2(\\overline T^+))\\simeq \\overline\\Psi^\\nu_2(\\overline T)$. Thus we get $\\mathrm{Ind}_{2,+}^{2,\\nu}\\mathrm{KZ}^+_2(P)\\simeq \\overline\\Psi^\\nu_2(\\overline T)$.\n\\end{itemize}\n\nWe obtain\n$$\n\\mathrm{KZ}^\\nu_d(^\\mathcal{O}\\mathrm{Ind}^{\\Gamma_d}_{W_H}P)\\simeq \\mathrm{Ind}_{2,+}^{d,\\nu}(\\mathrm{KZ}^+_2(P))\\simeq \\mathrm{Ind}_{2,\\nu}^{d,\\nu}(\\overline\\Psi^\\nu_2(\\overline T))\\simeq \\Psi^\\nu_d(F^{d-2}(\\overline T)).\n$$\nHere the first isomorphism follows from \\cite[(6.1)]{RSVV}, the third isomorphism follows from Lemma \\ref{ch3:lem_com-F-calA-Hecke}.\n\\end{proof}\n\nNow, composing the equivalences of categories in Propositions \\ref{ch3:prop_equiv-Cher-bfA}, \\ref{ch3:prop_constr-of-equiv} we obtain the following result.\n\n\\smallskip\n\\begin{coro}\n\\label{ch3:coro_constr-of-equiv-A-A}\nAssume that $\\nu_r\\geqslant d$ for each $r\\in[1,l]$ and $e>2$.\nThere is an equivalence of categories $\\theta'_d\\colon \\mathbf{A}^\\nu_R[d]\\simeq \\mathcal{A}^\\nu_R[d]$ such that we have the following isomorphism of functors $\\overline\\Psi^\\nu_d\\circ\\theta'_d\\simeq\\Psi^\\nu_d$.\\qed\n\\end{coro}\nFor each $\\alpha\\in Q^+_I$ such that $|\\alpha|=d$ the equivalence $\\theta'_d\\colon \\mathbf{A}^\\nu_R[d]\\simeq \\mathcal{A}^\\nu_R[d]$ restricts to an equivalence $\\theta'_\\alpha\\colon \\mathbf{A}^\\nu_R[\\alpha]\\simeq \\mathcal{A}^\\nu_R[\\alpha]$ such that we have an isomorphism of functors $\\overline\\Psi^\\nu_\\alpha\\circ\\theta'_\\alpha\\simeq\\Psi^\\nu_\\alpha$.\n\n\n\n\n\n\n\n\nFrom now on we work over the field $\\mathbb{C}$. Recall that we fixed an isomorphism $R^{\\Lambda}_{\\alpha}\\simeq H^\\nu_{\\alpha}(\\zeta_e)$.\nThe following lemma can be proved by the method used in \\cite[Sec.~5.9]{RSVV}.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_F-comm-KZ-e}\nAssume that $\\nu_r>|\\alpha|$ for each $r\\in[1,l]$.\nThe following diagrams are commutative modulo an isomorphism of functors.\n$$\n\\begin{CD}\n\\mathbf{A}^\\nu[\\alpha]@>{F_k}>> \\mathbf{A}^\\nu[\\alpha+\\alpha_k]\\\\\n@V{\\Psi^\\nu_\\alpha}VV @V{\\Psi^\\nu_{\\alpha+\\alpha_k}}VV\\\\\n\\mathrm{mod}(R^\\Lambda_\\alpha) @>{F^\\Lambda_k}>> \\mathrm{mod}(R^\\Lambda_{\\alpha+\\alpha_k})\n\\end{CD}\n$$\n$$\n\\begin{CD}\n\\mathcal{A}^\\nu[\\alpha]@>{F_k}>> \\mathcal{A}^\\nu[\\alpha+\\alpha_k]\\\\\n@V{\\overline\\Psi^\\nu_\\alpha}VV @V{\\overline\\Psi^\\nu_{\\alpha+\\alpha_k}}VV\\\\\n\\mathrm{mod}(R^\\Lambda_\\alpha) @>{F^{\\Lambda}_k}>> \\mathrm{mod}(R^\\Lambda_{\\alpha+\\alpha_k})\n\\end{CD}\n$$\n\\qed\n\\end{lem}\n\nNow, Theorem \\ref{ch3:thm_intro-main-decomp-functors} follows from the following one.\n\n\\smallskip\n\\begin{thm}\n\\label{ch3:thm_decomp_Fk-A}\nAssume that $\\nu_r>|\\alpha|$ for each $r\\in[1,l]$ and $e>2$.\nThen the following diagram is commutative\n$$\n\\begin{CD}\n\\mathcal{A}^\\nu[\\alpha]@>{F_k}>> \\mathcal{A}^\\nu[\\alpha+\\alpha_k]\\\\\n@A{\\theta'_\\alpha}AA @A{\\theta'_{\\alpha+\\alpha_k}}AA\\\\\n\\mathbf{A}^\\nu[\\alpha]@>{F_k}>> \\mathbf{A}^\\nu[\\alpha+\\alpha_k].\n\\end{CD}\n$$\n\\end{thm}\n\\begin{proof}[Proof]\nThe result follows from Corollary \\ref{ch3:coro_constr-of-equiv-A-A}, Lemma \\ref{ch3:lem_F-comm-KZ-e} and an argument similar to \\cite[Lem.~2.4]{Shan-Fock}.\n\\end{proof}\n\n\\section{Graded lifts of the functors}\n\\label{ch3:sec_gr-lifts}\n\n\\subsection{Graded categories}\nFor any noetherian ring $A$, let $\\mathrm{mod}(A)$ be the category of finitely generated left $A$-modules. For any noetherian $\\mathbb{Z}$-graded ring $A=\\bigoplus_{n\\in\\mathbb{Z}}A_n$, let $\\mathrm{grmod}(A)$ be the category of $\\mathbb{Z}$-graded finitely generated left $A$-modules. The morphisms in $\\mathrm{grmod}(A)$ are the morphisms which are homogeneous of degree zero. For each $M\\in\\mathrm{grmod}(A)$ and each $r\\in\\mathbb{Z}$ denote by $M_r$ the homogeneous component of degree $r$ in $M$. For $n\\in\\mathbb{Z}$ let $M\\langle n \\rangle$ be the $n$th shift of grading on $M$, i.e., we have $(M\\langle n \\rangle)_r=M_{r-n}$. For each $\\mathbb{Z}$-graded finite dimensional $\\mathbb{C}$-vector space $V$, let $\\dim_q V\\in\\mathbb{N}[q,q^{-1}]$ be its graded dimension, i.e., $\\dim_q V=\\sum_{r\\in \\mathbb{Z}}(\\dim V_r) q^r$.\n\nThe following lemma is proved in \\cite[Lem.~2.5.3]{BGS}.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem-grad-unique}\nAssume that $M$ is an indecomposable $A$-module of finite length. Then,\nif $M$ admits a graded lift, this lift is unique up to grading shift and isomorphism.\n\\qed\n\\end{lem}\n\n\\smallskip\n\\begin{df}\nA $\\mathbb{Z}$-\\emph{category} (or a \\emph{graded category}) is an additive category $\\widetilde\\mathcal{C}$ with a fixed auto-equivalence $T\\colon\\widetilde\\mathcal{C}\\to \\widetilde\\mathcal{C}$. We call $T$ the shift functor. For each $X\\in\\widetilde \\mathcal{C}$ and $n\\in\\mathbb{Z}$, we set $X\\langle n \\rangle=T^n(X)$. A functor of $\\mathbb{Z}$-categories is a functor commuting with the shift functor.\n\\end{df}\n\n\\smallskip\nFor a graded noetherian ring $A$ the category $\\mathrm{grmod}(A)$ is a $\\mathbb{Z}$-category where $T$ is the shift of grading, i.e., for $M=\\oplus_{n\\in\\mathbb{Z}}M_n\\in \\mathrm{grmod}(A), ~k\\in\\mathbb{Z}$, we have $T(M)_k=M_{k-1}$.\n\n\\smallskip\n\\begin{df}\n\\label{def_gr-ver}\nLet $\\mathcal{C}$ be an abelian category. We say that an abelian $\\mathbb{Z}$-category $\\widetilde\\mathcal{C}$ is a \\emph{graded version} of $\\mathcal{C}$ if there exists a functor $F_{\\mathcal{C}}\\colon\\widetilde \\mathcal{C}\\to\\mathcal{C}$ and a graded noetherian ring $A$ such that we have the following commutative diagram, where the horizontal arrows are equivalences of categories and the top horizontal arrow is a functor of $\\mathbb{Z}$-categories\n$$\n\\begin{CD}\n\\widetilde \\mathcal{C}\\ @>>> \\mathrm{grmod}(A)\\\\\n@V{F_\\mathcal{C}}VV @V\\mbox{forget}VV\\\\\n\\mathcal{C} @>>> \\mathrm{mod}(A).\n\\end{CD}\n$$\n\\end{df}\n\n\\smallskip\nIn the setup of Definition \\ref{def_gr-ver}, we say that an object $\\widetilde X\\in\\widetilde\\mathcal{C}$ is a \\emph{graded lift} of an object $X\\in\\mathcal{C}$ if we have $F_\\mathcal{C}(\\widetilde X)\\simeq X$. For objects $X, Y\\in\\mathcal{C}$ with fixed graded lifts $\\widetilde{X},\\widetilde{Y}$ the $\\mathbb{Z}$-module $\\mathrm{Hom}_\\mathcal{C}(X,Y)$ admits a $\\mathbb{Z}$-grading given by $\\mathrm{Hom}_\\mathcal{C}(X,Y)_n=\\mathrm{Hom}_{\\widetilde\\mathcal{C}}(\\widetilde X\\langle n\\rangle,\\widetilde Y)$. In the sequel we will often denote the object $X$ and its graded lift $\\widetilde X$ by the same symbol.\n\n\\smallskip\n\\begin{df}\nFor two abelian categories $\\mathcal{C}_1$, $\\mathcal{C}_2$ with graded versions $\\widetilde{\\mathcal{C}_1}$, $\\widetilde{\\mathcal{C}_2}$ we say that the functor of $\\mathbb{Z}$-categories $\\widetilde\\Phi\\colon\\widetilde\\mathcal{C}_1\\to\\widetilde\\mathcal{C}_2$ is a \\emph{graded lift} of a functor $\\Phi\\colon\\mathcal{C}_1\\to\\mathcal{C}_2$ if $F_{\\mathcal{C}_2}\\circ\\widetilde\\Phi=\\Phi\\circ F_{\\mathcal{C}_1}$.\n\\end{df}\n\n\\subsection{The truncated category $O$}\n\nWe can extend the Bruhat order $\\leqslant$ on $\\widetilde W$ to an order $\\leqslant$ on $\\widehat W$ in the following way. For each $w_1,w_2\\in\\widehat W$ we have $w_1\\leqslant w_2$ if and only if there exists $n\\in \\mathbb{Z}$ such that $w_1\\pi^n,w_2\\pi^n\\in \\widetilde W$ and we have $w_1\\pi^n\\leqslant w_2\\pi^n$ in $\\widetilde W$. Note that the order on $\\widehat W$ is defined in such a way that for $w_1,w_2\\in \\widehat W$ we can have $w_1\\leqslant w_2$ only if $\\widetilde W w_1=\\widetilde W w_2$.\n\nFix $\\mu=(\\mu_1,\\cdots,\\mu_e)\\in X_I[N]$. Let $W_\\mu$ be the stabilizer of the weight $1_\\mu\\in P$ in $\\widetilde W$ (or equivalently in $\\widehat W$). \nLet $J_{\\mu}$ (resp. $J_{\\mu,+}$) be the set of shortest (resp. longest) representatives of the cosets $\\widehat W\/W_\\mu$ in $\\widehat W$. For each $v\\in \\widehat W$ put $^vJ_\\mu=\\{w\\in J_\\mu;~w\\leqslant v\\}$ and $^vJ_{\\mu,+}=\\{w\\in J_{\\mu,+};~w\\leqslant v\\}$. They are finite posets.\n\nAssume that $R$ is a local deformation ring. Let $^vO_{\\mu,R}$ be the Serre subcategory of $O_{\\mu,R}$ generated by the modules $\\Delta^{w(1_\\mu)}_R$ with $w\\in {^vJ_\\mu}$. This is a highest weight category, see \\cite[Lem.~3.7]{SVV}.\nNote that the definition of the category $^vO_{\\mu,R}$ does not change if we replace $v$ by the minimal length element in $vW_\\mu$ (i.e., by an element of $J_\\mu$). However, in some situations it will be more convenient to assume that $v$ is maximal in $vW_\\mu$ (and not minimal).\n\nRecall the decomposition\n$$O_{\\mu,R}=\\bigoplus_{n\\in\\mathbb{Z}}\\mathcal{O}_{\\pi^n(1_\\mu),R}$$\nin (\\ref{ch3:eq_dec-O-What-Wtilde}). Note that the definition of the order on $\\widehat W$ implies that the category $^vO_{\\mu,R}$ lies in $\\mathcal{O}_{\\pi^n(1_\\mu),R}$, where $n\\in\\mathbb{Z}$ is such that $v\\in \\widetilde W\\pi^n$.\n\n\\subsection{Linkage}\nWe still consider the non-parabolic category $O$. In particular we have $l=N$.\n\nLet $\\bfk$ be a deformation ring that is a field. Recall that the affine Weyl group $\\widetilde W$ is generated by reflections $s_\\alpha$, where $\\alpha$ is a real affine root. Now we consider the following equivalence relation $\\sim_\\bfk$ on $P$. We define it as the equivalence relation generated by $\\lambda_1\\sim_\\bfk\\lambda_2$ when $\\widetilde\\lambda_1+\\widehat\\rho=s_\\alpha(\\widetilde\\lambda_2+\\widehat\\rho)$ for some real affine root $\\alpha$. The definition of $\\sim_\\bfk$ depends on $\\bfk$ because the definitions of $\\widetilde\\lambda$ and $\\widehat\\rho$ depend on the elements $\\tau_r,\\kappa\\in \\bfk$.\n\nNow, let $R$ be a deformation ring that is a local ring with residue field $\\bfk$. Then for $\\lambda_1,\\lambda_2\\in P$ we write $\\lambda_1\\sim_R\\lambda_2$ if and only if we have $\\lambda_1\\sim_\\bfk\\lambda_2$. Note that the definition of the equivalence relation above is motivated by \\cite[Thm.~3.2]{Fie-cen}.\n\nIn the particular case when $R$ is a local deformation ring, the equivalence relation $\\sim_R$ coincides with the equivalence relation $\\sim_\\mathbb{C}$ because we have $\\tau_r=0$ and $\\kappa=e$ in the residue field of $R$. The relation $\\sim_\\mathbb{C}$ can be easily described in terms of the $e$-action of $\\widehat W$ on $P$, introduced in Section \\ref{ch3:subs_ext-aff}. We have $\\lambda_1\\sim_\\mathbb{C}\\lambda_2$ if and only the elements $\\lambda_1+\\rho$ and $\\lambda_2+\\rho$ of $P^{(e)}$ are in the same $\\widetilde W$-orbit.\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk-small-orbits}\nLet $\\bfk$ be as above.\n\n$(a)$ Assume that for each $r,t\\in[1,l]$ such that $r\\ne t$ we have $\\tau_r-\\tau_t\\not\\in \\mathbb{Z}$. In this case the equivalence relation $\\sim_\\bfk$ is the equality.\n\n$(b)$ Assume that we have $\\tau_r-\\tau_t\\in\\mathbb{Z}$ for a unique couple $(r,t)$ as above. In this case each equivalence class with respect to $\\sim_\\bfk$ contains at most two elements. \n\n$(c)$ Let $R$ be as local deformation ring in general position with the field of fractions $K$. By $(a)$, the equivalence relation $\\sim_K$ is just the equality. Now, let $\\mathfrak{p}$ be a prime ideal of height $1$ in $R$. In this case, each equivalence class with respect to $\\sim_{R_\\mathfrak{p}}$ contains at most two elements (this follows from \\cite[Prop.~5.22~$(a)$]{RSVV}, $(a)$ and $(b)$). \n\\end{rk}\n\n\\smallskip\nThe relation $\\sim_R$ yields a decomposition of the category $O_{-e,R}$ in a direct sum of subcategories, see \\cite[Prop.~2.8]{Fie-cen}. More precisely, let $\\Lambda$ be an $\\sim_R$-equivalence class in $P$. Let $\\mathcal{O}_{\\Lambda,R}$ be the Serre subcategory of $O_{-e,R}$ generated by $\\Delta(\\lambda)$ for $\\lambda\\in\\Lambda$. Then we have\n\\begin{equation}\n\\label{ch3:eq_block-decomp-Fie}\nO_{\\mu,R}=\\bigoplus_{\\Lambda\\subset P[\\mu]-\\rho}\\mathcal{O}_{\\Lambda,R}.\n\\end{equation}\nFor example, if $R$ is a local deformation ring, then this decomposition coincides with (\\ref{ch3:eq_dec-O-What-Wtilde}).\nThe following lemma explains what happens after the base change, see \\cite[Lem.~2.9,~Cor.~2.10]{Fie-cen}.\n\n\\smallskip\n\\begin{lem}\nThe $R$ and $T$ be deformation rings that are local and let $R\\to T$ be a ring homomorphism.\n\n$(a)$ The equivalence relation $\\sim_T$ is finer than the relation $\\sim_R$.\n\n$(b)$ Let $\\Lambda$ be an equivalence class with respect to $\\sim_R$. Then $T\\otimes_R \\mathcal{O}_{\\Lambda,R}$ is equal to $\\bigoplus_{\\Lambda'}\\mathcal{O}_{\\Lambda',T}$, where the sum is taken by all $\\sim_T$-equivalence classes $\\Lambda'$ in $\\Lambda$.\n\\qed\n\\end{lem}\n\n\\smallskip\n\\begin{df}\n\\label{ch3:def_generic}\nWe say that the category $\\mathcal{O}_{\\Lambda,R}$ is \\emph{generic} if $\\Lambda$ contains a unique element and \\emph{subgeneric} if it contains exactly two elements.\n\\end{df}\n\\smallskip\n\nMore details about the structure of generic and subgeneric categories can be found in \\cite[Sec.~3.1]{Fie-str}.\n\n\\subsection{Centers}\nWe assume that $R$ is a deformation ring that is a local ring with the residue field $\\bfk$ and the field of fractions $K$. Recall that we have $l=N$ because we consider the non-parabolic category $O$.\n\nLet $\\Lambda$ be an equivalence class in $P$ with respect to $\\sim_R$. Consider the category $\\mathcal{O}_{\\Lambda,R}$ as in (\\ref{ch3:eq_block-decomp-Fie}). There is a partial order $\\leqslant$ on $\\Lambda$ such that $\\lambda_1\\leqslant \\lambda_2$ when $\\widetilde\\lambda_2-\\widetilde\\lambda_1$ is a sum of simple roots. There exists an element $\\lambda\\in\\Lambda$ such that $\\Lambda$ is minimal in $\\Lambda$ with respect to this order. Assume that $\\Lambda$ is finite.\n\n\\smallskip\n\\begin{df}\nThe \\emph{antidominant projective module in $\\mathcal{O}_{\\Lambda,R}$} is the projective cover in $\\mathcal{O}_{\\Lambda,R}$ of the simple module $L_R(\\lambda)$, where $\\lambda$ is the minimal element in $\\Lambda$. (The existence of the protective cover as above is explained in \\cite[Thm.~2.7]{Fie-cen}.)\n\\end{df}\n\n\\smallskip\nThis notion has no sense if $\\Lambda$ is infinite. However we can consider the truncated version. Fix $v\\in \\widehat W$. We have a truncation of the decomposition (\\ref{ch3:eq_block-decomp-Fie}):\n\\begin{equation}\n\\label{ch3:eq_block-decomp-Fie-trunc}\n^vO_{\\mu,R}=\\bigoplus_{\\Lambda}{^v\\mathcal{O}}_{\\Lambda,R},\n\\end{equation}\nwhere we put ${^v\\mathcal{O}}_{\\Lambda,R}=\\mathcal{O}_{\\Lambda,R}\\cap{^vO}_{\\mu,R}$.\n\nBy \\cite[Thm.~2.7]{Fie-cen} each simple module in $^vO_{\\mu,R}$ has a projective cover.\nAs above, we denote by $\\lambda$ the element of $\\Lambda$ that is minimal in $\\Lambda$ with respect to the order $\\leqslant$.\n\n\\smallskip\n\\begin{df}\nThe \\emph{antidominant projective module in $^v\\mathcal{O}_{\\Lambda,R}$} is the projective cover in $^v\\mathcal{O}_{\\Lambda,R}$ of the simple module $L_R(\\lambda)$.\n\\end{df}\n\n\\smallskip\nFrom now on we assume that $R$ is a local deformation ring in general position, see Section \\ref{ch3:subs_def-ring}. Let $\\bfk$ and $K$ be the residue field and the field of fractions of $R$ respectively. We set $h_0=\\tau_l-\\tau_1-\\kappa+e$ and $h_r=\\tau_{r+1}-\\tau_{r}$ for $r\\in[1,l-1]$. We have $h_r\\ne 0$ for each $r\\in[0,l-1]$ because the ring is assumed to be in general position. Under the assumption on $R$, the decomposition (\\ref{ch3:eq_block-decomp-Fie-trunc}) contains only one term. Let $^vP^\\mu_R$ be the antidominant projective module in $^vO_{\\mu,R}$, i.e., $^vP^\\mu_R$ is the projective cover of $L_R^{\\pi^n(1_\\mu)}$, where $n$ is such that we have $\\pi^n\\leqslant v$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_antid-proj}\n$(a)$ The module $^vP^\\mu_R$ has a $\\Delta$-filtration such that each Verma module in the category $^vO_{\\mu,R}$ appears exactly ones as a subquoitent in this $\\Delta$-filtration.\n\n$(b)$ For each base change $R'\\otimes_R\\bullet$, where $R'$ is a deformation ring that is local, the module $R'\\otimes_R {^vP}^\\mu_{R}$ splits into a direct sum of anti-dominant projective modules in the blocks of the category $^vO_{\\mu,R'}$.\n\n\\end{lem}\n\\begin{proof}[Proof]\nThe first part in $(a)$ holds by \\cite[Thm.~2~(2)]{Fie-str} and the second part in $(a)$ holds by the proof of \\cite[Lem.~4]{Fie-str}. Finally, $(b)$ follows from \\cite[Rem.~5]{Fie-str}.\n\\end{proof}\n\n\\smallskip\nWe will need the following lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_center-of-modA}\nLet $A$ be a ring. Then the center $Z(\\mathrm{mod}(A))$ of the category $\\mathrm{mod}(A)$ is isomorphic to the center $Z(A)$ of the ring $A$.\n\\end{lem}\n\\begin{proof}\nThere is an obvious injective homomorphism $Z(A)\\to Z(\\mathrm{mod}(A))$. We need only to check that it is also surjective.\n\nLet $z$ be an element of the center of $\\mathrm{mod}(A)$. By definition, $z$ consists of an endomorphism $z_M$ of $M$ for each $M\\in\\mathrm{mod}(A)$ such that these endomorphisms commute with all morphisms between the modules in $\\mathrm{mod}(A)$. Then $z_A$ is an endomorphism of the $A$-module $A$ that commutes with each other endomorphism of the $A$-module $A$. Thus $z_A$ is a multiplication by an element $a$ in the center of $A$.\n\nNow we claim that for each module $M\\in\\mathrm{mod}(A)$ the endomorphism $z_M$ is the multiplication by $a$. Fix $m\\in M$. Let $\\phi\\colon A\\to M$ be the morphism of $A$-modules sending $1$ to $m$. We have $\\phi\\circ z_A=z_M\\circ\\phi$. Then\n$$\nz_M(m)=z_M\\circ \\phi(1)=\\phi\\circ z_A(1)=\\phi(a)=am.\n$$\nThis completes the proof.\n\\end{proof}\n\n\\smallskip\nLet $Z_{\\mu,R}$ (resp. $^vZ_{\\mu,R}$) be the center of the category $O_{\\mu,R}$ (resp. $^vO_{\\mu,R}$).\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_eval-on-proj}\nThe evaluation homomorphism $^vZ_{\\mu,R}\\to \\mathrm{End}(^vP^\\mu_R)$ is an isomorphism.\n\\qed\n\\end{prop}\n\\begin{proof}[Proof]\nThe statement is proved in \\cite[Lem.~5]{Fie-str}. There are however some subtle points that we explain.\n\nFirstly, the statement of \\cite[Lem.~5]{Fie-str} announces the result for the non-truncated category $\\mathcal{O}$. But in fact, the main point of the proof of \\cite[Lem.~5]{Fie-str} is to show the statement first in the truncated case.\n\nSecondly, \\cite[Lem.~5]{Fie-str} assumes that the deformation ring $R$ is the localization of the symmetric algebra $S(\\widehat\\mathfrak{h})$ at the maximal ideal generated by $\\widehat\\mathfrak{h}$. Let us sketch the argument of \\cite[Lem.~5]{Fie-str} to show that it works well for our assumption on $R$.\n\nDenote by $ev_R\\colon {^vZ}_{\\mu,R}\\to \\mathrm{End}(^vP^\\mu_R)$ the homomorphism in the statement.\nLet $I(R)$ be the set of prime ideals of height $1$ in $R$.\nWe claim that we have\n\\begin{equation}\n\\label{ch3:eq_Z-as-inter-loc}\n^vZ_{\\mu,R}=\\bigcap_{\\mathfrak{p}\\in I(R)}{^vZ}_{\\mu,R_\\mathfrak{p}},\n\\end{equation}\nwhere the intersection is taken inside of $^vZ_{\\mu,K}$.\nReally, let $^vA_{\\mu,R}$ be the endomorphism algebra of the minimal projective generator of $^vO_{\\mu,R}$. We have an equivalence of categories $^vO_{\\mu,R}\\simeq \\mathrm{mod}(^vA_{\\mu,R})$. By Lemma \\ref{ch3:lem_center-of-modA} we have an algebra isomorphism $^vZ_{\\mu,R}\\simeq Z(^vA_{\\mu,R})$. The same is true if we replace $R$ by $R_\\mathfrak{p}$ or $K$. By \\cite[Prop.~2.4]{Fie-cen} we have $^vA_{\\mu,R_\\mathfrak{p}}\\simeq R_\\mathfrak{p} \\otimes_R{^vA}_{\\mu,R}$, $^vA_{\\mu,K}\\simeq K \\otimes_R{^vA}_{\\mu,R}$. The algebra ${^vA}_{\\mu,R}$ is free over $R$ as an endomorphism algebra of a projective module in $^vO_{\\mu,R}$. Thus we have ${^vA}_{\\mu,R}=\\bigcap_{\\mathfrak{p}\\in I(R)}{^vA}_{\\mu,R_\\mathfrak{p}}$, where the intersection is taken in ${^vA}_{\\mu,K}$. If we intersect each term in the previous formula with $^vZ_{\\mu,K}=Z({^vA}_{\\mu,K})$, we get (\\ref{ch3:eq_Z-as-inter-loc}).\n\n\n Similarly, we have\n$$\n\\mathrm{End}(^vP^\\mu_{R})=\\bigcap_{\\mathfrak{p}\\in I(R)}\\mathrm{End}(R_\\mathfrak{p}\\otimes_R{^vP}^\\mu_{R})\n$$\nincide of $\\mathrm{End}(K\\otimes_R{^vP}^\\mu_{R})$.\n\nTo conclude, we only need to show that the evaluation homomorphisms\n$$\nev_{R_\\mathfrak{p}}\\colon{^vZ}_{\\mu,R_\\mathfrak{p}}\\to\\mathrm{End}(R_\\mathfrak{p}\\otimes_R{^vP}^{\\mu}_{R}),\\qquad ev_K\\colon{^vZ}_{\\mu,K}\\to\\mathrm{End}(K\\otimes_R{^vP}^{\\mu}_R)\n$$\nare isomorphisms for each $\\mathfrak{p}\\in I(R)$ and that the following diagram is commutative\n$$\n\\begin{CD}\n\\mathrm{End}(R_\\mathfrak{p}\\otimes_R{^vP}^{\\mu}_{R})@>>> \\mathrm{End}(K\\otimes_R{^vP}^{\\mu}_R)\\\\\n@A{ev_{R_\\mathfrak{p}}}AA @A{ev_K}AA\\\\\n^vZ_{\\mu,R_\\mathfrak{p}}@>>> ^vZ_{\\mu,K}\\\\\n\\end{CD}\n$$\n\nThe commutativity of the diagram is obvious. Since $R$ is in general position, the category $^vO_{\\mu,K}$ is semisimple, see Remark \\ref{ch3:rk-small-orbits}. Moreover, for each $\\mathfrak{p}\\in I(R)$, the category $^vO_{\\mu,R_\\mathfrak{p}}$ is a direct sum of blocks with at most two Verma modules in each, see Remark \\ref{ch3:rk-small-orbits}. Similarly, by Lemma \\ref{ch3:lem_antid-proj} $(b)$ the localisation $R_\\mathfrak{p}\\otimes_R{^vP}^\\mu_{R}$ of the antidominant projective module splits into a direct sum of antidominant projective modules. Now, the invertibility of $ev_{R_\\mathfrak{p}}$ and $ev_{K}$ follows from \\cite[Lem.~2]{Fie-str}. \n\n\\end{proof}\n\n\\smallskip\nWe will need the following lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_trunc-categ-gen}\nAssume that $\\mathcal{C}_1$ is an abelian category and $\\mathcal{C}_2$ is an abelian subcategory of $\\mathcal{C}_1$. Let $i\\colon \\mathcal{C}_2\\to \\mathcal{C}_1$ be the inclusion functor. For each object $M\\in\\mathcal{C}_1$ we assume that $M$ has a maximal quotient that is in $\\mathcal{C}_2$ and we denote this quotient by $\\tau(M)$. Then we have the following.\n\n$(a)$ The functor $\\tau\\colon\\mathcal{C}_2\\to\\mathcal{C}_1$ is left adjoint to $i$.\n\n$(b)$ Let $L$ be a simple object in $\\mathcal{C}_2$. Assume that $L$ has a projective cover $P$ in $\\mathcal{C}_1$. Then $\\tau(P)$ is a projective cover of $L$ in $\\mathcal{C}_2$.\n\\end{lem}\n\\begin{proof}\nTake $M\\in\\mathcal{C}_1$ and $N\\in\\mathcal{C}_2$. For each homomorphism $f\\colon M\\to N$ we have $M\/\\mathrm{Ker} f\\simeq \\mathrm{Im} f\\in\\mathcal{C}_2$. Thus $\\mathrm{Ker} f$ must contain the kernel of $M\\to\\tau(M)$. This implies that each homomorphism $f\\colon M\\to N$ factors through $\\tau(M)$. This proves $(a)$.\n\nNow, we prove $(b)$. We have a projective cover $p\\colon P\\to L$ in $\\mathcal{C}_1$. First, we clam that the object $\\tau(P)$ is projective in $\\mathcal{C}_2$. Really, by $(a)$ the functors from $\\mathcal{C}_2$ to the category of $\\mathbb{Z}$-modules $\\mathrm{Hom}_{\\mathcal{C}_2}(\\tau(P),\\bullet)$ and $\\mathrm{Hom}_{\\mathcal{C}_1}(P,\\bullet)$ are isomorphic. Thus the first of them should be exact because the second one is exact by the projectivity of $P$. This shows that $\\tau(P)$ is projective in $\\mathcal{C}_2$. Denote by $\\overline p$ the surjection $\\overline p\\colon \\tau(P)\\to L$ induced by $p\\colon P\\to L$. Let $t$ be the surjection $t\\colon P\\to\\tau(P)$. We have $p=\\overline p\\circ t$. Now we must prove that each proper submodule $K\\subset \\tau(P)$ is in $\\ker \\overline p$. Really, if this is not true for some $K$, then $p(t^{-1}(K))$ is nonzero. Then we have $p(t^{-1}(K))=L$ because the module $L$ is simple. Then by the definition of a projective cover we must have $t^{-1}(K)=P$. This is impossible because $t$ is surjective and $K$ is a proper submodule of $\\tau(P)$.\n\\end{proof}\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_grade-trunc-gen}\nLet $A$ be a graded noetherian ring. Let $I\\subset A$ be a homogeneous ideal. Put $\\mathcal{C}_1=\\mathrm{mod}(A)$, $\\mathcal{C}_2=\\mathrm{mod}(A\/I)$, $\\widetilde\\mathcal{C}_1=\\mathrm{grmod}(A)$, $\\widetilde\\mathcal{C}_2=\\mathrm{grmod}(A\/I)$. There is an obvious inclusion of categories $i\\colon\\mathcal{C}_2\\to\\mathcal{C}_1$ and it has an obvious graded lift $\\widetilde i\\colon\\widetilde\\mathcal{C}_2\\to\\widetilde\\mathcal{C}_1$. The left adjoint functor $\\tau$ to $i$ is defined by $\\tau(M)=M\/IM$ and the left adjoint functor $\\widetilde\\tau$ to $\\widetilde i$ is also defined by $\\widetilde\\tau(M)=M\/IM$. This implies that the functor $\\widetilde\\tau$ is a graded lift of $\\tau$.\n\\end{rk}\n\n\n\n\n\n\n\n\n\\smallskip\nRecall that we denote by $s_0,\\cdots,s_{N-1}$ the simple reflections in $\\widetilde W$.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_eval-on-Verma-full}\nWe have an isomorphism\n\\begin{equation}\n\\label{ch3:eq_eval-on-Verma-full}\nZ_{\\mu,R}\\simeq\\left\\{(z_w)\\in\\prod_{w\\in J_\\mu}R;~z_{w}\\equiv z_{s_r w}~\\mathrm{mod}~h_r~\\forall r\\in[0,N-1],w\\in {J_\\mu}\\cap {s_rJ_\\mu}\\right\\}\n\\end{equation}\nwhich maps an element $z\\in {Z}_{\\mu,R}$ to the tuple $(z_w)_{w\\in {J_\\mu}}$ such that $z$ acts on the Verma module $\\Delta^{w(1_\\mu)}_R$ by $z_w$.\n\\end{prop}\n\\begin{proof}[Proof]\nThe statement is proved in \\cite[Thm.~3.6]{Fie-cen} in the case where $R$ is the localization of the symmetric algebra $S(\\widehat\\mathfrak{h})$ at the maximal ideal generated by $\\widehat\\mathfrak{h}$. Similarly to Proposition \\ref{ch3:prop_eval-on-proj}, the same proof works under our assumption on the ring $R$.\n\\end{proof}\n\n\\smallskip\nProposition \\ref{ch3:prop_eval-on-Verma-full} has the following truncated version that can be proved in the same way using the approach of localizations at the prime ideals of height 1. (See, for example, the proof of Proposition \\ref{ch3:prop_eval-on-proj}). For each such localization the result becomes clear by \\cite[Coro.~3.5]{Fie-cen} and Remark \\ref{ch3:rk-small-orbits}.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_eval-on-Verma-trucn}\nWe have an isomorphism\n\\begin{equation}\n\\label{ch3:eq_eval-on-Verma-trunc}\n^vZ_{\\mu,R}\\simeq\\left\\{(z_w)\\in\\prod_{w\\in ^vJ_\\mu}R;~z_{w}\\equiv z_{s_r w}~\\mathrm{mod}~h_r~\\forall r\\in[0,N-1],w\\in {^vJ_\\mu}\\cap s_r{^vJ_\\mu}\\right\\}\n\\end{equation}\nwhich maps an element $z\\in {^vZ}_{\\mu,R}$ to the tuple $(z_w)_{w\\in {^vJ_\\mu}}$ such that $z$ acts on the Verma module $\\Delta^{w(1_\\mu)}_R$ by $z_w$.\n\\qed\n\\end{prop}\n\n\n\n\n\\smallskip\nFor each $v\\in\\widehat W$, set $^vJ=\\{w\\in \\widehat W;~w\\leqslant v\\}$ and\n$$\n^vZ_{R}\\simeq\\left\\{(z_w)\\in\\prod_{w\\in {^vJ}}R;~z_{w}\\equiv z_{s_r w}~\\mathrm{mod}~h_{r}~\\forall r\\in[0,N-1],w\\in {^vJ}\\cap s_r{^vJ}\\right\\}.\n$$\nIf $v$ is in $J_{\\mu,+}$, then the group $W_\\mu$ acts on $^vZ_R$ by $w(z)=z'$ where $z'_x=z_{xw^{-1}}$ for each $x\\in {^vJ}$. Note that the algebra $^vZ^{W_\\mu}_{R}$ of $W_\\mu$-invariant elements in $^vZ_{R}$ is obviously isomorphic to the right hand side in (\\ref{ch3:eq_eval-on-Verma-trunc}). Thus Proposition \\ref{ch3:prop_eval-on-Verma-trucn} identifies the center $^vZ_{\\mu,R}$ of $^vO_{\\mu,R}$ with $^vZ^{W_\\mu}_{R}$.\n\n\n\\subsection{The action on standard and projetive modules}\nAs above, we fix $k\\in[0,e-1]$ and set $\\mu'=\\mu-\\alpha_k$.\nFrom now on, we assume that $R$ is an analytic local deformation ring in general position of dimension $\\leqslant 2$ with resudue field $\\bfk=\\mathbb{C}$, see Section \\ref{ch3:subs_def-ring}.\n\nFrom now on we always assume that we have\n\\begin{equation}\n\\label{ch3:eq_assum_inclusion-stab}\nW_\\mu\\subset W_{\\mu'}.\n\\end{equation}\nThis happens if and only if we have $\\mu_k=1$. In this case we have $J_{\\mu'}\\subset J_\\mu$ and $J_{\\mu',+}\\subset J_{\\mu,+}$. From now on we always assume that the element $v$ is in $J_{\\mu',+}$ (thus $v$ is also in $J_{\\mu,+}$). We have an inclusion of algebras ${^vZ}_{\\mu',R}\\subset {^vZ}_{\\mu,R}$ because ${^vZ}_{\\mu',R}\\simeq {^vZ}_{R}^{W_{\\mu'}}$ and ${^vZ}_{\\mu,R}\\simeq {^vZ}_{R}^{W_{\\mu}}$. Let $\\mathrm{Res}\\colon \\mathrm{mod}(^vZ_{\\mu,R})\\to \\mathrm{mod}(^vZ_{\\mu',R})$ and $\\mathrm{Ind}\\colon \\mathrm{mod}(^vZ_{\\mu',R})\\to \\mathrm{mod}(^vZ_{\\mu,R})$ be the restriction and the induction functors. We may write $\\mathrm{Res}_{\\mu}^{\\mu'}$ and $\\mathrm{Ind}_{\\mu'}^{\\mu}$ to specify the parameters.\n\nIt is easy to see on Verma modules using two lemmas below that the functors $E_k$ and $F_k$ restrict to functors of truncated categories\n$$\nF_k\\colon {^vO}^\\Delta_{\\mu,R}\\to {^vO}^\\Delta_{\\mu',R},\\qquad E_k\\colon {^vO}^\\Delta_{\\mu',R}\\to {^vO}^\\Delta_{\\mu,R}.\n$$\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Fk-Verma-spcase1}\nFor each $w\\in\\widehat W$, we have $F_k(\\Delta^{w(1_\\mu)}_R)\\simeq\\Delta^{w(1_{\\mu'})}_R$.\n\\end{lem}\n\\begin{proof}\nSince $\\mu_k=1$, the weight $w(1_\\mu)\\in P$ has a unique coordinate containing an element congruent to $k$ modulo $e$. Let $r\\in[1,N]$ be the position number of this coordinate.\nBy Proposition \\ref{ch3:prop_functors-on-O-gen} $(e)$, we have $[F_k(\\Delta^{w(1_\\mu)}_R)]=[\\Delta^{w(1_\\mu)+\\epsilon_r}_R]$. The equality of classes in the Grothendieck group implies that we have an isomorphism of modules $F_k(\\Delta^{w(1_\\mu)}_R)\\simeq \\Delta^{w(1_{\\mu})+\\epsilon_r}_R$. Finally, since $w(1_\\mu)+\\epsilon_r=w(1_{\\mu'})$, we get $F_k(\\Delta^{w(1_\\mu)}_R)\\simeq\\Delta^{w(1_{\\mu'})}_R$.\n\\end{proof}\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Ek-Verma-spcase1}\nFor each $w\\in\\widehat W$, we have $[E_k(\\Delta^{w(1_{\\mu'})}_R)]=\\sum_{z\\in W_{\\mu'}\/W_\\mu}[\\Delta^{wz(1_{\\mu})}_R]$.\n\\end{lem}\n\\begin{proof}\nBy Proposition \\ref{ch3:prop_functors-on-O-gen} $(e)$, we have\n\\begin{equation}\n\\label{ch3:eq_Ek-on-Vermas-1}\n[E_{k}(\\Delta^{w(1_{\\mu'})}_R)]=\\sum_{r} [\\Delta^{w(1_{\\mu'})-\\epsilon_r}_R],\n\\end{equation}\nwhere the sum in taken by all indices $r\\in[1,N]$ such that the $r$th coordinate of $w(1_\\mu)$ is congruent to $k+1$ modulo $e$. For each such $r$ we have $w(1_{\\mu'})-\\epsilon_r=wz(1_\\mu)$ for a unique element $z\\in W_{\\mu'}\/W_{\\mu}$. Moreover, the obtained map $r\\mapsto z$ is a bijection from the set of possible indices $r$ to $W_{\\mu'}\/W_{\\mu}$. Thus $(\\ref{ch3:eq_Ek-on-Vermas-1})$ can be rewritten as\n$$\n[E_{k}(\\Delta^{w(1_{\\mu'})}_R)]=\\sum_{z\\in W_{\\mu'}\/W_{\\mu}} [\\Delta^{wz(1_{\\mu})}_R].\n$$\n\\end{proof}\n\n\n\\smallskip\n\\begin{lem}\nWe have $E_{k}(^vP^{\\mu'}_R)\\simeq {^vP}^{\\mu}_R$.\n\\end{lem}\n\\begin{proof}[Proof]\nBy Lemma \\ref{ch3:lem_antid-proj} $(a)$, the class $[^vP^{\\mu}_R]$ of $^vP^{\\mu}_R$ in the Grothendieck group of $^vO_{\\mu,R}^\\Delta$ is the sum of all classes of Verma modules of the category $^vO^{\\Delta}_{\\mu,R}$ and similarly for $[^vP^{\\mu'}_R]$. Taking the sum in the equality in Lemma \\ref{ch3:lem_Ek-Verma-spcase1} over all $w\\in {^vJ_{\\mu'}}$, we get $[E_{k}(^vP^{\\mu'}_R)]=[^vP^{\\mu}_R]$. Finally, this yields an isomorphism $E_{k}(^vP^{\\mu'}_R)\\simeq {^vP}^{\\mu}_R$ because the modules $E_{k}(^vP^{\\mu'}_R)$ and $^vP^{\\mu}_R$ are projective.\n\\end{proof}\n\n\\smallskip\nFix an isomorphism $E_{k}(^vP^{\\mu'}_R)\\simeq {^vP}^{\\mu}_R$ as above. Then by functoriality it yields an algebra homomorphism $\\mathrm{End}(^vP^{\\mu'}_R)\\to \\mathrm{End}(^vP^{\\mu}_R)$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_diag-cent-proj-mu'-mu}\nThe following diagram of algebra homomorphisms is commutative\n$$\n\\begin{CD}\n\\mathrm{End}(^vP^{\\mu'}_R)@>>> \\mathrm{End}(^vP^{\\mu}_R)\\\\\n@AAA @AAA\\\\\n^vZ_{\\mu',R}@>>> {^vZ}_{\\mu,R},\n\\end{CD}\n$$\nwhere the top horizontal map is as above, the bottom horizontal map is the inclusion and the vertical maps are the isomorphisms from Proposition \\ref{ch3:prop_eval-on-proj}.\n\\end{lem}\n\\begin{proof}[Proof]\nNote that each element in $\\mathrm{End}(^vP^{\\mu}_R)$ is induced by the center ${^vZ}_{\\mu,R}$. In partilucar, each endomorphism of $^vP^{\\mu}_R$ preserves each submodule of $^vP^{\\mu}_R$. Moreover, by Lemma \\ref{ch3:lem_antid-proj} $(a)$, each Verma module in $^vO_{\\mu,R}$ is isomorphic to a subquotient of $^vP^{\\mu}_R$. Thus, by Proposition \\ref{ch3:prop_eval-on-proj} and Proposition \\ref{ch3:prop_eval-on-Verma-trucn}, an element of $\\mathrm{End}(^vP^{\\mu}_R)$ is determined by its action on the subquotients of a $\\Delta$-filtration of $^vP^{\\mu}_R$.\n\nFix an element $z=(z_w)$ in $^vZ_{\\mu',R}$, see Proposition \\ref{ch3:prop_eval-on-Verma-trucn}. Fix also a $\\Delta$-filtration of $^vP^{\\mu'}_R$. The element $z$ acts on $^vP^{\\mu'}_R$ in such a way that it preserves each component of the $\\Delta$-filtration and the induced action on the subquotient $\\Delta^{w(1_{\\mu'})}_R$ of $^vP^{\\mu'}_R$ is the multiplication by $z_w$.\n\nFor each $w\\in \\widehat W$, the module $E_{k}(\\Delta^{w(1_{\\mu'})}_R)$ is $\\Delta$-filtered. The subquotients in this $\\Delta$-filtration can be described by Lemma \\ref{ch3:lem_Ek-Verma-spcase1}.\nSince the functor $E_k$ is exact, the $\\Delta$-filtration of $^vP^{\\mu'}_R$ induces a $\\Delta$-filtration of $^vP^{\\mu}_R\\simeq E_{k}(^vP^{\\mu'}_R)$. Thus the image of $z$ by\n$$\n^vZ_{\\mu',R}\\to\\mathrm{End}(^vP^{\\mu'}_R)\\to\\mathrm{End}(^vP^{\\mu}_R)\n$$\nacts on the subquotients of the $\\Delta$-filtration of $^vP^{\\mu}_R$ in the following way: it acts on the subquotient $\\Delta^{w(1_{\\mu})}_R$ of $^vP^{\\mu}_R$ by the multiplication by $z_w$. On the other hand, the image of $z$ by\n$$\n^vZ_{\\mu',R}\\to{^vZ}_{\\mu,R}\\to\\mathrm{End}(^vP^{\\mu}_R)\n$$\nacts on the subquotients in the same way. This proves the statement because an element of $\\mathrm{End}(^vP^{\\mu}_R)$ is determined by its action on the subquotients of a $\\Delta$-filtration of $^vP^{\\mu}_R$.\n\n\\end{proof}\n\n\\subsection{The functor $\\mathbb{V}$}\nNow, we assume that $v$ is an arbitrary elements of $\\widehat W$. \nWe have a functor\n$$\n\\mathbb{V}_{\\mu,R}\\colon {^vO}_{\\mu,R}\\to \\mathrm{mod}(^vZ_{\\mu,R}), \\quad M\\mapsto \\mathrm{Hom}(^vP_R^\\mu,M).\n$$\nSet $^vZ_{\\mu}=\\mathbb{C}\\otimes_R{^vZ}_{\\mu,R}$ and $^vZ=\\mathbb{C}\\otimes_R{^vZ}_{R}$. By \\cite[Prop.~2.6]{Fie-cen} we have $\\mathbb{C}\\otimes_R{^vP}^\\mu_{R}={^vP}^\\mu$. Next, \\cite[Prop.~2.7]{Fie-cen} yields an algebra isomorphism $^vZ_{\\mu}\\simeq \\mathrm{End}(^vP_{\\mu})$. Now, consider the functor\n$$\n\\mathbb{V}_{\\mu}\\colon {^vO}_{\\mu}\\to \\mathrm{mod}(^vZ_{\\mu}),\\quad M\\mapsto \\mathrm{Hom}(^vP^\\mu,M).\n$$\n\nA Koszul grading on the category $^vO_{\\mu}$ is constructed in \\cite{SVV}. Let us denote by $^v\\widetilde O_{\\mu}$ the graded version of this category.\n\nThe functor $\\mathbb{V}$ above has following properties.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_Vk}\n$(a)$ The functor $\\mathbb{V}_{\\mu,R}$ is fully faithful on $^vO_{\\mu,R}^\\Delta$.\n\n$(b)$ The functor $\\mathbb{V}_{\\mu}$ is fully faithful on projective objects in $O_{\\mu}$.\n\n$(c)$ The functor $\\mathbb{V}_{\\mu}$ admits a graded lift $\\widetilde\\mathbb{V}_{\\mu}\\colon {^v\\widetilde O}_{\\mu}\\to \\mathrm{grmod}(^vZ_{\\mu})$.\n\\end{prop}\n\\begin{proof}[Proof]\nPart $(a)$ is \\cite[Prop.~2]{Fie-str} (1). Part $(b)$ is \\cite[Prop.~4.50]{SVV} $(b)$. Part $(c)$ is given in the proof of \\cite[Lem.~5.10]{SVV}.\n\\end{proof}\n\n\\subsection{The cohomology of Schubert varieties}\n\nAll cohomology groups in this section have coefficients in $\\mathbb{C}$.\n\nSet $G=GL_N$. Let $B\\subset G(\\mathbb{C}((t)))$ be the standard Borel subgroup. Let $P_\\mu\\subset G(\\mathbb{C}((t)))$ be the parabolic subgroup with Lie algebra $\\widehat\\mathbf{p}_{\\mu}$.\nLet $X_\\mu$ be the partial affine flag ind-scheme $G(\\mathbb{C}((t)))\/P_\\mu$. The affine Bruhat cells in $X_\\mu$ are indexed by $J_\\mu$. For $w\\in J_\\mu$ we denote by $X_{\\mu,w}$ (resp. $\\overline X_{\\mu,w}$) the corresponding finite dimensional affine Bruhat cell (resp. Schubert variety). Note that we have $X_{\\mu,w}\\simeq \\mathbb{C}^{\\ell(w)}$. The following statement is proved in \\cite[Prop.~4.43~(a)]{SVV}.\n\n\\smallskip\n\\begin{lem}\nAssume $v\\in J_\\mu\\cap \\widetilde W$. There is an isomorphism of graded algebras between $^vZ_{\\mu}$ and the cohomology $H^*(\\overline X_{\\mu,v})$.\n\\qed\n\\end{lem}\n\n\\smallskip\nNow we are going to extend the notions $X_{\\mu,w}$ and $\\overline X_{\\mu,w}$ to an arbitrary $w\\in J_\\mu$ in order to get an extended version of the previous lemma.\n\nLet $\\pi$ be the cyclic shift the of Dynkin diagram of type $A_{N-1}^{(1)}$ that takes the root $\\alpha_i$ to the root $\\alpha_{i-1}$ for $i\\in\\mathbb{Z}\/N\\mathbb{Z}$. It yields an automorphism $\\pi\\colon G\\to G$. \nThen for each $n\\in\\mathbb{Z}$ we have a parabolic subgroup $\\pi^n(P_\\mu)\\subset G(\\mathbb{C}((t)))$. Recall that the symbol $\\pi$ also denotes an element of $\\widehat W$, see Section \\ref{ch3:subs_ext-aff}. Let $X^n_\\mu$ be the partial affine flag ind-scheme defined in the same way as $X_\\mu$ with respect to the parabolic subgroup $\\pi^n(P_\\mu)\\subset G(\\mathbb{C}((t)))$ instead of $P_\\mu$. In particular, we have $X_\\mu=X^0_\\mu$. Let us use the abbreviation $\\pi^n(W_\\mu)$ for the subgroup $\\pi^nW_\\mu\\pi^{-n}$ of $\\widetilde W$. Note that the group $\\pi^n(W_\\mu)$ is the Weyl group of the Levi of $\\pi^n(P_\\mu)$. The Bruhat cells and the Schubert varieties in $X_\\mu^n$ are indexes by the shortest representatives of the cosets in $\\widetilde W\/\\pi^n(W_{\\mu})$. For such a representative $w$ let $X_{\\mu,w}^n$ (resp. $\\overline X_{\\mu,w}^n$) be the Bruhat cell (resp. Schubert variety) in $X^n_\\mu$.\n\nAssume that $v\\in J_\\mu$. Then $v$ has a unique decomposition of the form $v=w\\pi^n$, such that $w$ is minimal in $w\\pi^{n}(W_{\\mu})$. Then we set $X_{\\mu,v}=X^n_{\\mu,w}$ and $\\overline X_{\\mu,v}=\\overline X^n_{\\mu,w}$. Note that for $v\\in J_\\mu$ we have $X_{\\mu,v}\\simeq \\mathbb{C}^{\\ell(v)}$. \nWe get the following generalization of the lemma above.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_coh-Schub-extended}\nAssume $v\\in J_\\mu$. There is an isomorphism of graded algebras between $^vZ_{\\mu}$ and the cohomology $H^*(\\overline X_{\\mu,v})$.\n\\end{lem}\n\\begin{proof}\nConsider the decomposition $v=w\\pi^n$ as above. By definition, the truncated category $^vO_{\\mu}$ is a Serre subcategory of $\\mathcal{O}_{\\pi^n(1_\\mu)}$. It is generated by modules $L^{x\\pi^n(1_\\mu)}$, where $x\\in \\widetilde W$ is such that $x\\leqslant w$. Note also that the stabilizer of the weight $\\pi^n(1_\\mu)$ in $\\widetilde W$ is $\\pi^n(W_\\mu)$. Then, by \\cite[Prop.~4.43~(a)]{SVV}, we have an isomorphism of graded algebras $^vZ_{\\mu}=H^*(\\overline X_{\\mu,w}^n)$. On the other hand the variety $\\overline X_{\\mu,v}$ is defined as $\\overline X_{\\mu,w}^n$.\n\\end{proof}\n\n\\smallskip\nNow, assume that $v\\in J_{\\mu',+}$. Recall that in this case we have an inclusion of algebras ${^vZ}_{\\mu',R}\\subset {^vZ}_{\\mu,R}$ because of the assumption $W_{\\mu}\\subset W_{\\mu'}$. We want to show that after the base change we get an inclusion of algebras ${^v}Z_{\\mu'}\\subset {^vZ}_{\\mu}$. However, this is not obvious because the functor $\\mathbb{C}\\otimes_R\\bullet$ is not left exact. But this fact can be justified using geometry. The injectivity of the homomorphism ${^v}Z_{\\mu'}\\to {^vZ}_{\\mu}$ is a consequence of Lemma \\ref{ch3:lem_Zmu-Zmu'-mod} below.\n\nDenote by $w_\\mu$ the longest elements in $W_{\\mu}$. The shortest elements in $vW_{\\mu}$ and $vW_{\\mu'}$ are respectively $vw_\\mu$ and $vw_{\\mu'}$. By Lemma \\ref{ch3:lem_coh-Schub-extended}, we have algebra isomorphisms ${^vZ}_{\\mu}\\simeq H^*(\\overline X_{\\mu,vw_\\mu})$ and ${^vZ}_{\\mu'}\\simeq H^*(\\overline X_{\\mu',vw_{\\mu'}})$.\n\nThe group $\\widetilde W$ is a Coxeter group. In particular we have a length function $\\ell\\colon \\widetilde W\\to \\mathbb{N}$. We can extend it to $\\widehat W$ be setting $\\ell(w\\pi^n)=\\ell(w)$ for each $n\\in\\mathbb{Z}$ and $w\\in\\widetilde W$. Now we are ready to prove the following result.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Zmu-Zmu'-mod}\nThere is the following isomorphism of graded $^vZ_{\\mu'}$-modules\n$$\n^vZ_{\\mu}\\simeq \\bigoplus_{r=0}^{\\mu_{k+1}}{^vZ}_{\\mu'} \\langle 2r\\rangle.\n$$\n\\end{lem}\n\\begin{proof}\nLet $J_{\\mu'}^\\mu$ be the set of shortest representatives of classes in $W_{\\mu'}\/W_\\mu$.\nWe have the following decomposition into affine cells \n$$\n\\overline X_{\\mu,vw_\\mu}=\\coprod_{w\\in {^{v}J}_\\mu}X_{\\mu,w}=\\coprod_{w\\in {^{v}J}_{\\mu'}}\\coprod_{x\\in J_{\\mu'}^\\mu}X_{\\mu,wx}.\n$$\nThis yields\n$$\n\\begin{array}{rcl}\n^vZ_{\\mu}&\\simeq & H^*(\\overline X_{\\mu,vw_\\mu})\\\\\n&\\simeq&\\bigoplus_{w\\in {^vJ}_\\mu}H^*(X_{\\mu,w})\\langle 2\\ell(vw_\\mu)-2\\ell(w)\\rangle\\\\\n&\\simeq& \\bigoplus_{w\\in {^vJ}_{\\mu'}}\\bigoplus_{x\\in J_{\\mu'}^\\mu}H^*(X_{\\mu',wx})\\langle 2\\ell(vw_\\mu)-2\\ell(w)-2\\ell(x)\\rangle.\n\\end{array}\n$$\nWe also have\n$\\overline X_{\\mu,v}=\\coprod_{w\\in {^vJ}_\\mu} X_{\\mu,w}$. \nThis implies\n$$\n\\begin{array}{rcl}\n^vZ_{\\mu'}&\\simeq& H^*(\\overline X_{\\mu',vw_{\\mu'}})\\\\\n&\\simeq& \\bigoplus_{w\\in {^vJ}_{\\mu'}}H^*(X_{\\mu',w})\\langle 2\\ell(vw_{\\mu'})-2\\ell(w)\\rangle\n\\end{array}\n$$\n\nNote that we have $\\ell(w_{\\mu'})-\\ell(w_\\mu)=\\mu_{k+1}$. Moreover, for each $w\\in {^vJ}_{\\mu'}$ and $x\\in J_{\\mu'}^\\mu$ the variety $X_{\\mu,wx}$ is an affine fibration over $X_{\\mu',w}$. This implies\n$$\n^vZ_{\\mu}\\simeq \\bigoplus_{x\\in J_{\\mu'}^\\mu}{^vZ}_{\\mu'}\\langle 2\\ell(vw_\\mu)-2\\ell(vw_{\\mu'})-2\\ell(x) \\rangle=\\bigoplus_{r=0}^{\\mu_{k+1}}{^vZ}_{\\mu'}\\langle 2r \\rangle.\n$$\n\\end{proof}\n\n\\smallskip\nWe will write $\\mathrm{Ind}$ and $\\mathrm{Res}$ for the induction and restriction functors\n$$\n\\mathrm{Ind}\\colon \\mathrm{mod}(^vZ_{\\mu'})\\to\\mathrm{mod}(^vZ_{\\mu}),\\qquad \\mathrm{Res}\\colon \\mathrm{mod}(^vZ_{\\mu})\\to\\mathrm{mod}(^vZ_{\\mu'}).\n$$\nWe fix the graded lifts of $\\widetilde\\mathrm{Res}$ and $\\widetilde\\mathrm{Ind}$ of the functors $\\mathrm{Res}$ and $\\mathrm{Ind}$ in the following way\n$$\n\\widetilde\\mathrm{Res}(M)=M\\langle -\\mu_{k+1} \\rangle,\\qquad\\widetilde\\mathrm{Ind}(M)={^vZ}_{\\mu}\\otimes_{{^vZ}_{\\mu'}} M.\n$$\n\nNow, Lemma \\ref{ch3:lem_Zmu-Zmu'-mod} implies the following.\n\n\\smallskip\n\\begin{coro}\n\\label{ch3:coro_Res-Ind-adj}\n$(a)$ The pair of functors $(\\mathrm{Res},\\mathrm{Ind})$ is biadjoint.\n\n$(b)$ The pairs of functors $(\\widetilde\\mathrm{Ind},\\widetilde\\mathrm{Res}\\langle{\\mu_{k+1}}\\rangle)$ and $(\\widetilde\\mathrm{Res},\\widetilde\\mathrm{Ind}\\langle{-\\mu_{k+1}}\\rangle)$ are adjoint.\n\n$(c)$\n$$\n\\widetilde\\mathrm{Res}\\circ\\widetilde\\mathrm{Ind} =\\mathrm{Id}^{\\oplus [\\mu_{k+1}+1]_q}:= \\bigoplus_{r=0}^{\\mu_{k+1}}\\mathrm{Id}\\langle{2r-\\mu_{k+1}}\\rangle,\n$$\nwhere $\\mathrm{Id}$ is the identity endofunctor of the category $\\mathrm{grmod}(Z_{\\mu'})$.\n\\qed\n\\end{coro}\n\n\\subsection{Graded lifts of the functors}\n\nAs above we assume $W_\\mu\\subset W_{\\mu'}$ and that $v\\in J_{\\mu',+}$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_diag-Fk-Res}\nThe following diagram of functors is commutative\n$$\n\\begin{CD}\n{^vO}_{\\mu,R}^{\\Delta} @>{F_{k}}>> {^vO}_{\\mu', R}^{\\Delta}\\\\\n@V{\\mathbb{V}_{\\mu, R}}VV @V{\\mathbb{V}_{\\mu',R}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu,R}) @>{\\mathrm{Res}}>> \\mathrm{mod}(^vZ_{\\mu',R}).\n\\end{CD}\n$$\n\\end{lem}\n\\begin{proof}\nLet $M$ be an object in $^vO_{\\mu,R}^{\\Delta}$.\nWe have the following chain of isomorphisms of $^vZ_{\\mu',R}$-modules.\n$$\n\\begin{array}{lll}\n\\mathbb{V}_{\\mu',R}\\circ F_{k}(M)&\\simeq& \\mathrm{Hom}(^vP^{\\mu'}_{R},F_{k}(M))\\\\\n&\\simeq&\\mathrm{Hom}(E_{k}(^vP^{\\mu'}_{R}),M)\\\\\n&\\simeq&\\mathrm{Hom}(^vP^{\\mu}_{R},M)\\\\\n&\\simeq&\\mathbb{V}_{\\mu,R}(M)\\\\\n\\end{array}\n$$\nHere, the $^vZ_{\\mu,R}$-modules in the last two lines are considered as $^vZ_{\\mu',R}$-modules with respect to the inclusion $^vZ_{\\mu',R}\\subset {^vZ}_{\\mu,R}$.\nThe third isomorphism in the chain is an isomorphism of $^vZ_{\\mu',R}$-modules by Lemma \\ref{ch3:lem_diag-cent-proj-mu'-mu}.\n\\end{proof}\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_diag-Ek-Ind}\nThe following diagram of functors is commutative\n$$\n\\begin{CD}\n^vO^{\\Delta}_{\\mu,R} @<{E_{k}}<< ^vO^\\Delta_{\\mu',R}\\\\\n@V{\\mathbb{V}_{\\mu,R}}VV @V{\\mathbb{V}_{\\mu',R}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu,R}) @<{\\mathrm{Ind}}<< \\mathrm{mod}(^vZ_{\\mu',R}).\n\\end{CD}\n$$\n\\end{lem}\n\\begin{proof}\nLet $M$ be an object in $^vO^\\Delta_{\\mu',R}$. We have the following chain of isomorphisms of $^vZ_{\\mu,R}$-modules.\n$$\n\\begin{array}{lll}\n\\mathbb{V}_{\\mu,R}\\circ E_k(M)&\\simeq&\\mathrm{Hom}(^vP^\\mu_R,E_k(M))\\\\\n&\\simeq&\\mathrm{Hom}(F_k(^vP^\\mu_R),M)\\\\\n&\\simeq&\\mathrm{Hom}(\\mathbb{V}_{\\mu',R}\\circ F_k(^vP^\\mu_R),\\mathbb{V}_{\\mu',R}(M))\\\\\n&\\simeq&\\mathrm{Hom}(\\mathrm{Res}\\circ\\mathbb{V}_{\\mu,R}(^vP^\\mu_R),\\mathbb{V}_{\\mu',R}(M))\\\\\n&\\simeq&\\mathrm{Hom}(^vZ_{\\mu,R},\\mathbb{V}_{\\mu',R}(M))\\\\\n&\\simeq&\\mathrm{Ind}\\circ\\mathbb{V}_{\\mu',R}(M)\\\\\n\\end{array}\n$$\nHere the third isomorphism holds by Proposition \\ref{ch3:prop_Vk} $(a)$, the fourth isomorphism holds by Lemma \\ref{ch3:lem_diag-Fk-Res}.\nThe last isomorphism holds because, by Corollary \\ref{ch3:coro_Res-Ind-adj} $(a)$, the functor $\\mathrm{Hom}_{^vZ_{\\mu',R}}(^vZ_{\\mu,R},\\bullet)$, which is obviously right adjoint to $\\mathrm{Res}$, is isomorphic to $\\mathrm{Ind}$.\n\\end{proof}\n\n\n\\smallskip\nNow, Lemmas \\ref{ch3:lem_diag-Fk-Res}-\\ref{ch3:lem_diag-Ek-Ind} imply the following.\n\n\\smallskip\n\\begin{coro}\n\\label{ch3:coro_diag-F-Res}\nThe following diagrams of functors are commutative\n$$\n\\begin{CD}\n^vO_{\\mu} @>{F_{k}}>> ^vO_{\\mu'}\\\\\n@V{\\mathbb{V}_{\\mu}}VV @V{\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @>{\\mathrm{Res}}>> \\mathrm{mod}(^vZ_{\\mu'}),\n\\end{CD}\n$$\n$$\n\\begin{CD}\n^vO_{\\mu} @<{E_{k}}<< ^vO_{\\mu'}\\\\\n@V{\\mathbb{V}_{\\mu}}VV @V{\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @<{\\mathrm{Ind}}<< \\mathrm{mod}(^vZ_{\\mu'}).\n\\end{CD}\n$$\n\\end{coro}\n\\begin{proof}\nPassage to the residue field in Lemma \\ref{ch3:lem_diag-Ek-Ind} implies that the diagrams in the statement are commutative on $\\Delta$-filtered objects. A standard argument (see for example the proof of Lemma \\ref{ch3:lem_gr-lift-E-F+adj-case1}) shows that the commutativity on $\\Delta$-filtered objects implies the commutativity. \n\\end{proof}\n\\smallskip\nLet $^vO_{\\mu}^{\\rm proj}$ and $^v\\widetilde O_{\\mu}^{\\rm proj}$ be the full subcategories of projective modules in $^vO_{\\mu}$ and $^v\\widetilde O_{\\mu}$ respectively. The fully faithfulness of the functor $\\mathbb{V}_{\\mu}$ on projective modules implies the fully faithfulness of the functor $\\widetilde\\mathbb{V}_{\\mu}$ on projective modules. These functors identify $^vO_{\\mu,}^{\\rm proj}$ and $^v\\widetilde O_{\\mu}^{\\rm proj}$ with some full subcategories in $\\mathrm{mod}(^vZ_{\\mu})$ and $\\mathrm{grmod}(^vZ_{\\mu})$ that we denote $\\mathrm{mod}(^vZ_{\\mu})^{\\rm proj}$ and $\\mathrm{grmod}(^vZ_{\\mu})^{\\rm proj}$ respectively. Since the functor $F_k$ takes projective modules to projective modules, the commutativity of the diagram in Corollary \\ref{ch3:coro_diag-F-Res} implies that the functor $\\mathrm{Res}$ takes the category $\\mathrm{mod}(^vZ_{\\mu})^{\\rm proj}$ to $\\mathrm{mod}(^vZ_{\\mu'})^{\\rm proj}$. This implies that its graded lift $\\widetilde\\mathrm{Res}$ takes $\\mathrm{grmod}(^vZ_{\\mu})^{\\rm proj}$ to $\\mathrm{grmod}(^vZ_{\\mu'})^{\\rm proj}$. Similar statements hold for $\\mathrm{Ind}$ and $\\widetilde\\mathrm{Ind}$.\n\n\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_gr-lift-E-F+adj-case1}\n$(a)$ The functors $E_k$ and $F_k$ admit graded lifts $\\widetilde E_k\\colon{^v\\widetilde O}_{\\mu'} \\to {^v\\widetilde O}_{\\mu}$ and $\\widetilde F_k\\colon{^v\\widetilde O}_{\\mu} \\to {^v\\widetilde O}_{\\mu'}$. They can be chosen in such a way that the condition below holds.\n\n$(b)$\nThe following pairs of functors are adjoint\n$$\n(\\widetilde F_{k},\\widetilde E_{k}\\langle -\\mu_{k+1} \\rangle),\\quad (\\widetilde E_{k},\\widetilde F_{k}\\langle \\mu_{k+1} \\rangle).\n$$\n\\end{lem}\n\\begin{proof}\nLet us prove $(a)$. We give the prove only for the functor $F_k$. The proof for $E_k$ is similar. The proof below is similar to the proof of \\cite[Lem.~5.10]{SVV}.\n\nAs explained above, the functor $\\widetilde\\mathrm{Res}$ restricts to a functor $\\mathrm{grmod}(^vZ_{\\mu})^{\\rm proj}\\to\\mathrm{grmod}(^vZ_{\\mu'})^{\\rm proj}$. Together with the equivalences of categories $^v\\widetilde O_{\\mu}^{\\rm proj}\\simeq\\mathrm{grmod}(^vZ_{\\mu})^{\\rm proj}$ and $^v\\widetilde O_{\\mu'}^{\\rm proj}\\simeq\\mathrm{grmod}(^vZ_{\\mu'})^{\\rm proj}$ obtained by restricting $\\widetilde\\mathbb{V}_{\\mu}$ and $\\widetilde\\mathbb{V}_{\\mu'}$ this yields a functor $\\widetilde F_k\\colon {^v\\widetilde O}_{\\mu}^{\\rm proj}\\to {^v\\widetilde O}_{\\mu'}^{\\rm proj}$. Next, we obtain a functor of homotopy categories $\\widetilde F_k\\colon K^b(^v\\widetilde O_{\\mu}^{\\rm proj})\\to K^b(^v\\widetilde O_{\\mu'}^{\\rm proj})$. Since the categories $^v\\widetilde O_{\\mu}$ and $^v\\widetilde O_{\\mu'}$ have finite global dimensions, we have equivalences of categories $K^b(^v\\widetilde O_{\\mu}^{\\rm proj})\\simeq D^b(^v\\widetilde O_{\\mu})$ and $K^b(^v\\widetilde O_{\\mu'}^{\\rm proj})\\simeq D^b(^v\\widetilde O_{\\mu'})$.\nThus we get a functor of triangulated categories $\\widetilde F_k\\colon D^b(^v\\widetilde O_{\\mu})\\to D^b(^v\\widetilde O_{\\mu'})$. If we repeat the same construction for non-graded categories, we obtain a functor $F_k\\colon D^b(^vO_{\\mu})\\to D^b(^vO_{\\mu'})$ that is the same as the functor between the bounded derived categories induced by the exact functor $F_k\\colon { ^vO}_{\\mu}\\to {^vO}_{\\mu'}$, see Corollary \\ref{ch3:coro_diag-F-Res}. This implies that the following diagram is commutative\n$$\n\\begin{CD}\nD^b(^v\\widetilde O_{\\mu})@>{\\widetilde F_k}>>D^b(^v\\widetilde O_{\\mu'})\\\\\n@V{\\rm forget}VV @V{\\rm forget}VV\\\\\nD^b(^vO_{\\mu})@>{F_k}>> D^b(^vO_{\\mu'})\\\\\n\\end{CD}\n$$\nSince the bottom functor takes $^vO_{\\mu}$ to $^vO_{\\mu'}$, the top functor takes $^v\\widetilde O_{\\mu}$ to $^v\\widetilde O_{\\mu'}$. This completes the proof of $(a)$.\n\n\nNow we prove $(b)$.\nThe functors $\\widetilde E_k$ and $\\widetilde F_k$ are constructed as unique functors such that we have the following commutative diagrams\n\\begin{equation}\n\\label{ch3:eq_diag-E-Res-grad}\n\\begin{CD}\n^vO_{\\mu} @>{\\widetilde F_{k}}>> ^vO_{\\mu'}\\\\\n@V{\\widetilde\\mathbb{V}_{\\mu}}VV @V{\\widetilde\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu'}) @>{\\widetilde\\mathrm{Res}}>> \\mathrm{mod}(^vZ_{\\mu'}),\n\\end{CD}\n\\qquad\n\\begin{CD}\n^vO_{\\mu} @<{\\widetilde E_{k}}<< ^vO_{\\mu'}\\\\\n@V{\\widetilde\\mathbb{V}_{\\mu}}VV @V{\\widetilde\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @<{\\widetilde\\mathrm{Ind}}<< \\mathrm{mod}(^vZ_{\\mu'}).\n\\end{CD}\n\\end{equation}\n\nBy Corollary \\ref{ch3:coro_Res-Ind-adj} $(b)$ and Proposition \\ref{ch3:prop_Vk} $(b)$, the restrictions of the pairs $(\\widetilde F_{k},\\widetilde E_{k}\\langle -\\mu_{k+1} \\rangle)$ and $(\\widetilde E_{k},\\widetilde F_{k}\\langle \\mu_{k+1} \\rangle)$ to the subcategories of projective objects are biadjoint. We can conclude using the lemma below.\n\\end{proof}\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_lift-adj-proj}\nLet $\\mathcal{C}_1$, $\\mathcal{C}_2$ be abelian categories of finite global dimension and let $\\mathcal{C}_1'$, $\\mathcal{C}_2'$ be the full subcategories of projective objects. Assume that $E\\colon \\mathcal{C}_1\\to \\mathcal{C}_2$, $F\\colon \\mathcal{C}_2\\to\\mathcal{C}_1$ are exact functors. Assume that $E$ and $F$ send projective objects to projective objects and denote $E'\\colon \\mathcal{C}_1'\\to \\mathcal{C}_2'$, $F'\\colon \\mathcal{C}_2'\\to \\mathcal{C}_1'$ the restrictions of $E$ and $F$. Assume that the pair $(E',F')$ is adjoint. Then the pair $(E,F)$ is adjoint.\n\\end{lem}\n\\begin{proof}[Proof]\nLet\n$$\n\\varepsilon'\\colon E'F'\\to{\\mathrm{Id}},\\qquad\n\\eta'\\colon\\mathrm{Id}\\to F'E'\n$$\nbe the counit and the unit of the adjoint pair $(E',F')$.\n\nWe can extend the functors $E'$ and $F'$ to functors $E'\\colon K^b(\\mathcal{C}_1')\\to K^b(\\mathcal{C}_2')$ and $F'\\colon K^b(\\mathcal{C}_2')\\to K^b(\\mathcal{C}_1')$ of the homotopy categories of bounded complexes. The counit $\\varepsilon'$ and the unit $\\eta'$ extend to natural transformations of functors of homotopy categories. These extended natural transformations still satisfy the properties of the counit and the unit of an adjuntion. Thus the extended pair $(E',F')$ is adjoint.\n\nSince the categories $\\mathcal{C}_1$ and $\\mathcal{C}_2$ have finite global dimensions, we have equivalences of categories\n\\begin{equation}\n\\label{ch3:eq_K(C')=D(C)}\nK^b(\\mathcal{C}_1')\\simeq D^b(\\mathcal{C}_1),\\qquad K^b(\\mathcal{C}_2')\\simeq D^b(\\mathcal{C}_2).\n\\end{equation}\nBy construction, the functors\n\\begin{equation}\n\\label{ch3:eq_E-F-D(C)}\nE\\colon D^b(\\mathcal{C}_1)\\to D^b(\\mathcal{C}_2),\\quad F\\colon D^b(\\mathcal{C}_2)\\to D^b(\\mathcal{C}_1)\n\\end{equation}\nobtained from functors $E'$ and $F'$ via the equivalences (\\ref{ch3:eq_K(C')=D(C)}) coincide with the functors induced from $E\\colon \\mathcal{C}_1\\to\\mathcal{C}_2$ and $F\\colon \\mathcal{C}_2\\to\\mathcal{C}_1$. The pair of functors $(E,F)$ in (\\ref{ch3:eq_E-F-D(C)}) is adjoint with a counit $\\varepsilon$ and a unit $\\eta$, obtained from $\\varepsilon'$ and $\\eta'$. These counit and unit restrict to natural transformations of functors of abelian categories $E\\colon \\mathcal{C}_1\\to\\mathcal{C}_2$ and $F\\colon \\mathcal{C}_2\\to\\mathcal{C}_1$. This proves the statement.\n\\end{proof}\n\n\\smallskip\nWe need the following lemma later.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_prod-FE-case1}\nWe have the following isomorphism of functors\n$$\n\\widetilde F_k\\widetilde E_k\\simeq \\mathrm{Id}^{\\oplus [\\mu_{k+1}+1]_q}:=\\bigoplus_{r=0}^{\\mu_{k+1}}\\mathrm{Id}\\langle 2r-\\mu_{k+1} \\rangle,\n$$\nwhere $\\mathrm{Id}$ is the identity endofunctor of the category $^v\\widetilde O_{\\mu'}$.\n\\end{lem}\n\\begin{proof}[Proof]\nBy Corollary \\ref{ch3:coro_Res-Ind-adj} $(c)$ we have $\\widetilde \\mathrm{Res}\\circ\\widetilde\\mathrm{Ind}\\simeq \\mathrm{Id}^{\\oplus [\\mu_{k+1}+1]_q}$. Then the diagrams (\\ref{ch3:eq_diag-E-Res-grad}) and Proposition \\ref{ch3:prop_Vk} $(b)$ yield an isomorphism $\\widetilde F_k\\widetilde E_k\\simeq \\mathrm{Id}^{\\oplus [\\mu_{k+1}+1]_q}$ on projective modules in $^v\\widetilde O_{\\mu'}$. This isomorphism can be extended to the category $^v\\widetilde O_{\\mu'}$ in the same way as in the proof of Lemma \\ref{ch3:lem_lift-adj-proj}.\n\\end{proof}\n\n\\subsection{The case $W_{\\mu'}\\subset W_{\\mu}$}\n\\label{ch3:subs_second-case}\nIn the sections above we assumed $W_{\\mu}\\subset W_{\\mu'}$ (or equivalently $\\mu_k=1$). In this section we announce similar results in the case $W_{\\mu'}\\subset W_{\\mu}$ (or equivalently $\\mu_{k+1}=0$). All the proofs are the same as in the previous case but the roles of $E_k$ and $F_k$ should be exchanged.\n\nHere we always assume that $v$ is in $J_{\\mu,+}$ (thus also in $J_{\\mu',+}$).\nIn contrast with the situation above, we have $^vZ_{\\mu'}\\subset {^vZ}_{\\mu}$. Consider the induction and the restriction functors $\\mathrm{Ind}\\colon \\mathrm{mod}(^vZ_{\\mu'})\\to \\mathrm{mod}(^vZ_{\\mu})$ and $\\mathrm{Res}\\colon\\mathrm{mod}(^vZ_{\\mu})\\to \\mathrm{mod}(^vZ_{\\mu'})$.\n\nSimilarly to Corollary \\ref{ch3:coro_diag-F-Res} we can prove the following statement.\n\n\\smallskip\n\\begin{lem}\nThe following diagrams of functors are commutative\n$$\n\\begin{CD}\n^vO_{\\mu} @>{F_{k}}>> ^vO_{\\mu'}\\\\\n@V{\\mathbb{V}_{\\mu}}VV @V{\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @>{\\mathrm{Ind}}>> \\mathrm{mod}(^vZ_{\\mu'}),\n\\end{CD}\n$$\n$$\n\\begin{CD}\n^vO_{\\mu} @<{E_{k}}<< ^vO_{\\mu'}\\\\\n@V{\\mathbb{V}_{\\mu}}VV @V{\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @<{\\mathrm{Res}}<< \\mathrm{mod}(^vZ_{\\mu'}).\n\\end{CD}\n$$\n\\qed\n\\end{lem}\n\n\\smallskip\nNext, similarly to Lemmas \\ref{ch3:lem_gr-lift-E-F+adj-case1}, \\ref{ch3:lem_prod-FE-case1} we can deduce the following result.\n\n\\smallskip\n\\begin{lem}\n$(a)$ The functors $E_k$ and $F_k$ admit graded lifts $\\widetilde E_k\\colon{^v\\widetilde O}_{\\mu'} \\to {^v\\widetilde O}_{\\mu}$. They can be chosen in such a way that the conditions below hold.\n\n$(b)$ The following pairs of functors are adjoint\n$$\n(\\widetilde F_{k},\\widetilde E_{k}\\langle \\mu_{k}-1 \\rangle),\\quad (\\widetilde E_{k},\\widetilde F_{k}\\langle -\\mu_{k}+1 \\rangle).\n$$\n\n$(c)$ We have the following isomorphism of functors\n$$\n\\widetilde E_k\\widetilde F_k\\simeq \\mathrm{Id}^{\\oplus [\\mu_{k}]_q}:=\\bigoplus_{r=0}^{\\mu_{k}-1}\\mathrm{Id}\\langle 2r-\\mu_{k}+1 \\rangle,\n$$\nwhere $\\mathrm{Id}$ is the identity endofunctor of the category $^v\\widetilde O_{\\mu}$.\n\\end{lem}\n\n\\section{Koszul duality}\n\\label{ch3:sec-Koszul}\n\n\\subsection{Bimodules over a semisimple basic algebra}\n\\label{ch3:subs_bimod}\nLet $B$ be a $\\mathbb{C}$-algebra isomorphic to a finite direct sum of copies of $\\mathbb{C}$. We have $B=\\bigoplus_{\\lambda\\in\\Lambda}\\mathbb{C} e_\\lambda$ for some idempotents $e_\\lambda$.\n\n\\smallskip\n\\begin{df}\nLet $\\mathrm{bmod}(B)$ be the category of finite dimensional $(B,B)$-bimodules.\n\\end{df}\n\n\\smallskip\nA bimodule $M\\in \\mathrm{bmod}(B)$ can be viewed just as a collection of finite dimensional $\\mathbb{C}$-vector spaces $e_\\lambda M e_\\mu$ for $\\lambda,\\mu\\in\\Lambda$.\nTo each bimodule $M\\in\\mathrm{bmod}(B)$ we can associate a bimodule $M^\\star\\in\\mathrm{bmod}(M)$ as follows $M^\\star=\\mathrm{Hom}_{\\mathrm{bmod}(B)}(M,B\\otimes_{\\mathbb{C}}B)$. The bimodule structure on $M^\\star$ is defined in the following way. For $f\\in M^\\star$, $m\\in M$, $b_1,b_2\\in B$ we have $b_1fb_2(m)=f(b_2mb_1)$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_bimod}\nAssume that $M,N\\in \\mathrm{bmod}(B)$, $X,Y\\in\\mathrm{mod}(B)$, $Z\\in\\mathrm{mod}(B)^{\\rm op}$. Then we have the following isomorphisms:\n\n{\\rm (a)} $\\mathrm{Hom}_{\\mathrm{bmod}(B)}(M,N)\\simeq\\bigoplus_{\\lambda,\\mu\\in\\Lambda}\\mathrm{Hom}_\\mathbb{C}(e_\\lambda M e_\\mu,e_\\lambda M e_\\mu)$,\n\n{\\rm (b)} $\\mathrm{Hom}_B(X,Y)\\simeq \\bigoplus_{\\lambda\\in\\Lambda}\\mathrm{Hom}_\\mathbb{C}(e_\\lambda M,e_\\lambda M)$,\n\n{\\rm (c)} $X\\otimes_B Z=\\bigoplus_{\\lambda\\in\\Lambda}Xe_\\lambda\\otimes_\\mathbb{C} e_\\lambda Z$,\n\n{\\rm (d)} $e_\\lambda M^\\star e_\\mu\\simeq (e_\\mu Me_\\lambda)^*$, where $\\bullet^*$ is the usual duality for $\\mathbb{C}$-vector spaces,\n\n{\\rm (e)} $\\mathrm{Hom}_B(M^\\star\\otimes_BX,Y)\\simeq\\mathrm{Hom}_B(X,M\\otimes_BY)$,\n\n{\\rm (f)} $(M\\otimes_B N)^\\star\\simeq N^\\star\\otimes_B M^\\star$.\n\\end{lem}\n\\begin{proof}[Proof]\nParts (a), (b), (c) are obvious. Part (d) follows from (a). Part (e) follows from (b), (c), (d). Part (f) follows from (c), (d).\n\\end{proof}\n\n\n\\subsection{Quadratic dualities}\n\\label{ch3:subs_quad-dual}\n\nLet $A=\\oplus_{n\\in \\mathbb{N}}A_n$ be a finite dimensional $\\mathbb{N}$-graded algebra over $\\mathbb{C}$. Assume that $A_0$ is semisimple and basic. Let $T_{A_0}(A_1)=\\bigoplus_{n\\in N}A_1^{\\otimes n}$ be the tensor algebra of $A_1$ over $A_0$, here $A_1^{\\otimes n}$ means $A_1\\otimes_{A_0}A_1\\otimes_{A_0}\\cdots \\otimes_{A_0}A_1$ with $n$ components $A_1$. The algebra $A$ is said to be \\emph{quadratic} if it is generated by elements of degree $0$ and $1$ with relations in degree $2$, i.e., the kernel of the obvious map $T_{A_0}(A_1)\\to A$ is generated by elements in $A_1\\otimes_{A_0}A_1$.\n\n\\smallskip\n\\begin{df}\nConsider the ($A_0$,$A_0$)-bimodule morphism $\\phi\\colon A_1\\otimes_{A_0}A_1\\to A_2$ given by the product in $A$. Let $\\phi^\\star \\colon A_2^\\star\\to A_1^\\star\\otimes_{A_0} A_1^\\star$ be the dual morphism to $\\phi$, see Lemma \\ref{ch3:lem_bimod}, here $\\bullet^\\star$ is as in Section \\ref{ch3:subs_bimod} with respect to $B=A_0$.\nThe \\emph{quadratic dual algebra} to $A$ is the quadratic algebra $A^!=T_{A_0}(A_1^\\star)\/(\\mathrm{Im}~\\phi^\\star)$.\n\\end{df}\n\n\\smallskip\n\\begin{rk}\nIn the previous definition we do not assume that the algebra $A$ is quadratic itself. However, if it is true, we have a graded $\\mathbb{C}$-algebra isomorphism $(A^!)^!\\simeq A$.\n\\end{rk}\n\n\\smallskip\n\nLet $\\mathcal{C}$ be an abelian category such that its objects are graded modules.\nDenote by $\\mathrm{Com}^\\downarrow(\\mathcal{C})$ the category of complexes $X^\\bullet$ in $\\mathcal{C}$ such that the $j$th graded component of $X^i$ is zero when $i>>0$ or $i+j<<0$. Similarly, let $\\mathrm{Com}^\\uparrow(\\mathcal{C})$ the category of complexes $X^\\bullet$ in $\\mathcal{C}$ such that the $j$th graded component of $X^i$ is zero when $i<<0$ or $i+j>>0$. Denote by $D^\\downarrow(\\mathcal{C})$ and $D^\\uparrow(\\mathcal{C})$ the corresponding derived categories of such complexes.\nWe will use the following abbreviations\n$$\nD^\\downarrow(A)=D^\\downarrow(\\mathrm{grmod}(A)),\\quad D^\\uparrow(A)=D^\\uparrow(\\mathrm{grmod}(A)),\\quad\nD^b(A)=D^b(\\mathrm{grmod}(A)).\n$$\n\nIn the situation above we have the following functors\n$\\mathcal{K}\\colon D^\\downarrow(A)\\to D^\\uparrow(A^!)$ and $\\mathcal{K}'\\colon D^\\uparrow(A^!)\\to D^\\downarrow(A)$ called \\emph{quadratic duality functors}. See \\cite[Sec.~5]{MOS} for more details.\n\n\\subsection{Koszul algebras}\n\\label{ch3:subs_Koszul-alg}\nLet $A=\\bigoplus_{n\\in\\mathbb{N}}A_n$ be a finite dimensional $\\mathbb{N}$-graded $\\mathbb{C}$-algebra such that $A_0$ is semisimple. We identify $A_0$ with the left graded $A$-module $A_0\\simeq A\/{\\oplus_{n>0}A_n}$.\n\n\\smallskip\n\\begin{df}\nThe graded algebra $A$ is \\emph{Koszul} if the left graded $A$-module $A_0$ admits a projective resolution $\\cdots\\to P^2\\to P^1\\to P^0\\to A_0$ such that $P^r$ is generated by its degree $r$ component.\n\\end{df}\n\n\\smallskip\nIf $A$ is Koszul, we consider the graded $\\mathbb{C}$-algebra $A^!=\\mathrm{Ext}^*_A(A_0,A_0)^{\\rm op}$ and we call it the \\emph{Koszul dual} algebra to $A$.\nThe following is well-known, see \\cite{BGS}.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_Koszul-duality}\nLet $A$ be a Koszul $\\mathbb{C}$-algebra. Assume that $A$ and $A^!$ are finite dimensional. Then, the following holds.\n\n$(a)$ The algebra $A$ is quadratic. The Koszul dual algebra $A^!$ coincides with the quadratic dual algebra.\n\n$(b)$ The algebra $A^!$ is also Koszul and there is a graded algebra isomorphism $(A^!)^!\\simeq A$.\n\n$(c)$ There is an equivalence of categories\n$$\n\\mathcal{K}\\colon {D}^b(A)\\to {D}^b(A^!), \\qquad M\\mapsto \\mathrm{RHom}_A(A_0,M).\n$$\n \\qed\n\\end{prop}\n\n\\smallskip\nIf $A$ is Koszul, then the functors $\\mathcal{K}$ and $\\mathcal{K}'$ from the previous section are mutually inverse. Moreover, the equivalence $\\mathcal{K}$ of bounded derived categories in Proposition \\ref{ch3:prop_Koszul-duality} $(c)$ is the restriction of the functor $\\mathcal{K}$ from the previous section.\n\n\\smallskip\n\\begin{df}\nLet $A$ and $B$ be Koszul algebras. We say that the functor $\\Phi\\colon D^b(A)\\to D^b(B)$ is Koszul dual to the functor $\\Psi\\colon D^b(A^!)\\to D^b(B^!)$ if the following diagram of functor is commutative\n$$\n\\begin{CD}\nD^b(A) @>{\\Psi}>> D^b(B)\\\\\n@V{\\mathcal{K}}VV @V{\\mathcal{K}}VV\\\\\nD^b(A^!)@>{\\Phi}>> D^b(B^!).\n\\end{CD}\n$$\n\\end{df}\n\n\\subsection{Categories of linear complexes}\nIn this section we recall some results from \\cite{MOS} about linear complexes. Let $A$ be as in Section \\ref{ch3:subs_quad-dual}.\n\n\\smallskip\n\\begin{df}\nLet $\\mathcal{LC}(A)$ be the category of complexes $\\cdots\\to\\calX^{k-1}\\to\\calX^{k}\\to\\calX^{k+1}\\to\\cdots$ of projective modules in $\\mathrm{grmod}(A)$ such that for each $k\\in\\mathbb{Z}$ each indecomposable direct factor $P$ of $\\calX^k$ is a direct factor of $A\\langle k\\rangle$.\n\\end{df}\n\n\\smallskip\n\\begin{prop}\nThere is an equivalence of categories $\\epsilon_A\\colon \\mathcal{LC}(A)\\simeq \\mathrm{grmod}(A^!)$.\\qed\n\\end{prop}\n\n\\smallskip\nLet us describe the construction of $\\epsilon^{-1}_A$. Let $M=\\oplus_{n\\in\\mathbb{Z}}M_n$ be in $\\mathrm{grmod}(A^!)$. The graded $A^!$-module structure yields morphisms of $A_0$-modules $f'_n\\colon A^!_1\\otimes M_n\\to M_{n+1}$ for each $n\\in\\mathbb{Z}$. We have\n\\begin{eqnarray*}\n\\mathrm{Hom}_{A_0}(A^!_1\\otimes_{A_0} M_n,M_{n+1})&=&\\mathrm{Hom}_{A_0}(M_n,(A^!_1)^\\star\\otimes_{A_0} M_{n+1})\\\\\n&=&\\mathrm{Hom}_{A_0}(M_n,A_1\\otimes_{A_0} M_{n+1}).\n\\end{eqnarray*}\n\nLet $f_n\\colon\\mathrm{Hom}_{A_0}(M_n,A_1\\otimes_{A_0} M_{n+1})$ be the image of $f'_n$ by the chain of isomorphisms above.\n\nWe have $\\epsilon_A^{-1}(M)=\\cdots\\stackrel{\\partial_{k-2}}{\\to}\\calX^{k-1}\\stackrel{\\partial_{k-1}}{\\to}\\calX^{k}\\stackrel{\\partial_k}{\\to}\\calX^{k+1}\\stackrel{\\partial_{k+1}}{\\to}\\cdots$ with $\\calX^k=A\\langle k\\rangle\\otimes_{A_0} M_k$ and\n$$\n\\partial_k\\colon A\\langle k\\rangle\\otimes_{A_0} M_k\\to A\\langle k+1 \\rangle\\otimes_{A_0} M_{k+1},\\quad a\\otimes m\\mapsto (a\\otimes \\mathrm{Id})(f_k(m)).\n$$\n\nThe quadratic duality functor discussed in the previous section can be characterizes as follows, see \\cite[Prop.~21]{MOS}.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Kos-Tot}\nUp to isomorphism of functors, the following diagram is commutative:\n$$\n\\begin{diagram}\n\\node[2]{D^\\uparrow(\\mathcal{LC}(A))} \\arrow{sw,l}{\\mathrm{Tot}} \\\\\n\\node{D^\\downarrow(A)}\n\\node[2]{D^\\uparrow(A^!)} \\arrow[2]{w,b}{\\mathcal{K}'} \\arrow{nw,t}{\\epsilon^{-1}_A}\\\\\n\\end{diagram},\n$$\nwhere $\\mathrm{Tot}$ is the functor taking the total complex.\n\\qed\n\\end{lem}\n\n\\subsection{The main lemma about Koszul dual functors}\n\\label{ch3:subs_key-lem}\nLet $\\{e_\\lambda;\\lambda\\in\\Lambda\\}$ be the set of indecomposable idempotents of $A_0$, i.e., we have $A_0=\\bigoplus_{\\lambda\\in\\Lambda}\\mathbb{C} e_\\lambda$. Denote by $e^!_\\lambda$ the corresponding idempotent of $A^!_0$ via the identification $A_0\\simeq A^!_0$. For each subset $\\Lambda'\\subset\\Lambda$ set $e_{\\Lambda '}=\\sum_{\\lambda\\in\\Lambda '}e_\\lambda$. Consider the graded algebras\n$$\nA_{\\Lambda'}=e_{\\Lambda'}Ae_{\\Lambda'}, \\qquad _{\\Lambda'}A=A\/(e_{\\Lambda\\backslash\\Lambda'}).\n$$\nSimilarly, we can define $A^!_{\\Lambda'}$ and $_{\\Lambda'}A^!$.\n\nWe have a functor $F\\colon\\mathrm{grmod}(A_{\\Lambda'})\\to\\mathrm{grmod}(A)$, $M\\mapsto Ae_{\\Lambda'}\\otimes_{A_{\\Lambda'}} M$. Note also that the category $\\mathrm{grmod}(_{\\Lambda'}A^!)$ can be viewed as a subcategory of $\\mathrm{grmod}(A^!)$ containing modules that are killed by $e_{\\Lambda\\backslash\\Lambda'}$. Let $\\iota\\colon\\mathrm{grmod}(_{\\Lambda'}A^!)\\to \\mathrm{grmod}(A^!)$ be the inclusion. The following proposition is proved in \\cite[Thm.~28]{MOS}.\n\n\\smallskip\n\\begin{prop}\n\\label{ch3:prop_dual-F-G}\n{\\rm (a)} The quadratic dual algebra to $A_{\\Lambda'}$ is isomorphic to $_{\\Lambda'}A^!$.\n\n{\\rm (b)} The following diagram commutes up to isomorphism of functors.\n$$\n\\begin{CD}\nD^\\downarrow(A) @<\\mathcal{K}'<< D^\\uparrow(A^!)\\\\\n@AFAA @A\\iota AA\\\\\nD^\\downarrow(A_{\\Lambda'}) @<\\mathcal{K}'<< D^\\uparrow(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\n\\end{prop}\n\\begin{proof}[Idea of proof of (b)]\nBy Lemma \\ref{ch3:lem_Kos-Tot} it is enough to proof the commutativity of the following diagram.\n$$\n\\begin{CD}\n\\mathcal{LC}(A) @<\\epsilon_A^{-1}<< \\mathrm{grmod}(A^!)\\\\\n@AFAA @A\\iota AA\\\\\n\\mathcal{LC}(A_{\\Lambda'}) @<\\epsilon_{A_{\\Lambda'}}^{-1}<< \\mathrm{grmod}(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\n\\end{proof}\n\n\\smallskip\nWe can generalize this result as follows.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_key}\nLet $A'$ be a finite dimensional $\\mathbb{N}$-graded $\\mathbb{C}$-algebra. Assume that for some subset $\\Lambda'\\subset\\Lambda$ there is a graded (unitary) homomorphism $\\psi\\colon A'\\to A_{\\Lambda'}$ such that\n\\begin{itemize}\n \\item[\\rm{(a)}] $\\psi$ is an isomorphism in degrees $0$ and $1$,\n \\item[\\rm{(b)}] $\\psi$ induces an isomorphism between the kernel of $A'_1\\otimes_{A'_0}A'_1\\to A'_2$ and the kernel of $(A_{\\Lambda'})_1\\otimes_{(A_{\\Lambda'})_0}(A_{\\Lambda'})_1\\to (A_{\\Lambda'})_2$.\n\\end{itemize}\nThen the quadratic dual of $A'$ is isomorphic to $_{\\Lambda'}A$.\n\nConsider the graded $(A,A')$-bimodule $Ae_{\\Lambda'}$, where the right $A'$-module structure is obtained from the right $A_{\\Lambda'}$-module structure using $\\psi$. Consider the functor $T\\colon \\mathrm{grmod}(A')\\to\\mathrm{grmod}(A)$, $M\\mapsto Ae_{\\Lambda'}\\otimes_{A'}M$. Then the following diagram commutes up to an isomorphism of functors.\n$$\n\\begin{CD}\nD^\\downarrow(A) @<\\mathcal{K}'<< D^\\uparrow(A^!)\\\\\n@ATAA @A\\iota AA\\\\\nD^\\downarrow(A') @<\\mathcal{K}'<< D^\\uparrow(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\n\\end{lem}\n\\begin{proof}[Proof]\nBy definition, the quadratic dual of $A'$ depends only on the algebra $A'_0$, the $(A'_0,A'_0)$-bimodule $A'_1$ and the kernel of $A'_1\\otimes_{A'_0}A'_1\\to A'_2$. Thus the quadratic dual algebras of $A'$ and $A_{\\Lambda'}$ are isomorphic. Finally, Proposition \\ref{ch3:prop_dual-F-G} (a) implies that the quadratic dual of $A'$ is isomorphic to $_{\\Lambda'}A$.\n\nNow, by Lemma \\ref{ch3:lem_Kos-Tot} is enough to prove the commutativity of the following diagram up to an isomorphism of functors.\n$$\n\\begin{CD}\n\\mathcal{LC}(A) @<\\epsilon^{-1}_A<< \\mathrm{grmod}(A^!)\\\\\n@ATAA @A\\iota AA\\\\\n\\mathcal{LC}(A') @<\\epsilon^{-1}_{A'}<< \\mathrm{grmod}(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\n\nBy analogy with the definition of the functor $T$, consider the functor $\\Phi\\colon \\mathrm{grmod}(A')\\to\\mathrm{grmod}(A_{\\Lambda'})$, $M\\mapsto A_{\\Lambda'}\\otimes_{A'}M$. For each $\\lambda\\in\\Lambda'$ let $e'_\\lambda$ be the idempotent in $A'_0$ such that $\\psi(e'_\\lambda)=e_\\lambda$. We have $\\Phi(A'e'_\\lambda)=A_{\\Lambda'}e_\\lambda$ for each $\\lambda\\in\\Lambda'$. In particular $\\Phi$ induces a bijection between the indecomposable direct factors of $A'$ and $A_{\\Lambda'}$. Thus $\\Phi$ induces a functor $\\Phi\\colon \\mathcal{LC}(A')\\to \\mathcal{LC}(A_{\\Lambda'})$. Note that by definition the boundary maps in the complexes of the category $\\mathcal{LC}(\\bullet)$ are of degree $1$. Thus, by (a) and (b) the functor $\\Phi$ induces an equivalence of categories $\\Phi\\colon \\mathcal{LC}(A')\\to \\mathcal{LC}(A_{\\Lambda'})$.\n\n\nConsider the following diagram, where the functor $F$ is as before Proposition \\ref{ch3:prop_dual-F-G}.\n$$\n\\begin{CD}\n\\mathcal{LC}(A) @<\\mathrm{Id}<< \\mathcal{LC}(A) @<\\epsilon^{-1}_A<< \\mathrm{grmod}(A^!)\\\\\n@ATAA @AFAA @A\\iota AA\\\\\n\\mathcal{LC}(A') @<\\Phi^{-1}<< \\mathcal{LC}(A_{\\Lambda'}) @<\\epsilon^{-1}_{A_{\\Lambda'}}<< \\mathrm{grmod}(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\nThe right square commutes by the proof of Proposition \\ref{ch3:prop_dual-F-G} and the commutativity of the left square is obvious. To conclude we need only to check that $\\epsilon^{-1}_{A'}=\\Phi^{-1}\\circ\\epsilon^{-1}_{A_{\\Lambda'}}$.\n\nLet us check that $\\Phi\\circ\\epsilon^{-1}_{A'}=\\epsilon^{-1}_{A_{\\Lambda'}}$.\nThis is clear on objects because\n$$\n\\begin{array}{ccccccccccc}\n\\epsilon^{-1}_{A_{\\Lambda'}}(M)&= \\cdots &\\stackrel{\\partial'_{k-1}}{\\to}& A_{\\Lambda'}\\langle k \\rangle \\otimes_{(A_{\\Lambda'})_0}M_k &\\stackrel{\\partial'_k}{\\to}& A_{\\Lambda'}\\langle k+1 \\rangle \\otimes_{(A_{\\Lambda'})_0}M_{k+1}&\\stackrel{\\partial'_{k+1}}{\\to}&\\cdots,\\\\\n\\epsilon^{-1}_{A'}(M)&= \\cdots &\\stackrel{\\partial''_{k-1}}{\\to}& A'\\langle k \\rangle \\otimes_{A'_0}M_k &\\stackrel{\\partial''_k}{\\to}& A'\\langle k+1 \\rangle \\otimes_{A'_0}M_{k+1}&\\stackrel{\\partial''_{k+1}}{\\to}&\\cdots.\n\\end{array}\n$$\n\nThe boundary maps are defined as follows\n$$\n\\begin{array}{rll}\n\\partial'_k\\colon& A_{\\Lambda'}\\langle k\\rangle\\otimes_{(A_{\\Lambda'})_0} M_k\\to A_{\\Lambda'}\\langle k+1 \\rangle\\otimes_{(A_{\\Lambda'})_0} M_{k+1},&\\quad a\\otimes m\\mapsto (a\\otimes \\mathrm{Id})(f^1_n(m)),\\\\\n\\partial''_k\\colon& A'\\langle k\\rangle\\otimes_{A'_0} M_k\\to A'\\langle k+1 \\rangle\\otimes_{A'_0} M_{k+1},&\\quad a\\otimes m\\mapsto (a\\otimes \\mathrm{Id})(f^2_n(m)),\n\\end{array}\n$$\nwhere $f^1_n\\colon M_n\\to (A_{\\Lambda'})_1\\otimes_{(A_{\\Lambda'})_0} M_{n+1}$ and $f^2_n\\colon M_n\\to A'_1\\otimes_{A'_0} M_{n+1}$ are defined in the same way as $f_n$ in the definition of $\\epsilon^{-1}$. Thus it is also clear that $\\Phi$ commutes with the boundary maps.\n\\end{proof}\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk_cond-b}\nCondition (b) is necessary only to deduce that $(A')^!\\simeq {(A_{\\Lambda'})^!}$. Without this condition we know only that the algebra $(A')^!$ is isomorphic to a quotient of ${(A_{\\Lambda'})^!}$. Thus condition (b) can be replaced by the requirement $\\dim (A')^!=\\dim {_{\\Lambda'}A^!}$.\n\\end{rk}\n\n\\smallskip\nWe can reformulate Lemma \\ref{ch3:lem_key} in the following way.\n\n\\smallskip\n\\begin{coro}\n\\label{ch3:coro_main-Koszul}\nLet $A'$ be an $\\mathbb{N}$-graded finite dimensional $\\mathbb{C}$-algebra with basic $A'_0$ such that the indecomposable idempotents of $A'_0$ are parameterized by a subset $\\Lambda'$ of $\\Lambda$, i.e., we have $A'_0=\\bigoplus_{\\lambda\\in\\Lambda'}\\mathbb{C} e'_\\lambda$. Assume that $\\dim (A')^!=\\dim {_{\\Lambda'}A^!}$. Assume also that there is an exact functor $T\\colon \\mathrm{grmod}(A')\\to\\mathrm{grmod}(A)$ such that\n\\begin{itemize}\n \\item[\\rm{(a)}] $T(A'e'_\\lambda)= Ae_\\lambda$ $\\forall \\lambda\\in\\Lambda'$,\n \\item[\\rm{(b)}] the functor $T$ yields an isomorphism $\\mathrm{Hom}_{A'}(A'e'_\\lambda\\langle 1\\rangle,A'e'_\\mu)\\simeq \\mathrm{Hom}_{A}(Ae_\\lambda\\langle 1\\rangle,Ae_\\mu)$.\n\\end{itemize}\nThen the quadratic dual for $A'$ is $_{\\Lambda'}A^!$ and the following diagram commutes up to isomorphism of functors.\n$$\n\\begin{CD}\nD^\\downarrow(A) @<\\mathcal{K}'<< D^\\uparrow(A^!)\\\\\n@ATAA @A\\iota AA\\\\\nD^\\downarrow(A') @<\\mathcal{K}'<< D^\\uparrow(_{\\Lambda'}A^!)\\\\\n\\end{CD}\n$$\n\\end{coro}\n\\begin{proof}[Proof]\n\nCondition (a) implies that the functor $T$ yields a homomorphism of graded algebras $\\psi\\colon A'\\to A_{\\Lambda'}$. Moreover, condition (b) implies that $\\psi$ satisfies condition (a) of Lemma \\ref{ch3:lem_key}. Finally, the assumption $\\dim (A')^!=\\dim {_{\\Lambda'}A}$ implies that $\\psi$ satisfies condition (b) of Lemma \\ref{ch3:lem_key}, see Remark \\ref{ch3:rk_cond-b}. The functor $T$ hare can be identified with the functor $T=Ae_{\\Lambda'}\\otimes_{A'}\\bullet$ in the statement of Lemma \\ref{ch3:lem_key}, see \\cite[Lem.~3.4]{Str}. Thus the statement follows from Lemma \\ref{ch3:lem_key}.\n\\end{proof}\n\n\\subsection{Zuckerman functors}\n\\label{ch3:subs_Zuck}\nFix $v\\in \\widehat W$. Let $\\nu_1$ and $\\nu_2$ be two different parabolic types such that $W_{\\nu_1}\\subset W_{\\nu_2}$. By definition of the parabolic category $\\mathcal{O}$, there is an inclusion of categories ${^v}O^{\\nu_2}_\\mu\\subset {^v}O^{\\nu_1}_\\mu$. We denote by $\\mathrm{inc}$ the inclusion functor. We may write $\\mathrm{inc}=\\mathrm{inc}_{\\nu_2}^{\\nu_1}$ to specify the parameters. The functor $\\mathrm{inc}$ admits a left adjoint functor $\\tr$. For $M\\in {^v}O^{\\nu_1}_\\mu$, the object $\\tr(M)$ is the maximal quotient of $M$ that is in ${^v}O^{\\nu_2}_\\mu$, see Lemma \\ref{ch3:lem_trunc-categ-gen} $(a)$. We call the functor $\\tr$ the \\emph{parabolic truncation} functor. We may write $\\tr_{\\nu_1}^{\\nu_2}$ to specify the parameters.\n\nNow, we assume that $\\nu_1$ and $\\nu_2$ are two arbitrary parabolic types. Then there is a parabolic type $\\nu_3$ such that we have $W_{\\nu_3}=W_{\\nu_1}\\cap W_{\\nu_2}$. The \\emph{Zuckerman functor} $\\mathrm{Zuc}_{\\nu_1}^{\\nu_2}$ (or simply $\\mathrm{Zuc}$) is the composition $\\mathrm{Zuc}_{\\nu_1}^{\\nu_2}=\\tr_{\\nu_3}^{\\nu_2}\\circ\\mathrm{inc}_{\\nu_1}^{\\nu_3}$.\n\nThe parabolic inclusion functor is exact. The parabolic truncation functor is only right exact. This implies that the Zuckerman functor is right exact.\n\n\nNow, we are going to grade Zuckerman functors. Let $^vA^\\nu_\\mu$ be the indomorphism algebra of the minimal projective generator of $^vO_\\mu^\\nu$ (or simply $^vA_\\mu$ in the non-parabolic case). We have $^vO_\\mu^\\nu\\simeq \\mathrm{mod}(^vA_\\mu^\\nu)$. The Koszul grading on $^vA^\\nu_\\mu$ is constructed in \\cite{SVV}. The graded version $^v\\widetilde O_\\mu^\\nu$ of $^vO_\\mu^\\nu$ is the category $\\mathrm{grmod}(^vA^\\nu_\\mu)$. Moreover, the algebra $^vA_\\mu^\\nu$ is the quotient of $^vA_\\mu$ by a homogeneous ideal $I_\\nu$. By construction, the grading on $^vA_\\mu^\\nu$ is induced from the grading on $^vA_\\mu$. Assume that $\\nu_1$ and $\\nu_2$ are such that $W_{\\nu_1}\\subset W_{\\nu_2}$. Then we have $I_{\\nu_1}\\subset I_{\\nu_2}$. This implies that the graded algebra $^vA_\\mu^{\\nu_2}$ is isomorphic to the quotient of the graded algebra $^vA^{\\nu_1}_\\mu$ by the homogeneous ideal $I_{\\nu_2}\/I_{\\nu_1}$. This yields an inclusion of graded categories ${^v}\\widetilde O^{\\nu_2}_\\mu\\subset {^v}\\widetilde O^{\\nu_1}_\\mu$. Let us denote by $\\widetilde\\mathrm{inc}_{\\nu_2}^{\\nu_1}$ (or simply $\\widetilde\\mathrm{inc}$) the inclusion functor. It is a graded lift of the functor $\\mathrm{inc}$. Similarly, its left adjoint functor $\\widetilde\\tr_{\\nu_1}^{\\nu_2}$ is a graded lift of the functor $\\tr$, see Remark \\ref{ch3:rk_grade-trunc-gen}. Thus we get graded lifts $\\widetilde\\mathrm{Zuc}_{\\nu_1}^{\\nu_2}$ of the Zuckerman functor $\\mathrm{Zuc}_{\\nu_1}^{\\nu_2}$ for arbitrary parabolic types $\\nu_1$ and $\\nu_2$.\n\nSimilarly, we can define the parabolic inclusion functor, the parabolic truncation functor, the Zuckerman functor and their graded versions for the affine category $\\mathcal{O}$ at a positive level.\n\n\n\n\\subsection{The Koszul dual functors in the category $O$}\n\\label{ch3:subs_dual-funct-in-O}\n\nAs above, we assume $W_{\\mu}\\subset W_{\\mu'}$. Set $J^\\nu_\\mu=\\{w\\in J_\\mu;~w(1_\\mu)\\in P^\\nu\\}$. Note that the inclusion $J_{\\mu'}\\subset J_\\mu$ induces an inclusion $J^\\nu_{\\mu'}\\subset J^\\nu_\\mu$. For $v\\in\\widehat W$ we set $^vJ^\\nu_\\mu=\\{w\\in J^\\nu_\\mu;~w\\leqslant v\\}$.\n\nAs in Section \\ref{ch3:sec_gr-lifts} we assume that we have $W_\\mu\\subset W_{\\mu'}$. Fix a parabolic type $\\nu=(\\nu_1,\\cdots,\\nu_l)\\in X_l[N]$. \n\nAssume $v\\in J_{\\mu'}^\\nu w_{\\mu'}$. The functors\n$$\nF_k\\colon {^vO}_{\\mu}\\to {^vO}_{\\mu'},\\qquad E_k\\colon {^vO}_{\\mu'}\\to {^vO}_{\\mu}\n$$\nrestrict to functors of parabolic categories\n$$\nF_k\\colon {^vO}^\\nu_{\\mu}\\to {^vO}^\\nu_{\\mu'},\\qquad E_k\\colon {^vO}^\\nu_{\\mu'}\\to {^vO}^\\nu_{\\mu}.\n$$\nThe restricted functors still satisfy the properties announced in Lemmas \\ref{ch3:lem_gr-lift-E-F+adj-case1}, \\ref{ch3:lem_prod-FE-case1}.\n\n\n\nAssume that $w\\in {^vJ}^\\nu_\\mu$. Let $^vP^{w(1_\\mu)}$ be the projective cover of $L^{w(1_\\mu)}$ in $^vO^\\nu_\\mu$. (Note that we do not indicate the parabolic type $\\nu$ in our notations for modules to simplify the notations.) We fix the grading on $L^{w(1_\\mu)}$ such that it is concentrated in degree zero when we consider $L^{w(1_\\mu)}$ as an $^vA^\\nu_\\mu$-module (see Section \\ref{ch3:subs_Zuck} for the definition of $^vA^\\nu_\\mu$). A standard argument shows that the modules $^vP^{w(1_\\mu)}$ and $\\Delta^{w(1_\\mu)}$ admit graded lifts. (The graded lift of $^vP^{w(1_\\mu)}$ can be constructed as the projective cover of the graded lift of $L^{w(1_\\mu)}$ in $^v\\widetilde O^\\nu_\\mu$. The existence of graded lifts of projective modules implies the existence of graded lifts of Verma modules, see \\cite[Cor.~4]{MO}.) We fix the graded lifts of $^vP^{w(1_\\mu)}$ and $\\Delta^{w(1_\\mu)}$ such that the surjections $^vP^{w(1_\\mu)}\\to L^{w(1_\\mu)}$ and $\\Delta^{w(1_\\mu)}\\to L^{w(1_\\mu)}$ are homogeneous of degree zero, see also Lemma \\ref{ch3:lem-grad-unique}.\n\nThe following lemma is stated in the parabolic category $O$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Ek-on-proj-spcase1}\n$(a)$ For each $w\\in {^vJ}_{\\mu'}^\\nu$, we have $E_k({^vP}^{w(1_{\\mu'})})={^vP}^{w(1_{\\mu})}$.\n\n$(b)$ For each $w\\in {^vJ}_{\\mu}^\\nu$, we have\n$$\nF_k(L^{w(1_{\\mu})})=\n\\left\\{\n\\begin{array}{ll}\nL^{w(1_{\\mu'})} &\\mbox{\\rm if } w\\in {^vJ}^\\nu_{\\mu'},\\\\\n0 &\\mbox{\\rm else}.\n\\end{array}\n\\right.\n$$\n\\end{lem}\n\\begin{proof}\nFirst, we prove $(a)$ in the non-parabolic situation (i.e., for $\\nu=(1,1,\\cdots,1)$). The modules $E_k({^vP}^{w(1_{\\mu'})})$ and ${^vP}^{w(1_{\\mu})}$ are both projective. Thus it is enough to show that their classes in the Grothendieck group are the same. To show this, we compare the multiplicities of Verma modules in the $\\Delta$-filtrations of $E_k({^vP}^{w(1_{\\mu'})})$ and ${^vP}^{w(1_{\\mu})}$.\n\nWe need to show that for each $x\\in {^vJ}_{\\mu'}$ we have\n$$\n[E_k({^vP}^{w(1_{\\mu'})}),\\Delta^{x(1_{\\mu'})}]=[{^vP}^{w(1_{\\mu})},\\Delta^{x(1_{\\mu})}].\n$$\nBy Lemma \\ref{ch3:lem_Ek-Verma-spcase1}, for each $x\\in {^vJ}_\\mu$, the multiplicity $[E_k({^vP}^{w(1_{\\mu'})}),\\Delta^{x(1_{\\mu})}]$ is equal to the multiplicity $[{^vP}^{w(1_{\\mu'})},\\Delta^{x(1_{\\mu'})}]$. So, we need to prove the equality\n$$\n[{^vP}^{w(1_{\\mu'})},\\Delta^{x(1_{\\mu'})}]=[{^vP}^{w(1_{\\mu})},\\Delta^{x(1_{\\mu})}].\n$$\nThe last equality is obvious because both of these multiplicities are given by the same parabolic Kazhdan-Lusztig polynomial. See, for example, \\cite[App.~A]{Mak-Koszul} for more details about multiplicities in the parabolic category $\\mathcal{O}$ for $\\widehat{\\mathfrak{gl}}_N$.\n\nNow, we prove $(b)$. Since the set of simple modules in the parabolic category $O$ is a subset of the set of simple modules of the non-parabolic casegory $O$, it is enough to prove $(b)$ in the non-parabolic case.\n\nFor each $w\\in {^vJ}_{\\mu}$ and $x\\in{^vJ}_{\\mu'}$, we have\n$$\n\\begin{array}{rcl}\n\\mathrm{Hom}(^vP^{x(1_{\\mu'})},F_k(L^{w(1_\\mu)}))&\\simeq& \\mathrm{Hom}(E_k(^vP^{x(1_{\\mu'})}),L^{w(1_\\mu)})\\\\\n&\\simeq&\n\\mathrm{Hom}(^vP^{x(1_{\\mu})},L^{w(1_\\mu)}).\n\\end{array}\n$$\nThis implies that we have $\\dim\\mathrm{Hom}(^vP^{x(1_{\\mu'})},F_k(L^{w(1_\\mu)}))=\\delta_{x,w}$.\nSince $\\dim\\mathrm{Hom}(^vP^{x(1_{\\mu'})},M)$ counts the multiplicity of the simple module $L^{x(1_{\\mu'})}$ in the module $M$ (this fact can be proved in the same way as \\cite[Thm.~ 3.9~(c)]{BGG}), this proves $(b)$.\n\nFinally, we prove $(a)$ in the parabolic situation. For each $w\\in {^vJ}_{\\mu'}^\\nu$ and each $x\\in {^vJ}_{\\mu}^\\nu$ we have\n$$\n\\begin{array}{rcl}\n\\mathrm{Hom}(E_k({^vP}^{w(1_{\\mu'})}),L^{x(1_{\\mu})})&\\simeq& \\mathrm{Hom}({^vP}^{w(1_{\\mu'})},F_k(L^{x(1_{\\mu})}))\\\\\n&\\simeq&\n\\left\\{\n\\begin{array}{lll}\n\\mathrm{Hom}({^vP}^{w(1_{\\mu'})},L^{x(1_{\\mu'})})& \\mbox{if } x\\in {^vJ}_{\\mu'}^\\nu \\\\\n0& \\mbox{else},\n\\end{array}\n\\right.\n\\end{array}\n$$\nwhere the second isomorphism follows from $(b)$. This implies that we have $\\dim\\mathrm{Hom}(E_k({^vP}^{w(1_{\\mu'})}),L^{x(1_{\\mu})})=\\delta_{w,x}$. Thus we have $E_k({^vP}^{w(1_{\\mu'})})\\simeq {^vP}^{w(1_{\\mu})}$.\n\\end{proof}\n\n\n\nThe definitions of the graded lifts $\\widetilde E_k$ and $\\widetilde F_k$ in Lemma \\ref{ch3:lem_gr-lift-E-F+adj-case1} depend on the choice of the graded lift $\\widetilde\\mathbb{V}_\\mu$ of $\\mathbb{V}_\\mu$. Note that we have the following isomorphism of ${^vZ}_\\mu$-modules $\\mathbb{V}_\\mu(^vP^{\\mu})\\simeq {^vZ}_\\mu$ for all $\\mu\\in X_I[N]$. By Lemma \\ref{ch3:lem-grad-unique}, for each choice of the graded lift $\\widetilde\\mathbb{V}_\\mu$, we have $\\widetilde\\mathbb{V}_\\mu(^vP^{\\mu})\\simeq {^vZ}_\\mu\\langle r\\rangle$ for some $r\\in\\mathbb{Z}$. From now on, we always assume that the graded lift $\\widetilde\\mathbb{V}_\\mu$ is chosen in such a way that we have an isomorphism of graded ${^vZ}_\\mu$-modules $\\widetilde\\mathbb{V}_\\mu(^vP^{\\mu})\\simeq {^vZ}_\\mu$ (without any shift $r$).\n\nIn the following statement we consider the non-parabolic situation.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Ek-on-Verma-spcase1-grad}\nFor each $w\\in {^vJ_{\\mu'}}$, the graded module $\\widetilde E_k(\\Delta^{w(1_{\\mu'})})$ has a graded $\\Delta$-filtration with constituents $\\Delta^{wz(1_\\mu)}\\langle\\ell(z)\\rangle$ for $z\\in J^\\mu_{\\mu'}$.\n\n\\end{lem}\n\\begin{proof}\nFirst, we prove that $\\widetilde E_k$ takes the graded anti-dominant projective module to the graded anti-dominant projective module, i.e., that we have $\\widetilde E_k(^vP^{\\mu'})\\simeq {^vP}^{\\mu}$.\n\n\nBy Lemma \\ref{ch3:lem-grad-unique}, the graded lift of ${^vP}^{\\mu}$ is unique up to graded shift. Thus, by Lemma \\ref{ch3:lem_Ek-on-proj-spcase1}, we have $\\widetilde E_k({^vP}^{\\mu'})={^vP}^{\\mu}\\langle r \\rangle$ for some $r\\in\\mathbb{Z}$. We need to prove that $r=0$.\n\nRecall that the graded lift $\\widetilde E_k$ of $E_k$ is constructed in the proof of Lemma \\ref{ch3:lem_gr-lift-E-F+adj-case1} in such a way that the following diagram is commutative\n$$\n\\begin{CD}\n^vO_{\\mu} @<{\\widetilde E_{k}}<< ^vO_{\\mu'}\\\\\n@V{\\widetilde\\mathbb{V}_{\\mu}}VV @V{\\widetilde\\mathbb{V}_{\\mu'}}VV\\\\\n\\mathrm{mod}(^vZ_{\\mu}) @<{\\widetilde\\mathrm{Ind}}<< \\mathrm{mod}(^vZ_{\\mu'}).\n\\end{CD}\n$$\nMoreover, by definition, we have the following isomorphisms of graded modules\n$$\n\\widetilde\\mathbb{V}_{\\mu}({^vP}^{w(1_{\\mu})})\\simeq{^vZ}_{\\mu}, \\quad \\widetilde\\mathbb{V}_{\\mu'}({^vP}^{w(1_{\\mu'})})\\simeq{^vZ}_{\\mu'},\\quad\n\\widetilde\\mathrm{Ind}(^vZ_{\\mu'})={^vZ}_{\\mu}.\n$$\nThis implies that we have $r=0$.\n\nNow we prove the statement of the lemma. The module $\\widetilde E_k(\\Delta^{w(1_{\\mu'})})$ has a graded $\\Delta$-filtartion because it has a $\\Delta$-filtration as an ungraded module, see \\cite[Rem.~2.13]{Mak-Koszul}. The constituents (up to graded shifts) are $\\Delta^{wz(1_\\mu)}$, $z\\in W_{\\mu'}\/W_{\\mu}$ by Lemma \\ref{ch3:lem_Ek-Verma-spcase1}. We need only to identify the shifts. The graded multiplicities of Verma modules in projective modules are given in terms of Kazhdan-Lusztig polynomials in \\cite[App.~A]{Mak-Koszul}. In particular, \\cite[Lem.~A.4~(d)]{Mak-Koszul} implies that, for each $w\\in {^vJ}_\\mu$, the module $\\Delta^{w(1_\\mu)}$ appears as a constituent in a graded $\\Delta$-filtration of $^vP^\\mu$ once with the graded shift by $\\ell(w)$. Similarly, for each $w\\in{^vJ}_{\\mu'}$, the module $\\Delta^{w(1_{\\mu'})}$ appears as a constituent in a graded $\\Delta$-filtration of $^vP^{\\mu'}$ once with the graded shift by $\\ell(w)$. Now, since $\\widetilde E_k({^vP}^{\\mu'})\\simeq {^vP}^{\\mu}$, we see that, for each $w\\in {^vJ}_{\\mu'}$ and each $z\\in J_{\\mu'}^{\\mu}$, the module $\\Delta^{wz(1_\\mu)}$ appears in the $\\Delta$-filtration of $\\widetilde E_k(\\Delta^{w(1_{\\mu'})})$ with the graded shift by $\\ell(z)$.\n\\end{proof}\n\n\\smallskip\nIn the following lemma me consider the general (i.e., parabolic) situation.\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_Ek-on-proj-grad-spcase1}\nFor each $w\\in {^vJ}_{\\mu'}^\\nu$, we have $\\widetilde E_k({^vP}^{w(1_{\\mu'})})={^vP}^{w(1_{\\mu})}$.\n\\end{lem}\n\\begin{proof}\nBy Lemmas \\ref{ch3:lem-grad-unique} and \\ref{ch3:lem_Ek-on-proj-spcase1}, we have $\\widetilde E_k({^vP}^{w(1_{\\mu'})})={^vP}^{w(1_{\\mu})}[r]$ for some integer $r$. We must show that the shift $r$ is zero.\n\nFirst, we prove this in the non-parabolic case. \nThe module $\\Delta^{w(1_\\mu')}$ (resp. $\\Delta^{w(1_\\mu)}$) is contained in each $\\Delta$-filtration of ${^vP}^{w(1_{\\mu'})}$ (resp. ${^vP}^{w(1_{\\mu})}$) only once and without a graded shift. Moreover, by Lemma \\ref{ch3:lem_Ek-on-Verma-spcase1-grad} the module $\\Delta^{w(1_\\mu)}$ is contained in each $\\Delta$-filtration of $\\widetilde E_k(\\Delta^{w(1_\\mu')})$ only once and without a graded shift. This implies that the graded shift $r$ is zero.\n\nThe parabolic case follows from the non-parabolic case. Really, the projective covers of simple modules in the parabolic category $O$ are quotients of protective covers in the non-parabolic category $O$ (see Lemma \\ref{ch3:lem_trunc-categ-gen} $(b)$). Thus the shift $r$ should be zero in the parabolic case because it is zero in the non-parabolic case.\n\\end{proof}\n\nLet us check that the functor $\\widetilde E_k\\colon {^v\\widetilde O}^\\nu_{\\mu}\\to {^v\\widetilde O}^\\nu_{\\mu'}$\nsatisfies the hypotheses of Corollary \\ref{ch3:coro_main-Koszul}. Condition $(a)$ follows from Lemma \\ref{ch3:lem_Ek-on-proj-grad-spcase1}.\n\nLet $P$ and $Q$ be projective covers of simple modules in $^v\\widetilde O_{\\mu}$ graded as above. To check $(b)$, we have to show that we have an isomorphism\n$$\n\\mathrm{Hom}(\\widetilde E_{k}(P)\\langle 1\\rangle,\\widetilde E_{k}(Q))\\simeq \\mathrm{Hom}(P\\langle 1 \\rangle,Q).\n$$\nWe have\n$$\n\\begin{array}{rcl}\n\\mathrm{Hom}(\\widetilde E_{k}(P)\\langle 1\\rangle,\\widetilde E_{k}(Q))&\\simeq &\\mathrm{Hom}(P,\\widetilde F_{k+1}\\widetilde E_{k+1}(Q)\\langle \\mu_{k+1}-1 \\rangle)\\\\\n&\\simeq&\\mathrm{Hom}(P,[\\mu_{k+1}+1]_q(Q)\\langle \\mu_{k+1}-1 \\rangle)\\\\\n&\\simeq&\\mathrm{Hom}(P,Q\\langle -1 \\rangle)\\bigoplus \\oplus_{r=1}^{\\mu_{k+1}}\\mathrm{Hom}(P,Q\\langle 2r-1 \\rangle)\\\\\n&\\simeq&\\mathrm{Hom}(P\\langle 1 \\rangle,Q).\n\\end{array}\n$$\nHere the first isomorphism follows from Lemma \\ref{ch3:lem_gr-lift-E-F+adj-case1} $(b)$, the second isomorphism follows from Lemma \\ref{ch3:lem_prod-FE-case1}. The last isomorphism holds because $\\mathrm{Hom}(P,Q\\langle r \\rangle)$ is zero for $r>0$ because the $\\mathbb{Z}$-graded algebra\n$$\n\\mathrm{End}(\\bigoplus_{w\\in {^vJ^\\nu_\\mu}}{^vP}^{w(1_{\\mu'})})\n$$\nhas zero negative homogeneous components (as it is Koszul).\n\nFor each $\\mu=(\\mu_1,\\cdots,\\mu_e)$ we set $\\mu^{\\rm op}=(\\mu_e,\\cdots,\\mu_1)$.\nWe can define the positive level version $O_{\\mu,+}^\\nu$ of the category $O_{\\mu}^\\nu$ in the following way. For each $\\lambda\\in P$ we set $\\widetilde\\lambda^+=\\lambda+z_\\lambda\\delta+(e-N)\\Lambda_0$,\nwhere $z_\\lambda=(\\lambda,2\\rho+\\lambda)\/2{e}$. For each $\\lambda\\in P^\\nu$ denote by $^+\\Delta(\\lambda)$ the Verma module with highest weight $\\widetilde\\lambda^+$ and denote by $^+L(\\lambda)$ its simple quotient. We will also abbreviate $^+\\Delta^\\lambda={^+\\Delta}(\\lambda-\\rho)$ and $^+L^\\lambda={^+L}(\\lambda-\\rho)$.\n Let $O_{\\mu,+}^\\nu$ be the Serre subcategory of $O^\\nu$ generated by the simple modules $^+L^\\lambda$ for $\\lambda\\in P^\\nu[\\mu^{\\rm op}]$. Similarly to the negative $e$-action of $\\widehat W$ on $P$ described in Section \\ref{ch3:subs_ext-aff} we can consider the positive $e$-action on $P$. We define the positive $e$-action in the following way: the element $w\\in\\widehat W$ sends $\\lambda$ to $-w(-\\lambda)$ (where $w(-\\lambda)$ corresponds to the negative $e$-action). The notion of the positive $e$-action of $\\widehat W$ on $P$ is motivated by the fact that the map\n$$\nP\\to \\widehat\\mathbf{h}^*,\\quad \\lambda\\mapsto\\widetilde{\\lambda-\\rho}^++\\widehat\\rho\n$$\nis $\\widehat W$-invariant. \nWe say that an element $\\lambda\\in P$ is $e$-\\emph{dominant} if we have $\\lambda_1\\geqslant\\lambda_2\\geqslant\\cdots\\geqslant\\lambda_N\\geqslant\\lambda_1-e$. Fix an $e$-dominant element $1^+_\\mu\\in P[\\mu^{\\rm op}]$. (We can take for example $1^+_\\mu=(e^{\\mu_1},\\cdots,1^{\\mu_e})$). Note that the stabilizer of $1^+_\\mu$ in $\\widehat W$ with respect to the positive $e$-action is $W_\\mu$. From now on, each time when we write $w(1^+_\\mu)$ we mean the positive $e$-action on $P$ and each time when we write $w(1_\\mu)$ we mean the negative $e$-action.\n\nRecall that $J_{\\mu,+}$ is the subset of $\\widehat W$ containing all $w$ such that $w$ is maximal in $wW_\\mu$. Set $J^\\nu_{\\mu,+}=\\{w\\in J_{\\mu,+};~w(1^+_\\mu)\\in P^\\nu\\}$. Note that the inclusion $J_{\\mu'}\\subset J_\\mu$ induces an inclusion $J^\\nu_{\\mu'}\\subset J^\\nu_\\mu$. For $v\\in\\widehat W$ we set $^vJ^\\nu_\\mu=\\{w\\in J^\\nu_\\mu;~w\\leqslant v\\}$ and $^vJ^\\nu_{\\mu,+}=\\{w\\in J^\\nu_{\\mu,+};~w\\leqslant v\\}$.\n\nWe have the following lemma. \n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_comb_index_set_O}\n$(a)$ There is a bijection $J_\\mu^\\nu\\to J_{\\nu,+}^\\mu$ given by $w\\mapsto w^{-1}$.\n\n$(b)$ For each $v\\in J_\\mu^\\nu$, there is a bijection $^vJ_\\mu^\\nu\\to {^{v^{-1}}}J_{\\nu,+}^\\mu$ given by $w\\mapsto w^{-1}$.\n\\end{lem}\n\\begin{proof} \nPart $(a)$ follows from \\cite[Cor.~3.3]{SVV}. Part $(b)$ follows from part $(a)$.\n\\end{proof}\n\n\\smallskip\nSimilarly to the truncated version $^vO_{\\mu}^\\nu$ of $O_\\mu^\\nu$, we can define the truncated version $^vO_{\\mu,+}^\\nu$ of $O_{\\mu,+}^\\nu$. We define $^vO_{\\mu,+}^\\nu$ as the Serre quotient of $O_{\\mu,+}^\\nu$, where we kill the simple module $^+L^{w(1^+_\\mu)}$ for each $w\\in J_{\\mu,+}^\\nu-{^vJ}_{\\mu,+}^\\nu$.\n\nBy \\cite[Thm.~3.12]{SVV}, for $v\\in J_\\mu^\\nu$, the category ${^v\\widetilde O}_{\\mu}^\\nu$ is Koszul dual to the category ${^{v^{-1}}\\widetilde O}_{\\nu,+}^\\mu$. The bijection between the simple modules in ${^v\\widetilde O}_{\\mu}^\\nu$ and the indecomposable projective modules in ${^{v^{-1}}\\widetilde O}_{\\nu,+}^\\mu$ given by the Koszul functor $\\mathcal{K}$ is such that for each $w\\in {^vJ}_\\mu^\\nu$ the module $L^{w(1_\\mu)}$ corresponds to the projective cover of $^+L^{w^{-1}(1^+_\\nu)}$. \n\nWe should make a remark about our notation. Usually, we denote by $e$ the number of components in $\\mu$ and we denote by $l$ the number of components in $\\nu$. So, when we exchange the roles of $\\mu$ and $\\nu$ and we consider the category $O_{\\nu,+}^\\mu$, we mean that this category is defined with respect to the level $l-N$ (and not $e-N$).\n\nNow, assume again that $v$ is in $J_{\\mu'}^\\nu w_{\\mu'}$. Then we have $vw_\\mu\\in J_\\mu^\\nu$ and $vw_{\\mu'}\\in J_{\\mu'}^\\nu$. In this case the Koszul dual categories to $^vO_\\mu^\\nu$ and $^vO_{\\mu'}^\\nu$ are $^{w_\\mu v^{-1}}O_{\\nu,+}^\\mu$ and $^{w_{\\mu'}v^{-1}}O_{\\nu,+}^{\\mu'}$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem-can_change_trunc_pos_level}\n$(a)$ We have \n$$^{w_{\\mu'}v^{-1}}J_{\\nu,+}^{\\mu'}={^{w_\\mu v^{-1}}}J_{\\nu,+}^\\mu\\cap J_{\\nu,+}^{\\mu'}.$$\n\n$(b)$ We have \n$$^{w_{\\mu'}v^{-1}}J_{\\nu,+}^{\\mu'}=^{w_{\\mu}v^{-1}}J_{\\nu,+}^{\\mu'}.$$\n\\end{lem}\n\\begin{proof}\n\n\nLet us prove $(a)$. By Lemma \\ref{ch3:lem_comb_index_set_O} the statement is equivalent to $$^{vw_{\\mu'}}J^{\\nu}_{\\mu'}={^{vw_\\mu}}J^{\\nu}_\\mu\\cap J^{\\nu}_{\\mu'}.$$ Moreover, by definition, we have $^{vw_{\\mu'}}J^{\\nu}_{\\mu'}={^v}J^{\\nu}_{\\mu'}$ and ${^{vw_\\mu}}J^{\\nu}_\\mu={^{v}}J^{\\nu}_\\mu$. Thus, the statement is equivalent to $^{v}J^{\\nu}_{\\mu'}={^{v}}J^{\\nu}_\\mu\\cap J^{\\nu}_{\\mu'}$. The last equality is obvious.\n\nPart $(b)$ follows from part $(a)$.\n\n\\end{proof}\n\n\\smallskip\nNow, put $u=w_{\\mu}v^{-1}$. The discussion above together with Lemma \\ref{ch3:lem-can_change_trunc_pos_level} shows that the Koszul dual categories to to $^vO_\\mu^\\nu$ and $^vO_{\\mu'}^\\nu$ are $^{u}O_{\\nu,+}^\\mu$ and $^{u}O_{\\nu,+}^{\\mu'}$.\n\nWe get the following result.\n\n\\smallskip\n\\begin{thm}\n\\label{ch3:thm_dual-func-O-spcase1}\nAssume that we have $W_{\\mu}\\subset W_{\\mu'}$.\n\n$(a)$ The functor $\\widetilde F_k\\colon D^b(^v\\widetilde O^\\nu_{\\mu})\\to D^b(^v\\widetilde O^\\nu_{\\mu'})$ is Koszul dual to the shifted parabolic truncation functor $\\widetilde\\tr\\langle \\mu_{k+1} \\rangle\\colon D^b(^u\\widetilde O_{\\nu,+}^\\mu)\\to D^b(^u\\widetilde O_{\\nu,+} ^{\\mu'})$.\n\n$(b)$ The functor $\\widetilde E_k\\colon D^b(^v\\widetilde O^\\nu_{\\mu'})\\to D^b(^v\\widetilde O^\\nu_{\\mu})$ is Koszul dual to the parabolic inclusion functor $\\widetilde\\mathrm{inc}\\colon D^b(^u\\widetilde O_{\\nu,+}^{\\mu'})\\to D^b(^u\\widetilde O_{\\nu,+} ^{\\mu})$.\n\\end{thm}\n\\begin{proof}\nWe have checked above that the functor $\\widetilde E_k\\colon {^v\\widetilde O}^\\nu_{\\mu}\\to {^v\\widetilde O}^\\nu_{\\mu'}$ satisfies the hypotheses of Corollary \\ref{ch3:coro_main-Koszul}. Thus Corollary \\ref{ch3:coro_main-Koszul} implies part $(b)$. Part $(a)$ follows from part $(b)$ by adjointness.\n\\end{proof}\n\n\\smallskip\nSimilarly to the situation $W_{\\mu}\\subset W_{\\mu'}$, we can do the same in the situation $W_{\\mu}\\subset W_{\\mu'}$ (see also Section \\ref{ch3:subs_second-case}). In this case we should take $v\\in J_{\\mu}^\\nu w_\\mu$ and put $u=w_{\\mu'}v^{-1}$. We get the following theorem.\n\n\\smallskip\n\\begin{thm}\n\\label{ch3:thm_dual-func-O-spcase2}\nAssume that we have $W_{\\mu'}\\subset W_{\\mu}$.\n\n$(a)$ The functor $\\widetilde F_k\\colon D^b(^v\\widetilde O^\\nu_{\\mu})\\to D^b(^v\\widetilde O^\\nu_{\\mu'})$ is Koszul dual to the parabolic inclusion functor $\\widetilde\\mathrm{inc}\\colon D^b(^u\\widetilde O_{\\nu,+}^\\mu)\\to D^b(^u\\widetilde O_{\\nu,+} ^{\\mu'})$.\n\n$(b)$ The functor $\\widetilde E_k\\colon D^b(^v\\widetilde O^\\nu_{\\mu'})\\to D^b(^v\\widetilde O^\\nu_{\\mu})$ is Koszul dual to the shifted parabolic truncation functor $\\widetilde\\tr\\langle \\mu_k-1\\rangle\\colon D^b(^u\\widetilde O_{\\nu,+}^{\\mu'})\\to D^b(^u\\widetilde O_{\\nu,+} ^{\\mu})$.\n\\qed\n\\end{thm}\n\n\n\\subsection{The restriction to the category $\\mathbf{A}$}\nThe goal of this section is to restrict the results of the previous section to category $\\mathbf{A}$.\n\nWe have seen that we can grade the functor $E_k$ and $F_k$ for category $O$ when we have $W_\\mu\\subset W_{\\mu'}$ or $W_{\\mu'}\\subset W_\\mu$. Let us show that in this cases we can also grade similar functors for the category $\\mathbf{A}$. We have $\\mathbf{A}^\\nu[\\alpha]\\subset {^vO^\\nu_\\mu}$ and $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]\\subset {^vO^\\nu_{\\mu'}}$. Denote by $h$ the inclusion functor from $\\mathbf{A}^\\nu[\\alpha]$ to ${^vO^\\nu_\\mu}$. Abusing the notation, we will use the same symbol for the inclusion functor from $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ to ${^vO^\\nu_{\\mu'}}$. Let $h^*$ and $h^!$ be the left and right adjoint functors to $h$.\nThe functor $F_k$ for the category $\\mathbf{A}$ is defined as the restriction of the functor $F_k$ for the category $O$. This restriction can be written as $h^!F_kh$. The functor $E_k$ for the category $O$ does not preserve the category $\\mathbf{A}$ in general. The functor $E_k$ for the category $\\mathbf{A}$ is defined in \\cite[Sec.~5.9]{RSVV} as $h^*E_kh$. It is easy to see that we can grade the functor $h$ and its adjoint functors in the same way as we graded Zuckerman functors. Thus we obtain graded lifts $\\widetilde E_k$ and $\\widetilde F_k$ of the functors $E_k$ and $F_k$ for the category $\\mathbf{A}$. Moreover, we still have the adjunctions $(\\widetilde E_k,\\widetilde F_k\\langle\\mu_{k+1}\\rangle)$ (when $W_\\mu\\subset W_{\\mu'}$) and $(\\widetilde E_k,\\widetilde F_k\\langle 1-\\mu_{k}\\rangle)$ (when $W_{\\mu'}\\subset W_{\\mu}$) in the category $\\mathbf{A}$.\n\nWe do not have adjunctions in other direction in general. However, if additionally we have $\\nu_r>|\\alpha|$ for each $r\\in[1,l]$, then the functors $E_k$ and $F_k$ for the category $\\mathbf{A}$ are biadjoint by \\cite[Lem.~7.6]{RSVV}. This means that there is no difference between $h^*E_kh$ and $h^!E_kh$. Thus we also get the adjunctions $(\\widetilde F_k,\\widetilde E_k\\langle-\\mu_{k+1}\\rangle)$ (when $W_\\mu\\subset W_{\\mu'}$) and $(\\widetilde F_k,\\widetilde E_k\\langle \\mu_{k}-1\\rangle)$ (when $W_{\\mu'}\\subset W_{\\mu}$) in the category $\\mathbf{A}$. (In fact, we always have the adjunctions in both directions if $k\\ne 0$ because in this case the functor $E_k$ for the category $\\mathbf{A}$ is just the restriction of the functor $E_k$ for the category $O$ and similarly for $\\widetilde E_k$.)\n\n\nWe start from a general lemma. Let $A$ be a finite dimensional Koszul algebra over $\\mathbb{C}$. Let $\\{e_\\lambda;\\lambda\\in\\Lambda\\}$ be the set of indecomposable idempotents in $A$. Fix a subset $\\Lambda'\\subset \\Lambda$. Assume that the algebra $_{\\Lambda'}A$ (see Section \\ref{ch3:subs_key-lem} for the notations) is also Koszul. Then we have an algebra isomorphism $(_{\\Lambda'}A)^!\\simeq(A^!)_{\\Lambda'}$. The graded algebra $_{\\Lambda'}A$ is a quotient of the graded algebra $A$ by a homogeneous ideal. In particular we have an inclusion of categories $\\iota\\colon\\mathrm{grmod}(_{\\Lambda'}A)\\to \\mathrm{grmod}(A)$. Moreover, there is a functor\n$$\n\\tau\\colon \\mathrm{grmod}(A^!)\\to \\mathrm{grmod}((A^!)_{\\Lambda'}),\\qquad M\\mapsto e^{!}_{\\Lambda'}M.\n$$\nThe functors $\\iota$ and $\\tau$ are both exact. They yield functors between derived categories $\\iota\\colon D^b(_{\\Lambda'}A)\\to D^b(A)$ and $\\tau\\colon D^b(A^!)\\to D^b((A^!)_{\\Lambda'})$.\n\nSince the algebra $A$ is Koszul, there is a functor $\\mathcal{K}\\colon D^b(A)\\to D^b(A^!)$ defined by $\\mathcal{K}=\\mathrm{RHom}(A_0,\\bullet)$, see Section \\ref{ch3:subs_Koszul-alg}. We will sometimes write $\\mathcal{K}_A$ to specify the algebra $A$.\n\nIn the following lemma we identify $(_{\\Lambda'}A)^!=(A^!)_{\\Lambda'}$.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem_calK-subcat}\nWe have the following isomorphism of functors $D^b(_{\\Lambda'}A)\\to D^b((A^!)_{\\Lambda'})$\n$$\n\\mathcal{K}_{_{\\Lambda'}A}\\simeq \\tau\\circ \\mathcal{K}_A\\circ \\iota.\n$$\n\\end{lem}\n\\begin{proof}\nFor a complex $M\\in D^b(_{\\Lambda'}A)$, we have\n$$\n\\begin{array}{rcl}\n\\tau\\circ \\mathcal{K}_A\\circ i(M)&\\simeq&\\tau(\\mathrm{RHom}_A(A_0,M))\\\\\n&\\simeq&\\mathrm{RHom}_A(e_{\\Lambda'}A_0,M)\\\\\n&\\simeq&\\mathrm{RHom}_{_{\\Lambda'}A}({(_{\\Lambda'}A})_0,M)\\\\\n&\\simeq&\\mathcal{K}_{_{\\Lambda'}A}(M).\n\\end{array}\n$$\n\n\\end{proof}\n\n\\smallskip\nFix $\\alpha\\in Q^+_I$. Consider the category $\\mathbf{A}^\\nu[\\alpha]$ as in Section \\ref{ch3:subs_cat-bfA}. Let $\\mu$ be such that $\\mathbf{A}^\\nu[\\alpha]$ is a subcategory of $O^\\nu_\\mu$. (Then $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ is a subcategory of $O^\\nu_{\\mu'}$.) Assume that we have $W_{\\mu}\\subset W_{\\mu'}$. Assume that $v\\in J_{\\mu'}^\\nu w_{\\mu'}$ is such that $\\mathbf{A}^\\nu[\\alpha]$ is a subcategory of $^vO^\\nu_\\mu$ and $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ is a subcategory of $^vO^\\nu_{\\mu'}$. Put $u=w_\\mu v^{-1}$. The category $\\mathbf{A}^\\nu[\\alpha]$ is also Koszul. Denote by $\\widetilde\\mathbf{A}^\\nu[\\alpha]$ its graded version. The Koszul dual category to $\\mathbf{A}^\\nu[\\alpha]$ is a Serre quotient of the category $^uO_{\\nu,+}^\\mu$ (see \\cite[Rem.~3.15]{Mak-Koszul}). Let us denote this quotient and its graded version by $\\mathbf{A}_+^\\mu[\\alpha]$ and $\\widetilde\\mathbf{A}_+^\\mu[\\alpha]$ respectively. (We will also use similar notations for $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$.)\n\nFirst, we prove the following lemma.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem-Zuck_defined_A+}\nAssume that we have $W_{\\mu}\\subset W_{\\mu'}$ and $k\\ne 0$.\n\n$(a)$\nThe inclusion of categories ${^uO}_{\\nu,+}^{\\mu'}\\subset {^uO}_{\\nu,+}^{\\mu}$ yields an inclusion of categories $\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k]\\subset \\mathbf{A}_+^\\mu[\\alpha]$.\n\n$(b)$ \nThe inclusion of categories ${^u\\widetilde O}_{\\nu,+}^{\\mu'}\\subset {^u\\widetilde O}_{\\nu,+}^{\\mu}$ yields an inclusion of categories $\\widetilde\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k]\\subset \\widetilde\\mathbf{A}_+^\\mu[\\alpha]$.\n\nAssume that we have $W_{\\mu}\\supset W_{\\mu'}$ and $k\\ne 0$.\n\n$(c)$\nThe inclusion of categories ${^uO}_{\\nu,+}^{\\mu}\\subset {^uO}_{\\nu,+}^{\\mu'}$ yields an inclusion of categories $\\mathbf{A}_+^{\\mu}[\\alpha]\\subset \\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k]$.\n\n$(d)$ \nThe inclusion of categories ${^u\\widetilde O}_{\\nu,+}^{\\mu}\\subset {^u\\widetilde O}_{\\nu,+}^{\\mu'}$ yields an inclusion of categories $\\widetilde\\mathbf{A}_+^{\\mu'}[\\alpha]\\subset \\widetilde\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k]$.\n\\end{lem}\n\\begin{proof}\nDenote by $p_1$ and $p_2$ respectively the quotient functors \n$$\np_1\\colon{^uO}_{\\nu,+}^{\\mu'}\\to\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k],\\qquad p_2\\colon{^uO}_{\\nu,+}^{\\mu}\\to \\mathbf{A}^\\mu_+[\\alpha].\n$$\nTo prove $(a)$ and $(b)$, it is enough to prove that each simple module in ${^uO}_{\\nu,+}^{\\mu'}$ is killed by the functor $p_1$ if and only if it is killed by the functor $p_2$. We can get the combinatorial description of the simple modules killed by $p_1$ and $p_2$ respectively using \\cite[Rem.~2.18]{Mak-Koszul}.\n\nFor each $w\\in {^vJ}_{\\mu'}^{\\nu}$ (resp. $w\\in {^vJ}_{\\mu}^{\\nu}$), the simple module $^+L^{w^{-1}(1^+_{\\nu})}$ is killed by $p_1$ (resp. $p_2$) if and only if the simple module $L^{w(1_{\\mu'})}\\in {^v}O_{\\mu'}^\\nu$ is not in $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ (resp. the simple module $L^{w(1_\\mu)}\\in {^v}O_{\\mu}^\\nu$ is not in $\\mathbf{A}^\\nu[\\alpha]$). So, we need to show that for each $w\\in {^vJ}_{\\mu'}^\\nu$ the module $L^{w(1_{\\mu'})}\\in {^v}O_{\\mu'}^\\nu$ is in $\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ if and only if the module $L^{w(1_\\mu)}\\in {^v}O_{\\mu}^\\nu$ is in $\\mathbf{A}^\\nu[\\alpha]$. Finally, we have to show that for each $w\\in {^vJ}_{\\mu'}^\\nu$ we have $w(1_{\\mu'})\\geqslant \\rho_\\nu$ if and only if we have $w(1_{\\mu})\\geqslant \\rho_\\nu$. (Here the order is as in Section \\ref{ch3:subs_rank-ch-A}.) \n\n\nIt is obvious that $w(1_{\\mu})\\geqslant \\rho_\\nu$ implies $w(1_{\\mu'})\\geqslant \\rho_\\nu$ because we have $w(1_{\\mu'})\\geqslant w(1_{\\mu})$. Now, let us show the inverse statement. Note that we have $w(1_{\\mu'})=w(1_\\mu)+\\epsilon_r$, where $r\\in[1,N]$ is the unique index such that $w(1_\\mu)_r\\equiv k$ mod $e$. Assume that we have $w(1_{\\mu'})\\geqslant \\rho_\\nu$ but not $w(1_{\\mu})\\geqslant \\rho_\\nu$. Then we have $w(1_{\\mu'})_r=(\\rho_\\nu)_r$. Assume first that $(\\rho_\\nu)_r\\ne 1$. In particular this implies $r|\\alpha|$ for each $r\\in[1,l]$, $e>2$ and $k\\ne 0$.\n\n$(a)$ The functor $F_k\\colon \\mathbf{A}^\\nu[\\alpha]\\to\\mathbf{A}^\\nu[\\alpha+\\alpha_k]$ has a graded lift $\\widetilde F_k$ such that the functor $\\widetilde F_k\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_k])$ is Koszul dual to the shifted Zuckerman functor $\\widetilde\\mathrm{Zuc}_k^+\\langle\\mu_{k+1} \\rangle\\colon D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_k])$.\n\n$(b)$ The functor $E_k\\colon\\mathbf{A}^\\nu[\\alpha+\\alpha_k]\\to\\mathbf{A}^\\nu[\\alpha]$ has a graded lift such that the functor $\\widetilde E_k\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_k])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])$ is Koszul dual to the shifted Zuckerman functor $\\widetilde\\mathrm{Zuc}_k^-\\langle \\mu_{k}-1 \\rangle\\colon D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_k])$.\n\\end{thm}\n\\begin{proof}\nBy Theorem \\ref{ch3:thm_decomp_Fk-A} we have the following commutative diagram\n$$\n\\begin{diagram}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]} \\arrow{e,t}{\\overline F_k}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k]} \\arrow{e,t}{\\overline F_{k+1}}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k+\\overline\\alpha_{k+1}]} \\arrow{s,r}{} \\\\\n\\node{\\mathbf{A}^\\nu[\\alpha]} \\arrow{n,l}{}\n\\arrow[2]{e,b}{F_k} \\node[2]{\\mathbf{A}^\\nu[\\alpha+\\alpha_k]}\n\\end{diagram}\n$$\n\nHere the vertical maps are some equivalences of categories.\nBy unicity of Koszul grading (see \\cite[Cor.~2.5.2]{BGS}) there exist unique graded lifts of vertical maps such that they are equivalences of graded categories and they respect the chosen grading of simple modules (i.e., concentrated in degree $0$). Moreover, the top horizontal maps have graded lifts because for a suitable $v$ we have\n$$\n\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]\\subset {^v\\overline O}_{\\overline\\mu}^\\nu,\\qquad \\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k]\\subset {^v\\overline O}_{\\overline\\mu^0}^\\nu,\\qquad \\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_k+\\overline\\alpha_{k+1}]\\subset {^v\\overline O}_{\\overline\\mu'}^\\nu\n$$\nand $W_{\\overline\\mu}\\supset W_{\\overline\\mu^0}\\subset W_{\\overline\\mu'}$. This implies that there is a graded version $\\widetilde F_k$ of the functor $F_k$ such that it makes the graded version of the diagram above commutative.\n\nSince the categories $\\mathbf{A}^\\nu[\\alpha]$ and $\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]$ are equivalent, their Koszul dual categories are also equivalent. We can chose the equivalences $(\\mathbf{A}^\\nu[\\alpha])^!\\simeq \\mathbf{A}^\\mu_+[\\alpha]$ and $(\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha])^!\\simeq \\mathbf{A}^\\mu_+[\\alpha]$ in such a way that the vertical map in the diagram is Koszul dual to the identity functor. We can do the same with the categories in the right part of the diagram above.\n\nBy Theorem \\ref{ch3:thm_dual-func-A-spcases}, the left top functor in the graded version of the diagram above is Koszul dual to the parabolic inclusion functor $\\widetilde\\mathrm{inc}$ and the top right functor in the diagram is Koszul dual to the graded shift $\\widetilde\\tr\\langle\\mu_{k+1}\\rangle$ of the parabolic truncation functor. By definition (see Section \\ref{ch3:subs_Zuck}), the Zuckerman functor is the composition of the parabolic inclusion and the parabolic truncation functors. This implies that the functor $\\widetilde F_k\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_k])$ is Koszul dual to the shifted Zuckerman functor $\\widetilde\\mathrm{Zuc}_k^+\\langle\\mu_{k+1}\\rangle$. This proves $(a)$.\n\nWe can prove $(b)$ in the same way. By adjointness, the diagram above yields a similar diagram for the functor $E$. This diagram allows to grade the functor $E_k$. Then we deduce the Koszul dual functor to $E_k$ in the same way as in $(a)$.\n\\end{proof}\n\n\\subsection{The case $k=0$}\n\\label{ch3:subs_k=0}\nNow, we are going to get an analogue of Theorem \\ref{ch3:thm-final-F-E-dual-Zuck} in the case $k=0$. The main difficulty in this case is that we cannot define Zuckerman functors for the category $\\mathbf{A}_+$ in the same was as in Section \\ref{ch3:subs_Zuck-A} because Lemma \\ref{ch3:lem-Zuck_defined_A+} fails. To fix this problem we replace the category $\\mathbf{A}$ by a smaller category $A$.\n\nAssume that we have $k=0$ and $W_{\\mu}\\supset W_{\\mu'}$. In particular this implies $\\mu_1=0$.\n\nLet $A^\\nu[\\alpha+\\alpha_0]$ be the Serre subcategory of $\\mathbf{A}^\\nu[\\alpha+\\alpha_0]$ generated by simple modules $L^{\\lambda}$ such that the weight $\\lambda\\in P$ has no coordinates equal to $1$. It is a highest weight subcategory.\n\n\\smallskip\n\\begin{rk}\n\\label{ch3:rk-gr_from_bfA_to_A}\n$(a)$\nThe category $A^\\nu[\\alpha+\\alpha_0]$ inherits the Koszul grading from the category $\\mathbf{A}^\\nu[\\alpha+\\alpha_0]$ in the following way. We know that there is a Koszul algebra $A$ such that $\\mathbf{A}^\\nu[\\alpha+\\alpha_0]\\simeq \\mathrm{mod}(A)$. Let $\\{e_\\lambda;\\lambda\\in\\Lambda\\}$ be the set of indecomposable idempotents of $A_0$. Then by \\cite[Lem.~2.17]{Mak-Koszul} there is a subset $\\Lambda'\\subset \\Lambda$ such that we have $\\mathbf{A}^\\nu[\\alpha+\\alpha_0]\\simeq \\mathrm{mod}(_{\\Lambda'}A)$ (see Section \\ref{ch3:subs_key-lem} for the notations). Moreover, the Koszul dual algebra to $_{\\Lambda'}A$ is $A^!_{\\Lambda'}$.\n\nSince, we have $\\mathrm{mod}(A^!)\\simeq \\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0]$, the Koszul dual category $A^{\\mu'}_+[\\alpha+\\alpha_0]$ to $A^\\nu[\\alpha+\\alpha_0]$ is a Serre quotient of $\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0]$. The quotient functor $$a\\colon \\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0]\\to A^{\\mu'}_+[\\alpha+\\alpha_0]$$ can be seen as the functor \n$$\na\\colon\\mathrm{mod}(A^!)\\to\\mathrm{mod}(A^!_{\\Lambda'}),\\qquad M\\mapsto e^!_{\\Lambda'}M.\n$$\n\n$(b)$ The left adjoint functor $b\\colon A^{\\mu'}_+[\\alpha+\\alpha_0]\\to \\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0]$ to $a$ can be seen as\n$$\nb\\colon \\mathrm{mod}(A^!_{\\Lambda'})\\to \\mathrm{mod}(A^!),\\qquad M\\mapsto A^!e^!_{\\Lambda'}\\otimes _{A^!_{\\Lambda'}}M.\n$$\nThe functors $a$ and $b$ have obvious graded lifts\n$$\n\\widetilde a\\colon \\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0]\\to \\widetilde A^{\\mu'}_+[\\alpha+\\alpha_0],\\qquad \\widetilde b\\colon \\widetilde A^{\\mu'}_+[\\alpha+\\alpha_0]\\to \\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0].\n$$\n\nBy Proposition \\ref{ch3:prop_dual-F-G}, the functor $\\widetilde b$ is Koszul dual to the inclusion functor $\\widetilde A^\\nu[\\alpha+\\alpha_0]\\to \\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_0]$. Then, by adjointness, the functor $\\widetilde a$ is Koszul dual to the right adjoint functor to the inclusion functor above. \n\\end{rk}\n\n\n\\smallskip\nIt is easy to see from the action of $F_0$ on Verma modules (see Proposition \\ref{ch3:prop_functors-on-O-gen} $(e)$) that the image of the functor $F_0\\colon \\mathbf{A}^\\nu[\\alpha]\\to \\mathbf{A}^\\nu[\\alpha+\\alpha_0]$ is in $A^\\nu[\\alpha+\\alpha_0]$. Moreover, recall from Section \\ref{ch3:subs_cat-bfA} that the functor $E_0\\colon O_{\\mu'}^\\nu\\to O_{\\mu}^{\\nu}$ does not take $\\mathbf{A}^\\nu[\\alpha+\\alpha_0]$ to $\\mathbf{A}^\\nu[\\alpha]$. (The reader should pay attention to the fact that the functor $E_0$ for the category $\\mathbf{A}$ is not defined as the restriction of the functor $E_0$ for the category $O$.) However, it is easy to see from the action of $E_0$ on Verma modules (see Proposition \\ref{ch3:prop_functors-on-O-gen} $(e)$) that the functor $E_0$ for the category $O$ takes $A^\\nu[\\alpha+\\alpha_0]$ to $\\mathbf{A}^\\nu[\\alpha]$. Thus we get a functor $E_0\\colon A^\\nu[\\alpha+\\alpha_0]\\to\\mathbf{A}^\\nu[\\alpha]$. This functor also coincides with the restriction of the functor $E_0\\colon\\mathbf{A}^\\nu[\\alpha+\\alpha_0]\\to\\mathbf{A}^\\nu[\\alpha]$ to the category $A^\\nu[\\alpha+\\alpha_0]$. \n\n\nThe following statement can be proved in the same way as Lemma \\ref{ch3:lem-Zuck_defined_A+}.\n\n\\smallskip\n\\begin{lem}\n\\label{ch3:lem-Zuck_defined_A+_k=0}\nAssume that we have $W_{\\mu}\\supset W_{\\mu'}$.\n\n$(a)$\nThe inclusion of categories ${^uO}_{\\nu,+}^{\\mu}\\subset {^uO}_{\\nu,+}^{\\mu'}$ yields an inclusion of categories $\\mathbf{A}_+^{\\mu}[\\alpha]\\subset A_+^{\\mu'}[\\alpha+\\alpha_0]$.\n\n$(b)$ \nThe inclusion of categories ${^u\\widetilde O}_{\\nu,+}^{\\mu'}\\subset {^u\\widetilde O}_{\\nu,+}^{\\mu}$ yields an inclusion of categories $\\widetilde \\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_0]\\subset \\widetilde A_+^\\mu[\\alpha]$.\n\n\\qed\n\\end{lem}\n\n\n\n\n\n\n\nThe lemma above allows us to define the inclusion and the truncation functors $\\mathrm{inc}\\colon \\mathbf{A}_+^{\\mu}[\\alpha]\\to A_+^{\\mu'}[\\alpha+\\alpha_0]$, $\\tr\\colon A_+^{\\mu'}[\\alpha+\\alpha_0]\\to \\mathbf{A}_+^{\\mu}[\\alpha]$ and their graded versions $\\widetilde\\mathrm{inc}$, $\\widetilde\\tr$.\n\n\n\nWe still assume $k=0$ but we do not assume $W_{\\mu}\\supset W_{\\mu'}$ any more. We define the Zuckerman functors $\\mathrm{Zuc}^{\\pm}_0$ for this case. Let us identify $\\mathbf{A}_+^{\\mu}[\\alpha]\\simeq \\overline\\mathbf{A}_+^{\\overline\\mu}[\\beta+\\overline\\alpha]$ and $\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_k]\\simeq \\overline\\mathbf{A}_+^{\\overline\\mu'}[\\beta+\\overline\\alpha+\\overline\\alpha_k+\\overline\\alpha_{k+1}]$. By Lemmas \\ref{ch3:lem-Zuck_defined_A+}, \\ref{ch3:lem-Zuck_defined_A+_k=0} we have the following inclusions of categories\n$$\n\\overline\\mathbf{A}_+^{\\overline\\mu}[\\beta+\\overline\\alpha]\\subset \\overline A_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0],\\qquad \\overline\\mathbf{A}_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0]\\supset \\overline\\mathbf{A}_+^{\\overline\\mu'}[\\beta+\\overline\\alpha+\\overline\\alpha_0+\\overline\\alpha_{1}].\n$$\nWe define the Zuckerman functor $\\mathrm{Zuc}_0^+\\colon \\mathbf{A}_+^{\\mu}[\\alpha]\\to \\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_0]$ as the composition \n$$\n\\mathbf{A}_+^{\\mu}[\\alpha]\\stackrel{\\mathrm{inc}}{\\to}\\overline A_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0]\\stackrel{b}{\\to}\\overline \\mathbf{A}_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0]\\stackrel{\\tr}{\\to} \\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_0].\n$$\nSimilarly, we define the Zuckerman functor $\\mathrm{Zuc}_0^-\\colon \\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_0]\\to \\mathbf{A}_+^{\\mu}[\\alpha]$ as the composition\n$$\n\\mathbf{A}_+^{\\mu'}[\\alpha+\\alpha_0]\\stackrel{\\mathrm{inc}}{\\to}\\overline \\mathbf{A}_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0]\\stackrel{a}{\\to}\\overline A_+^{\\overline\\mu^0}[\\beta+\\overline\\alpha+\\overline\\alpha_0]\\stackrel{\\tr}{\\to}\\mathbf{A}_+^{\\mu}[\\alpha].\n$$\nReplacing the functors $\\mathrm{inc}$, $\\tr$, $a$, $b$ by their graded versions $\\widetilde\\mathrm{inc}$, $\\widetilde\\tr$, $\\widetilde a$, $\\widetilde b$ yields graded versions $\\widetilde\\mathrm{Zuc}_0^+$ and $\\widetilde\\mathrm{Zuc}_0^-$ of the Zuckerman functors.\n\nNow, similarly to Theorem \\ref{ch3:thm_dual-func-A-spcases} we can prove the following.\n\n\\smallskip\n\\begin{thm}\n\\label{ch3:thm_dual-func-A-spcases-k=0}\nAssume that we have $k=0$ and $W_{\\mu}\\supset W_{\\mu'}$.\n\n$(a)$ The functor $\\widetilde F_0\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])\\to D^b(\\widetilde A^\\nu[\\alpha+\\alpha_0])$ is Koszul dual to the parabolic inclusion functor $\\widetilde\\mathrm{inc}\\colon D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])\\to D^b(\\widetilde A^{\\mu'}_+[\\alpha+\\alpha_0])$.\n\n$(b)$ The functor $\\widetilde E_0\\colon D^b(\\widetilde A^\\nu[\\alpha+\\alpha_0])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])$ is Koszul dual to the shifted parabolic truncation functor $\\widetilde\\tr\\langle \\mu_{0}-1\\rangle\\colon D^b(\\widetilde A^{\\mu'}_+[\\alpha+\\alpha_0])\\to D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])$.\n\\qed\n\\end{thm}\n\n\\smallskip\nFinally, we get an analogue of Theorem \\ref{ch3:thm-final-F-E-dual-Zuck} in the case $k=0$. \n\n\\smallskip\n\\begin{thm}\n\\label{ch3:thm-final-F-E-dual-Zuck}\nAssume that we have $\\nu_r>|\\alpha|$ for each $r\\in[1,l]$ and $e>2$.\n\n$(a)$ The functor $F_0\\colon \\mathbf{A}^\\nu[\\alpha]\\to\\mathbf{A}^\\nu[\\alpha+\\alpha_0]$ has a graded lift $\\widetilde F_0$ such that the functor $\\widetilde F_0\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_0])$ is Koszul dual to the shifted Zuckerman functor $\\widetilde\\mathrm{Zuc}_0^+\\langle\\mu_{1} \\rangle\\colon D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0])$.\n\n$(b)$ The functor $E_0\\colon\\mathbf{A}^\\nu[\\alpha+\\alpha_0]\\to\\mathbf{A}^\\nu[\\alpha]$ has a graded lift $\\widetilde E_0$ such that the functor $\\widetilde E_0\\colon D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha+\\alpha_0])\\to D^b(\\widetilde\\mathbf{A}^\\nu[\\alpha])$ is Koszul dual to the shifted Zuckerman functor $\\widetilde\\mathrm{Zuc}_0^-\\langle \\mu_{0}-1 \\rangle\\colon D^b(\\widetilde\\mathbf{A}^\\mu_+[\\alpha])\\to D^b(\\widetilde\\mathbf{A}^{\\mu'}_+[\\alpha+\\alpha_0])$.\n\\end{thm}\n\\begin{proof}\nThe proof is similar to the proof of Theorem \\ref{ch3:thm-final-F-E-dual-Zuck}. To prove $(a)$ we should consider the diagram as in the proof of Theorem \\ref{ch3:thm-final-F-E-dual-Zuck} with an additional term.\n$$\n\\begin{diagram}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha]} \\arrow{e,t}{\\overline F_0}\n\\node{\\overline A^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_0]} \\arrow{e,t}{}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_0]} \\arrow{e,t}{\\overline F_{1}}\n\\node{\\overline\\mathbf{A}^\\nu[\\beta+\\overline\\alpha+\\overline\\alpha_0+\\overline\\alpha_{1}]} \\arrow{s,r}{} \\\\\n\\node{\\mathbf{A}^\\nu[\\alpha]} \\arrow{n,l}{}\n\\arrow[3]{e,b}{F_0} \\node[3]{\\mathbf{A}^\\nu[\\alpha+\\alpha_0]}\n\\end{diagram}\n$$\n\nWe prove $(b)$ in the same way by considering the diagram obtained from the diagram above by adjointness. Note that in this case we have the adjunction $(F_0,E_0)$ (and not only $(E_0,F_0)$) because of the assumption on $\\nu$. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\n\nWhen studying the homotopy theory of algebras over operads, a common question is that of rectification, i.e., determining when a weak equivalence $f:\\mathsf{O} \\to \\mathscr{P}$ of operads induces a Quillen equivalence between $\\mathsf{O}$-algebras and $\\mathscr{P}$-algebras. Rectification can be viewed as a generalization of change-of-rings results. A question that has received less attention, but which is an important part of the theory, regards changing the base model category. When does a Quillen equivalence $L:\\mathcal{M} \\hspace{-.1cm \\mathcal{N}:R$ lift to an equivalence on the level of algebras? This question was first studied in \\cite{ss03}, has been studied in a limited scope for $\\Sigma$-cofibrant operads in \\cite{fresse-book}, and has been studied for commutative monoids in \\cite{white-commutative}, but a general treatment is lacking in the literature.\n\nIn this paper, we provide a unified framework for answering questions of rectification and change of base model category simultaneously and in a great deal of generality. We work with $\\mathfrak{C}$-colored operads for any set $\\mathfrak{C}$, and we use model categories and semi-model categories as our setting for studying the homotopy theory of operad algebras. Relevant definitions are reviewed in Sections \\ref{sec:prelims} and \\ref{sec:colored}. Fundamentally, we are interested in studying the adjoint lifting diagram\n\\begin{equation*}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{\\overline{L}} \\ar@<2.5pt>[d]^-{U}\n& \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{R} \\ar@<2.5pt>[d]^-{U} \\\\\n\\M^{\\mathfrak{C}} \\ar@<2.5pt>[r]^-{L} \\ar@<2.5pt>[u]^-{\\mathsf{O} \\circ -} \n& \\N^{\\mathfrak{C}} \\ar@<2.5pt>[l]^-{R} \\ar@<2.5pt>[u]^-{\\mathscr{P} \\circ -}}\n\\end{equation*}\nfrom a homotopical perspective, in the context of Quillen equivalences and operadic algebras. We determine when a Quillen equivalence $(L,R)$ induces a Quillen equivalence $(\\overline{L}, R)$ of algebras. In Section \\ref{sec:main}, we prove our Main Theorem \\ref{main.theorem}, stated here:\n\n\\begin{main}[Lifting Quillen Equivalences]\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}).\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. The entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise weak equivalence in $\\mathcal{N}$. \n\\end{enumerate}\nThen the lifted adjunction \\eqref{lbar.ocomp.diagram}\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{R}}\\]\nis a Quillen equivalence between the semi-model categories of $\\mathsf{O}$-algebras in $\\mathcal{M}$ and of $\\mathscr{P}$-algebras in $\\mathcal{N}$ (Theorem \\ref{theorem623}).\n\\end{main}\n\nWe provide numerous examples of model categories satisfying the conditions of this theorem, and we prove a version of this theorem for $\\Sigma$-cofibrant colored operads (Theorem \\ref{main.theorem.Sigma}), where the hypotheses on the adjunction $(L,R)$ are effectively always satisfied in examples of interest.\n\nIn Section \\ref{sec:rect-and-change}, we specialize this general theorem and the $\\Sigma$-cofibrant version to obtain results about rectification, and about lifting Quillen equivalences to modules, (commutative) algebras, non-symmetric operads, generalized props, cyclic operads, and modular operads. We recover results from \\cite{ss03}, \\cite{muro11}, \\cite{muro14}, and \\cite{fresse-book} all as special cases of the same general theorem, and we then provide numerous new applications of this theorem. The results for non $\\Sigma$-cofibrant operads are entirely new.\n\nIn Section \\ref{sec:bous-loc}, we apply our main theorem in left Bousfield localizations $L_\\C(\\M)$ and $L_\\D(\\N)$, so that we may lift Quillen equivalences $L:\\mathcal{M} \\hspace{-.1cm \\mathcal{N}:R$ to local categories of algebras\n\\[\\overline{L}: \\mathsf{Alg}(\\mathsf{O};\\lcm) \\hspace{-.1cm \\mathsf{Alg}(\\sP;\\ldn):R.\\]\nIn Theorems \\ref{main.theorem.local} and \\ref{main.theorem.local.Sigma}, we provide checkable conditions allowing for the local application of our main results,Theorems \\ref{main.theorem} and \\ref{main.theorem.Sigma}. We provide examples of model categories where the conditions are satisfied. We specialize these local results to obtain results about local rectification, local change-of-rings, and local modules, (commutative) algebras, non-symmetric operads, generalized props, cyclic operads, and modular operads. \n\nLastly, in Section \\ref{sec:applications}, we recover results of \\cite{ss03}, \\cite{shipley-hz-spectra}, and \\cite{richter-shipley}, where chains of Quillen equivalences were manually lifted to categories of modules, (commutative) monoids, and $E_\\infty$-algebras. These examples include the Dold-Kan equivalence, the Quillen equivalence between DGAs and $HR$-algebra spectra, and the Quillen equivalence between commutative DGAs and commutative $HR$-algebra spectra. We demonstrate how our main theorems could have been used in these settings. In particular, as special cases of our main theorem in the characteristic $0$ setting, we obtain a zig-zag of three Quillen equivalences between commutative $H\\mathbb{Q}$-algebra spectra and commutative differential graded $\\mathbb{Q}$-algebras. This is a slightly improved version of \\cite{richter-shipley} (Corollary 8.4), which contains a zig-zag of six Quillen equivalences between the same end categories. We hope our unified approach will find many applications in future such settings.\n\n\n\n\n\n\n\\section{Model Categories}\n\\label{sec:prelims}\n\nIn this section we recall some key concepts in model category theory. Our main references here are \\cite{hirschhorn,hovey,ss,ss03}. In this paper, $(\\mathcal{M}, \\otimes, \\mathbb{1}, \\Hom)$ and $\\mathcal{N}$ will usually be a symmetric monoidal closed category with $\\otimes$-unit $\\mathbb{1}$ and internal hom $\\Hom$. We assume $\\mathcal{M}$ and $\\mathcal{N}$ have all small limits and colimits. Their initial and terminal objects are denoted by $\\varnothing$ and $*$, respectively.\n\n\\subsection{Monoidal Model Categories}\n\nWe assume the reader is familiar with basic facts about model categories as presented in \\cite{hirschhorn} and \\cite{hovey}. For a model category $\\mathcal{M}$, its subcategory of cofibrations is denoted by $\\M_{{\\scalebox{.5}{$\\mathrm{cof}$}}}$. When we work with model categories they will most often be \\emph{cofibrantly generated}; i.e., there is a set $I$ of cofibrations and a set $J$ of trivial cofibrations (i.e. maps which are both cofibrations and weak equivalences) which permit the small object argument (with respect to some cardinal $\\kappa$), and a map is a (trivial) fibration if and only if it satisfies the right lifting property with respect to all maps in $J$ (resp. $I$).\n\nLet $I$-cell denote the class of transfinite compositions of pushouts of maps in $I$, and let $I$-cof denote retracts of such. In order to run the small object argument, we will assume the domains $K$ of the maps in $I$ (and $J$) are $\\kappa$-small relative to $I$-cell (resp. $J$-cell); i.e., given a regular cardinal $\\lambda \\geq \\kappa$ and any $\\lambda$-sequence $X_0\\to X_1\\to \\cdots$ formed of maps $X_\\beta \\to X_{\\beta+1}$ in $I$-cell, the map of sets\n\\[\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\colim_{\\beta < \\lambda} \\mathcal{M}\\bigl(K,X_\\beta\\bigr) \\ar[r] \n& \\mathcal{M}\\bigl(K,\\colim_{\\beta < \\lambda} X_\\beta\\bigr)}\n\\]\nis a bijection. An object is \\emph{small} if there is some $\\kappa$ for which it is $\\kappa$-small. See Chapter 10 of \\cite{hirschhorn} for a more thorough treatment of this material.\n\nWe must now discuss the interplay between the monoidal structure and the model structure which we will require in this paper. This definition is taken from 3.1 in \\cite{ss}.\n\n\\begin{definition}\nA symmetric monoidal closed category $\\mathcal{M}$ equipped with a model structure is called a \\emph{monoidal model category} if it satisfies the following \\emph{pushout product axiom}: \n\n\\begin{itemize}\n\\item Given any cofibrations $f:X_0\\to X_1$ and $g:Y_0\\to Y_1$, the pushout corner map\n\\[\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\nX_0\\otimes Y_1 \\coprod\\limits_{X_0\\otimes Y_0}X_1\\otimes Y_0 \n\\ar[r]^-{f\\mathbin\\square g} & X_1\\otimes Y_1}\n\\]\nis a cofibration. If, in addition, either $f$ or $g$ is a weak equivalence then $f\\mathbin\\square g$ is a trivial cofibration.\n\\end{itemize}\n\\end{definition}\n\nNote that the pushout product axiom is equivalent to the statement that $-\\otimes-$ is a Quillen bifunctor.\n\n\n\\subsection{Semi-Model Categories}\n\nWhen attempting to study the homotopy theory of algebras over a colored operad, the usual method is to transfer a model structure from $\\mathcal{M}$ to this category of algebras along the free-forgetful adjunction (using Kan's Lifting Theorem \\cite{hirschhorn} (11.3.2)). Unfortunately, it is often the case that one of the conditions for Kan's theorem cannot be checked fully, so that the resulting homotopical structure on the category of algebras is something less than a model category. This type of structure was first studied in \\cite{hovey-monoidal} and \\cite{spitzweck-thesis}, and later in published sources such as \\cite{fresse} and \\cite{fresse-book} (12.1).\n\n\\begin{definition}\n\\label{defn:semi-model-cat}\nAssume there is an adjunction $F:\\mathcal{M} \\hspace{-.1cm \\mathcal{D}:U$ where $\\mathcal{M}$ is a cofibrantly generated model category, $\\mathcal{D}$ is bicomplete, and $U$ preserves colimits indexed by non-empty ordinals. \n\nWe say that $\\mathcal{D}$ is a \\emph{semi-model category} if $\\mathcal{D}$ has three classes of morphisms called \\emph{weak equivalences}, \\emph{fibrations}, and \\emph{cofibrations} such that the following axioms are satisfied. A \\emph{cofibrant} object $X$ means an object in $\\mathcal{D}$ such that the map from the initial object of $\\mathcal{D}$ to $X$ is a cofibration in $\\mathcal{D}$. Likewise, a \\emph{fibrant} object is an object for which the map to the terminal object in $\\mathcal{D}$ is a fibration in $\\mathcal{D}$.\n\n\\begin{enumerate}\n\\item $U$ preserves fibrations and trivial fibrations ($=$ maps that are both weak equivalences and fibrations).\n\\item $\\mathcal{D}$ satisfies the 2-out-of-3 axiom and the retract axiom of a model category.\n\\item Cofibrations in $\\mathcal{D}$ have the left lifting property with respect to trivial fibrations. Trivial cofibrations ($=$ maps that are both weak equivalences and cofibrations) in $\\mathcal{D}$ whose domain is cofibrant have the left lifting property with respect to fibrations.\n\\item Every map in $\\mathcal{D}$ can be functorially factored into a cofibration followed by a trivial fibration. Every map in $\\mathcal{D}$ whose domain is cofibrant can be functorially factored into a trivial cofibration followed by a fibration.\n\\item The initial object in $\\mathcal{D}$ is cofibrant.\n\\item Fibrations and trivial fibrations are closed under pullback.\n\\end{enumerate}\n\n\n$\\mathcal{D}$ is said to be \\textit{cofibrantly generated} if there are sets of morphisms $I'$ and $J'$ in $\\mathcal{D}$ such that the following conditions are satisfied.\n\\begin{enumerate}\n\\item\nDenote by $I'$-inj the class of maps that have the right lifting property with respect to maps in $I'$. Then $I'$-inj is the class of trivial fibrations.\n\\item\n$J'$-inj is the class of fibrations in $\\mathcal{D}$.\n\\item\nThe domains of $I'$ are small relative to $I'$-cell.\n\\item\nThe domains of $J'$ are small relative to maps in $J'$-cell whose domain is sent by $U$ to a cofibrant object in $\\mathcal{M}$.\n\\end{enumerate}\n\\end{definition}\n\n\nIn practice the weak equivalences (resp. fibrations) are morphisms $f$ such that $U(f)$ is a weak equivalence (resp. fibration) in $\\mathcal{M}$, and the generating (trivial) cofibrations of $\\mathcal{D}$ are maps of the form $F(I)$ and $F(J)$ where $I$ and $J$ are the generating (trivial) cofibrations of $\\mathcal{M}$.\n\n\nNote that the only difference between a semi-model structure and a model structure is that one of the lifting properties and one of the factorization properties requires the domain of the map in question to be cofibrant. Because fibrant and cofibrant replacements are constructed via factorization, (4) of a semi-model category implies that every object has a cofibrant replacement and that cofibrant objects have fibrant replacements. So one could construct a fibrant replacement functor which first does cofibrant replacement and then does fibrant replacement. These functors behave as they would in the presence of a full model structure. \n\n\n\n\\subsection{Quillen Adjunctions and Quillen Equivalences}\n\nAn adjunction with left adjoint $L$ and right adjoint $R$ is denoted by $L \\dashv R$.\n\n\\begin{definition}\nA \\emph{lax monoidal functor} $F : \\mathcal{M} \\to \\mathcal{N}$ between two monoidal categories is a functor equipped with structure maps\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{FX \\otimes FY \\ar[r]^-{F^2_{X,Y}} & F(X \\otimes Y), \\quad \\mathbb{1}^{\\mathcal{N}} \\ar[r]^-{F^0} & F\\mathbb{1}^{\\mathcal{M}}}\\]\nfor $X$ and $Y$ in $\\mathcal{M}$ that are associative and unital in a suitable sense \\cite{maclane} (XI.2). If, furthermore, $\\mathcal{M}$ and $\\mathcal{N}$ are symmetric monoidal categories, and $F^2$ is compatible with the symmetry isomorphisms, then $F$ is called a \\emph{lax symmetric monoidal functor}. If the structure maps $F^2$ and $F^0$ are isomorphisms, then $F$ is called a \\emph{strong monoidal functor}.\n\\end{definition}\n\nNote that what is called a lax monoidal functor here is simply called a monoidal functor in \\cite{maclane}.\n\n\\begin{definition}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is an adjunction between monoidal categories with $R$ a lax monoidal functor. For objects $X$ and $Y$ in $\\mathcal{M}$, the map\n\\begin{equation}\\label{comonoidal.map}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\left(X \\otimes Y\\right) \\ar[r]^-{L^2_{X,Y}} & LX \\otimes LY \\in \\mathcal{N},}\n\\end{equation}\ndefined as the adjoint of the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{X \\otimes Y \\ar[r]^-{(\\eta_X, \\eta_Y)} & RLX \\otimes RLY \\ar[r]^-{R^2_{X,Y}} & R\\bigl(LX \\otimes LY\\bigr),}\\]\nis called the \\emph{comonoidal structure map} of $L$ \\cite{ss03} (3.3). Here $\\eta$ is the unit of the adjunction.\n\\end{definition}\n\nThe following definition applies to both model categories and semi-model categories.\n\n\\begin{definition}\\label{quillen.pair}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is an adjunction between (semi) model categories.\n\\begin{enumerate}\n\\item We call $L \\dashv R$ a \\emph{Quillen adjunction} if the right adjoint $R$ preserves fibrations and trivial fibrations. In this case, we call $L$ a \\emph{left Quillen functor} and $R$ a \\emph{right Quillen functor}.\n\\item We call a Quillen adjunction $L \\dashv R$ a \\emph{Quillen equivalence} if, for each map $f : LX \\to Y \\in \\mathcal{N}$ with $X$ cofibrant in $\\mathcal{M}$ and $Y$ fibrant in $\\mathcal{N}$, $f$ is a weak equivalence in $\\mathcal{N}$ if and only if its adjoint $\\overline{f} : X \\to RY$ is a weak equivalence in $\\mathcal{M}$. In this case, we call $L$ a \\emph{left Quillen equivalence} and $R$ a \\emph{right Quillen equivalence}.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nFor model categories, our definition of a Quillen equivalence is of course the standard one \\cite{hovey} (1.3.12). On the other hand, for semi-model categories, our definition of a Quillen equivalence is actually slightly stronger than (and hence implies) the one given in \\cite{fresse-book} (12.1.8, p.191). To see that our definition of a Quillen equivalence between semi-model categories is stronger than Fresse's, simply use the usual proof \\cite{hovey} (1.3.13 (a) $\\Rightarrow$ (b)) that there are several equivalent ways to state a Quillen equivalence for model categories. As a consequence, the total derived functors of a Quillen equivalence between semi-model categories (as in Def. \\ref{quillen.pair} (2)) form adjoint equivalences between the homotopy categories.\n\\end{remark}\n\n\\begin{definition}\n\\label{def:weak.symmetric.monoidal}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a Quillen adjunction between monoidal categories that are also model categories in which $R$ is equipped with a lax (symmetric) monoidal structure. We call the Quillen adjunction $L \\dashv R$ a \\emph{weak (symmetric) monoidal Quillen adjunction} \\cite{ss03} (3.6) if the following two conditions hold.\n\\begin{enumerate}\n\\item For any cofibrant objects $X$ and $Y$ in $\\mathcal{M}$, the comonoidal structure map \\eqref{comonoidal.map} is a weak equivalence in $\\mathcal{N}$.\n\\item For some cofibrant replacement $q : Q\\mathbb{1}^{\\mathcal{M}} \\to \\mathbb{1}^{\\mathcal{M}}$ of the tensor unit in $\\mathcal{M}$, the composite\n\\begin{equation}\\label{unit.adjoint}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{LQ\\mathbb{1}^{\\mathcal{M}} \\ar[r]^-{Lq} & L\\mathbb{1}^{\\mathcal{M}} \\ar[r]^-{\\overline{R}^0} & \\mathbb{1}^{\\mathcal{N}}}\n\\end{equation}\nis a weak equivalence in $\\mathcal{N}$, in which $\\overline{R}_0$ is the adjoint of the structure map $R^0 : \\mathbb{1}^{\\mathcal{M}} \\to R\\mathbb{1}^{\\mathcal{N}}$.\n\\end{enumerate}\nIf, furthermore, $L \\dashv R$ is a Quillen equivalence, then we call it a \\emph{weak (symmetric) monoidal Quillen equivalence}.\n\\end{definition}\n\n\n\\begin{example} \\label{example:ss03}\nAs discussed in \\cite{ss03}, both\n\\begin{enumerate}\n\\item the Dold-Kan correspondence between simplicial modules and non-negatively graded dg modules over a characteristic $0$ field and\n\\item the adjunction between reduced rational simplicial Lie algebras and reduced rational dg Lie algebras \\cite{quillen} (p.211)\n\\end{enumerate}\nare weak symmetric monoidal Quillen equivalences.\n\\end{example}\n\n\n\\subsection{Model Category Assumptions}\n\n\\begin{definition}\n\\label{def:star}\nSuppose $\\mathcal{M}$ is a symmetric monoidal category and a model category. Define the following conditions.\n\\begin{quote}\n$(\\medstar)$ : Suppose $n \\geq 1$, $g : U \\to V \\in \\M^{\\Sigma^{\\smallop}_n}$ is a weak equivalence with $U$ and $V$ cofibrant in $\\mathcal{M}$, and $X \\in \\M^{\\Sigma_n}$ is cofibrant in $\\mathcal{M}$. Then the map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{U \\underset{\\Sigma_n}{\\otimes} X \\ar[r]^-{g \\underset{\\Sigma_n}{\\otimes} X} & V \\underset{\\Sigma_n}{\\otimes} X}\\]\nis a weak equivalence in $\\mathcal{M}$.\n\\end{quote}\n\\begin{quote}\n$(\\filledstar)$ : Suppose $n \\geq 1$ and $X \\in \\M^{\\Sigma_n}$ is cofibrant in $\\mathcal{M}$. Then the coinvariant $X_{\\Sigma_n} \\in \\mathcal{M}$ is also cofibrant.\n\\end{quote}\n\\end{definition}\n\n\\begin{example}\\label{star.examples}\nConditions $(\\medstar)$ and $(\\filledstar)$ both hold in the model categories of:\n\\begin{itemize}\n\\item simplicial modules over a characteristic $0$ field;\n\\item chain complexes, bounded or unbounded, over a characteristic $0$ field;\n\\item reduced rational simplicial Lie algebras;\n\\item reduced rational dg Lie algebras.\n\\end{itemize}\nIn fact, both $(\\medstar)$ and $(\\filledstar)$ hold whenever an object in $\\M^{\\Sigma_n}$ is cofibrant if and only if it is cofibrant as an object in $\\mathcal{M}$ (as in the above rational settings). To see this for $(\\medstar)$, one uses the left Quillen functor $(-)_{\\Sigma_n} : \\M^{\\Sigma_n} \\to \\mathcal{M}$ together with the pushout product axiom, \\cite{hirschhorn} (11.6.1 and 11.6.3), \\cite{bm06} (2.5.2), and Ken Brown's Lemma \\cite{hovey} (1.1.12). To see this for $(\\filledstar)$, it is enough to observe that $(-)_{\\Sigma_n}$ is a left Quillen functor.\n\\end{example}\n\nThe next definition is an equivariant version of Def. \\ref{def:weak.symmetric.monoidal}(1).\n\n\\begin{definition}\n\\label{def:sharp}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is an adjunction between symmetric monoidal categories that are also model categories with $R$ lax symmetric monoidal. Define the following condition.\n\\begin{quote}\n$(\\#)$ : Suppose $n \\geq 1$, $W \\in \\M^{\\Sigma^{\\smallop}_n}$ is cofibrant in $\\mathcal{M}$, and $X \\in \\M^{\\Sigma_n}$ is cofibrant in $\\mathcal{M}$. Then the map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt@C+.5cm{\\bigl[L(W \\otimes X)\\bigr]_{\\Sigma_n} \\ar[r]^-{(L^2_{W,X})_{\\Sigma_n}} & \\bigl[LW \\otimes LX\\bigr]_{\\Sigma_n}}\\]\nis a weak equivalence in $\\mathcal{N}$, where $L^2_{W,X}$ is the comonoidal structure map of $L$ \\eqref{comonoidal.map}. \n\\end{quote}\n\\end{definition}\n\nNote that $(\\#)$ is equivalent to the assertion that the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt@C+.5cm{L\\left(W \\underset{\\Sigma_n}{\\otimes} X\\right) \\ar[d]_-{\\cong} \\ar[r] & LW \\underset{\\Sigma_n}{\\otimes} LX\\\\\n\\bigl[L(W \\otimes X)\\bigr]_{\\Sigma_n} \\ar[r]^-{(L^2_{W,X})_{\\Sigma_n}} & \\bigl[LW \\otimes LX\\bigr]_{\\Sigma_n} \\ar[u]_-{=}}\\]\nis a weak equivalence in $\\mathcal{N}$. The isomorphism on the left comes from the fact that taking $\\Sigma_n$-coinvariant is a colimit, and the left adjoint $L$ commutes with colimits.\n\n\n\\begin{example}\nCondition $(\\#)$ holds when:\n\\begin{enumerate}\n\\item $L$ is strong symmetric monoidal.\n\\item $L \\dashv R$ is the Dold-Kan correspondence between simplicial $k$-modules and non-negatively graded chain complexes of $k$-modules for a characteristic $0$ field $k$.\n\\item $L \\dashv R$ is the adjunction between reduced rational simplicial Lie algebras and reduced rational dg Lie algebras \\cite{quillen} (p.211).\n\\end{enumerate}\n\\end{example}\n\n\n\\section{Colored Operads}\n\\label{sec:colored}\n\nIn this section we recall some results regarding colored operads that will be needed in later sections.\n\n\\subsection{Colors and Profiles}\n\nHere we recall from \\cite{jy2} some notations regarding colors that are needed to talk about colored objects.\n\n\n\\begin{definition}[Colored Objects]\n\\label{def:profiles}\nFix a non-empty set $\\mathfrak{C}$, whose elements are called \\textbf{colors}.\n\\begin{enumerate}\n\\item\nA \\emph{$\\mathfrak{C}$-profile} is a finite sequence of elements in $\\mathfrak{C}$, say,\n\\[\\underline{c} = (c_1, \\ldots, c_m) = c_{[1,m]}\\]\nwith each $c_i \\in \\mathfrak{C}$. If $\\mathfrak{C}$ is clear from the context, then we simply say \\emph{profile}. The empty $\\mathfrak{C}$-profile is denoted $\\emptyset$, which is not to be confused with the initial object in $\\mathcal{M}$. Write $|\\underline{c}|=m$ for the \\emph{length} of a profile $\\underline{c}$.\n\\item\nAn object in the product category $\\prod_{\\mathfrak{C}} \\mathcal{M} = \\mathcal{M}^{\\mathfrak{C}}$ is called a \\emph{$\\mathfrak{C}$-colored object in $\\mathcal{M}$}, and similarly for a map of $\\mathfrak{C}$-colored objects. A typical $\\mathfrak{C}$-colored object $X$ is also written as $\\{X_a\\}$ with $X_a \\in \\mathcal{M}$ for each color $a \\in \\mathfrak{C}$.\n\\item\nSuppose $X \\in \\calm^{\\fC}$ and $c \\in \\mathfrak{C}$. Then $X$ is said to be \\emph{concentrated in the color $c$} if $X_d = \\varnothing$ for all $c \\not= d \\in \\mathfrak{C}$.\n\\item\nSuppose $f : X \\to Y \\in \\mathcal{M}$ and $c \\in \\mathfrak{C}$. Then $f$ is said to be \\emph{concentrated in the color $c$} if both $X$ and $Y$ are concentrated in the color $c$.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\nNext we define the colored version of a $\\Sigma$-object, also known as a symmetric sequence.\n\n\n\n\\begin{definition}[Colored Symmetric Sequences]\n\\label{def:colored-sigma-object}\nFix a non-empty set $\\mathfrak{C}$.\n\\begin{enumerate}\n\\item\nIf $\\underline{a} = (a_1,\\ldots,a_m)$ and $\\underline{b}$ are $\\mathfrak{C}$-profiles, then a \\emph{map} (or \\emph{left permutation}) $\\sigma : \\underline{a} \\to \\underline{b}$ is a permutation $\\sigma \\in \\Sigma_{|\\underline{a}|}$ such that\n\\[\\sigma\\underline{a} = (a_{\\sigma^{-1}(1)}, \\ldots , a_{\\sigma^{-1}(m)}) = \\underline{b}\\]\nThis necessarily implies $|\\underline{a}| = |\\underline{b}| = m$.\n\\item\nThe \\emph{groupoid of $\\mathfrak{C}$-profiles}, with left permutations as the isomorphisms, is denoted by $\\Sigma_{\\frakC}$. The opposite groupoid $\\pofcop$ is regarded as the groupoid of $\\mathfrak{C}$-profiles with \\emph{right permutations}\n\\[\\underline{a}\\sigma = (a_{\\sigma(1)}, \\ldots , a_{\\sigma(m)})\\]\nas isomorphisms.\n\\item\nThe \\emph{orbit} of a profile $\\underline{a}$ is denoted by $[\\underline{a}]$. The maximal connected sub-groupoid of $\\Sigma_{\\frakC}$ containing $\\underline{a}$ is written as $\\Sigma_{\\smallbr{$[\\ua]$}}$. Its objects are the left permutations of $\\underline{a}$. There is a decomposition\n\\begin{equation}\n\\label{pofcdecomp}\n\\Sigma_{\\frakC} \\cong \\coprod_{[\\underline{a}] \\in \\Sigma_{\\frakC}} \\Sigma_{\\smallbr{$[\\ua]$}},\n\\end{equation}\nwhere there is one coproduct summand for each orbit $[\\underline{a}]$ of a $\\mathfrak{C}$-profile. By $[\\underline{a}] \\in \\Sigma_{\\frakC}$ we mean that $[\\underline{a}]$ is an orbit in $\\Sigma_{\\frakC}$.\n\\item\nDefine the diagram category\n\\[\\symseqc(\\calm) = \\mathcal{M}^{\\sigmacop \\times \\fC},\\]\nwhose objects are called \\emph{$\\mathfrak{C}$-colored symmetric sequences}. By the decomposition \\eqref{pofcdecomp}, there is a decomposition\n\\[\\symseqc(\\calm) \\cong \n\\prod_{([\\uc];d) \\in \\sigmacop \\times \\fC} \\mathcal{M}^{\\sigmabrcop \\times \\{d\\}},\\]\nwhere $\\sigmabrcop \\times \\{d\\} \\cong \\Sigma^{\\smallop}_{\\smallbr{$[\\uc]$}}$. \n\\item\nFor $X \\in \\symseqc(\\calm)$, we write\n\\[X\\singledbrc \\in \\mathcal{M}^{\\sigmabrcop \\times \\{d\\}} \\cong \\mathcal{M}^{\\sigmabrc^{\\smallop}}\\]\nfor its $([\\uc];d)$-component. For $(\\underline{c};d) \\in \\sigmacop \\times \\fC$ (i.e., $\\underline{c}$ is a $\\mathfrak{C}$-profile and $d \\in \\mathfrak{C}$), we write\n\\[X\\duc \\in \\mathcal{M}\\]\nfor the value of $X$ at $(\\underline{c};d)$.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\n\\begin{remark}\n\\label{soneobject}\nIn the one-colored case (i.e., $\\mathfrak{C} = \\{*\\}$), for each integer $n \\geq 0$, there is a unique $\\mathfrak{C}$-profile of length $n$, usually denoted by $[n]$. We have $\\Sigma_{[n]} = \\Sigma_n$, the symmetric group $\\Sigma_n$ regarded as a one-object groupoid. So we have\n\\[\n\\Sigma_{\\frakC} = \\coprod_{n \\geq 0} \\Sigma_n = \\Sigma\n\\qquad\\text{and}\\qquad\n\\symseqc(\\calm) = \\mathcal{M}^{\\sigmacop \\times \\fC} = \\mathcal{M}^{\\Sigma^{\\smallop}}.\n\\]\nIn other words, one-colored symmetric sequences are symmetric sequences (also known as $\\Sigma$-objects and collections) in the usual sense.\n\\end{remark}\n\n\n\nFrom now on, assume that $\\mathfrak{C}$ is a fixed non-empty set of colors, unless otherwise specified. \n\n\n\\subsection{Colored Circle Product}\n\nWe will define $\\mathfrak{C}$-colored operads as monoids with respect to the $\\mathfrak{C}$-colored circle product. To define the latter, we need the following definition.\n\n\n\n\\begin{definition}[Tensored over a Category]\n\\label{def:tensorover}\nSuppose $\\mathcal{D}$ is a small groupoid, $X \\in \\mathcal{M}^{\\mathcal{D}^{\\smallop}}$, and $Y \\in \\mathcal{M}^{\\mathcal{D}}$. Define the object $X \\otimes_{\\mathcal{D}} Y \\in \\mathcal{M}$ as the colimit of the composite\n\\[\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\n\\mathcal{D} \\ar[r]^-{\\cong \\Delta} \n& \\mathcal{D}^{\\smallop} \\times \\mathcal{D} \\ar[r]^-{(X,Y)}\n& \\mathcal{M} \\times \\mathcal{M} \\ar[r]^-{\\otimes}\n& \\mathcal{M},\n}\\]\nwhere the first map is the diagonal map followed by the isomorphism $\\mathcal{D} \\otimes \\mathcal{D} \\cong \\mathcal{D}^{\\smallop} \\times \\mathcal{D}$.\n\\end{definition}\n\n\nWe will mainly use the construction $\\otimes_{\\mathcal{D}}$ when $\\mathcal{D}$ is the finite connected groupoid $\\Sigma_{\\smallbr{$[\\uc]$}}$ for some orbit $[\\underline{c}] \\in \\Sigma_{\\frakC}$.\n\n\n\\begin{convention}\nFor an object $A \\in \\mathcal{M}$, $A^{\\otimes 0}$ is taken to mean $\\mathbb{1}$, the $\\otimes$-unit in $\\mathcal{M}$.\n\\end{convention}\n\n\n\n\\begin{definition}[Colored Circle Product]\n\\label{def:colored-circle-product}\nSuppose $X,Y \\in \\symseqc(\\calm)$, $d \\in \\mathfrak{C}$, $\\underline{c} = (c_1,\\ldots,c_m) \\in \\Sigma_{\\frakC}$, and $[\\underline{b}] \\in \\Sigma_{\\frakC}$ is an orbit.\n\\begin{enumerate}\n\\item\nDefine the object\n\\[Y^{\\underline{c}} \\in \\mathcal{M}^{\\pofcop} \\cong \\prod_{[\\underline{b}] \\in \\Sigma_{\\frakC}} \\mathcal{M}^{\\sigmabrb^{\\smallop}}\\]\nas having the $[\\underline{b}]$-component\n\\[Y^{\\underline{c}}([\\underline{b}]) =\n\\coprod_{\\substack{\\{[\\underline{b}_j] \\in \\Sigma_{\\frakC}\\}_{1 \\leq j \\leq m} \\,\\mathrm{s.t.} \\\\\n[\\underline{b}] = [(\\underline{b}_1,\\ldots,\\underline{b}_m)]}} \n\\Kan^{\\sigmabrb^{\\smallop}} \n\\left[\\bigotimes_{j=1}^m Y \\cjbrbj\\right] \n\\in \\mathcal{M}^{\\sigmabrb^{\\smallop}}.\n\\]\nThe above left Kan extension is defined as\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\prod_{j=1}^m \\sigmabrbj^{\\smallop} \n\\ar[d]_-{\\mathrm{concatenation}} \n\\ar[rr]^-{\\prod Y \\binom{c_j}{-}} \n&& \\mathcal{M}^{\\times m} \\ar[d]^-{\\otimes}\n\\\\\n\\sigmabrb^{\\smallop} \\ar[rr]_-{\\Kan^{\\sigmabrb^{\\smallop}}\\left[\\otimes Y(\\vdots)\\right]}^-{\\mathrm{left ~Kan~ extension}} && \\mathcal{M}.}\\]\n\\item\nBy allowing left permutations of $\\underline{c}$ above, we obtain\n\\[\nY^{[\\underline{c}]} \\in \\mathcal{M}^{\\pofcop \\times \\Sigma_{\\smallbr{$[\\uc]$}}} \\cong \\prod_{[\\underline{b}] \\in \\Sigma_{\\frakC}} \\mathcal{M}^{\\sigmabrb^{\\smallop} \\times \\Sigma_{\\smallbr{$[\\uc]$}}}\\]\nwith components\n\\[Y^{[\\underline{c}]}([\\underline{b}]) \\in \\mathcal{M}^{\\sigmabrb^{\\smallop} \\times \\Sigma_{\\smallbr{$[\\uc]$}}}.\\]\n\\item \nThe \\emph{$\\mathfrak{C}$-colored circle product}\n\\[X \\circ Y \\in \\symseqc(\\calm)\\]\nis defined to have components\n\\[(X \\circ Y)\\singledbrb \n= \\coprod_{[\\underline{c}] \\in \\Sigma_{\\frakC}} \nX\\singledbrc \\tensorover{\\Sigma_{\\smallbr{$[\\uc]$}}} Y^{[\\underline{c}]}([\\underline{b}]) \\in \\mathcal{M}^{\\Sigma^{\\smallop}_{\\smallbr{$[\\ub]$}} \\times \\{d\\}},\n\\]\nwhere the coproduct is indexed by all the orbits in $\\Sigma_{\\frakC}$, as $d$ runs through $\\mathfrak{C}$ and $[\\underline{b}]$ runs through all the orbits in $\\Sigma_{\\frakC}$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{proposition}[$=$ 3.2.18 in \\cite{white-yau}]\n\\label{circle-product-monoidal}\nWith respect to the $\\mathfrak{C}$-colored circle product $\\circ$, $\\symseqc(\\calm)$ is a monoidal category.\n\\end{proposition}\n\n\\begin{definition}\n\\label{def:colored-operad}\nFor a set $\\mathfrak{C}$ of colors, a \\emph{$\\mathfrak{C}$-colored operad} in $\\mathcal{M}$ is a monoid \\cite{maclane} (VII.3) in the monoidal category $(\\symseqc(\\calm), \\circ)$.\n\\end{definition}\n\n\n\\subsection{Algebras over Colored Operads}\n\n\\begin{definition}\n\\label{colored-operad-algebra}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad. The category of algebras over the monad \\cite{maclane} (VI.2)\n\\[\\mathsf{O} \\circ - : \\calm^{\\fC} \\to \\calm^{\\fC}\\]\nis denoted by $\\mathsf{Alg}(\\mathsf{O})$, whose objects are called \\emph{$\\mathsf{O}$-algebras} in $\\mathcal{M}$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def:asubc}\nSuppose $A = \\{A_c\\}_{c\\in \\mathfrak{C}} \\in \\mathcal{M}^{\\mathfrak{C}}$ and $\\underline{c} = (c_1,\\ldots,c_n) \\in \\Sigma_{\\frakC}$ with orbit $[\\underline{c}]$. Define the object\n\\[\nA_{\\underline{c}} = \\bigotimes_{i=1}^n A_{c_i} = A_{c_1} \\otimes \\cdots \\otimes A_{c_n} \\in \\mathcal{M}\n\\]\nand the diagram $A_{\\smallbr{$[\\uc]$}} \\in \\mathcal{M}^{\\Sigma_{\\smallbr{$[\\uc]$}}}$ with values\n\\[A_{\\smallbr{$[\\uc]$}}(\\underline{c}') = A_{\\underline{c}'}\\]\nfor each $\\underline{c}' \\in [\\underline{c}]$. All the structure maps in the diagram $A_{\\smallbr{$[\\uc]$}}$ are given by permuting the factors in $A_{\\underline{c}}$.\n\\end{definition}\n\nThere is a free-forgetful adjoint pair\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\calm^{\\fC} \\ar@<2pt>[r]^-{\\mathsf{O} \\circ -} \n& \\mathsf{Alg}(\\mathsf{O}) \\ar@<2pt>[l]}\\]\nfor each $\\mathfrak{C}$-colored operad $\\mathsf{O}$.\n\n\\begin{proposition}[$=$ 4.2.1 in \\cite{white-yau}]\n\\label{algebra-bicomplete}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad in $\\mathcal{M}$. Then the category $\\mathsf{Alg}(\\mathsf{O})$ has all small limits and colimits, with reflexive coequalizers and filtered colimits preserved and created by the forgetful functor $\\mathsf{Alg}(\\mathsf{O}) \\to \\calm^{\\fC}$.\n\\end{proposition}\n\n\\begin{definition}\\label{lifted.right.adjoint}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is an adjunction between symmetric monoidal categories with $R$ lax symmetric monoidal. Suppose $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ a $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ a $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. Here $R$ is applied entrywise to $\\mathscr{P}$; according to \\cite{jy2} (Theorem 12.11) $R\\mathscr{P}$ is a $\\mathfrak{C}$-colored opeard in $\\mathcal{M}$. \n\\begin{enumerate}\n\\item Define an induced functor\n\\begin{equation}\\label{R.alg}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) & \\mathsf{Alg}(\\sP) \\ar[l]_-{R}}\n\\end{equation}\nas follows. For a $\\mathscr{P}$-algebra $A$, apply $R$ entrywise to $A \\in \\N^{\\mathfrak{C}}$ to obtain $RA \\in \\M^{\\mathfrak{C}}$. Then $RA$ becomes an $\\mathsf{O}$-algebra with structure map the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O}\\duc \\otimes (RA)_{\\underline{c}} \\ar[r] \\ar[d]_-{(f,\\Id)} & RA_d\\\\\nR\\mathscr{P}\\duc \\otimes (RA)_{\\underline{c}} \\ar[r]^-{R^2} & R\\bigl(\\mathscr{P}\\duc \\otimes A_{\\underline{c}}\\bigr) \\ar[u]_-{R\\lambda}}\\]\nfor $d \\in \\mathfrak{C}$ and $\\underline{c} \\in \\Sigma_{\\frakC}$. The map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathscr{P}\\duc \\otimes A_{\\underline{c}} \\ar[r]^-{\\lambda} & A_d}\\]\nis a $\\mathscr{P}$-algebra structure map of $A$, and $R^2$ is a repeated application of the lax monoidal structure map of $R$. Note that if $R = \\Id$, then the functor $R : \\mathsf{Alg}(\\sP) \\to \\mathsf{Alg}(\\mathsf{O})$ is the restriction along $f$.\n\\item\nBy the Adjoint Lifting Theorem \\cite{borceux} (4.5.6), the functor $R$ \\eqref{R.alg} admits a left adjoint $\\overline{L} : \\mathsf{Alg}(\\mathsf{O}) \\to \\mathsf{Alg}(\\sP)$, which is in general \\emph{not} $L$ entrywise. The original adjoint pair $L \\dashv R$ is related to the lifted adjoint pair $\\overline{L} \\dashv R$ as follows:\n\\begin{equation}\\label{lbar.ocomp.diagram}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{\\overline{L}} \\ar@<2.5pt>[d]^-{U}\n& \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{R} \\ar@<2.5pt>[d]^-{U} \\\\\n\\M^{\\mathfrak{C}} \\ar@<2.5pt>[r]^-{L} \\ar@<2.5pt>[u]^-{\\mathsf{O} \\circ -} \n& \\N^{\\mathfrak{C}} \\ar@<2.5pt>[l]^-{R} \\ar@<2.5pt>[u]^-{\\mathscr{P} \\circ -}}\n\\end{equation}\nOn each side, the vertical arrows form a free-forgetful adjunction. The right adjoint diagram is commutative, i.e., $UR = RU$. By uniqueness the left adjoin diagram is also commutative, i.e., \n\\begin{equation}\\label{lbar.ocomp}\n\\overline{L}\\left(\\mathsf{O} \\circ X\\right) = \\mathscr{P} \\circ (LX)\n\\end{equation}\nin which $LX$ means $L$ is applied entrywise to $X \\in \\M^{\\mathfrak{C}}$.\n\\item For each $\\mathsf{O}$-algebra $A$, the unit $A \\to R\\overline{L} A \\in \\mathsf{Alg}(\\mathsf{O})$ of the adjunction, when regarded as a map in $\\M^{\\mathfrak{C}}$, has an entrywise adjoint\n\\begin{equation}\\label{comparison.map}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{LUA \\ar[r]^-{\\chi_A} & U\\overline{L} A \\in \\N^{\\mathfrak{C}}}\n\\end{equation}\ncalled the \\emph{comparison map}. We will usually write the comparison map as $LA \\to \\overline{L} A$, omitting the forgetful functors from the notation. \n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{example}\\label{comparison.initial}\nFor the initial $\\mathsf{O}$-algebra $\\varnothing_{\\sO} = \\left\\{\\mathsf{O}\\singledempty\\right\\}_{d \\in \\mathfrak{C}}$, we have\n\\[\\overline{L}\\varnothing_{\\sO} = \\varnothing_{\\sP} = \\left\\{\\mathscr{P}\\singledempty\\right\\}_{d \\in \\mathfrak{C}}\\]\nbecause $\\overline{L}$ is a left adjoint. In this case, the comparison map $\\chi_{\\varnothing_{\\sO}}$ is entrywise the adjoint of $f$,\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{(L\\varnothing_{\\sO})_d = L\\mathsf{O}\\singledempty \\ar[r]^-{\\overline{f}} & \\mathscr{P}\\singledempty = (\\overline{L}\\varnothing_{\\sO})_d \\in \\mathcal{N}}\\]\nfor $d \\in \\mathfrak{C}$.\n\\end{example}\n\n\n\\begin{example}\\label{comparison.adjoint}\nSuppose $A \\in \\mathsf{Alg}(\\mathsf{O})$, $B \\in \\mathsf{Alg}(\\sP)$, and $\\varphi : \\overline{L} A \\to B \\in \\mathsf{Alg}(\\sP)$. Then the adjoint of $\\varphi$ is a map $\\overline{\\varphi} : A \\to RB \\in \\mathsf{Alg}(\\mathsf{O})$. Considering $\\overline{\\varphi}$ as a map in $\\M^{\\mathfrak{C}}$, its entrywise adjoint is the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{LA \\ar[r]^-{\\chi_A} & \\overline{L} A \\ar[r]^-{\\varphi} & B \\in \\N^{\\mathfrak{C}}.}\\]\nThis example will be important in the proof of Theorem \\ref{main.theorem}. In the cases of monoids and of $1$-colored non-symmetric operads, the comparison map appeared in \\cite{ss03} (5.1) and \\cite{muro14} (7-2), respectively.\n\n\\end{example}\n\n\n\n\\subsection{Filtration for Pushouts of Colored Operadic Algebras}\n\n\n\n\\begin{definition}\nSuppose $X \\in \\symseqc(\\calm)$, $d \\in \\mathfrak{C}$, and $[\\underline{a}], [\\underline{b}], [\\underline{c}]$ are orbits in $\\pofc$. Define the diagram\n\\[X \\singledbrabrc \\in \\mathcal{M}^{\\sigmabra^{\\smallop} \\times \\sigmabrc^{\\smallop} \\times \\{d\\}}\\]\nas having the objects\n\\[X \\singledbrabrc(\\underline{a}'; \\underline{c}') \n= X \\singledaprimecprime \\in \\mathcal{M}\\]\nfor $\\underline{a}' \\in [\\underline{a}]$ and $\\underline{c}' \\in [\\underline{c}]$ and the structure maps of $X$.\n\\end{definition}\n\n\n\\begin{definition}[$\\mathsf{O}_A$ for $\\mathsf{O}$-algebras]\n\\label{oaalgebra}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad and $A \\in \\mathsf{Alg}(\\mathsf{O})$. Define $\\mathsf{O}_A \\in \\symseqc(\\calm)$ as follows. For $d \\in \\mathfrak{C}$ and orbit $[\\underline{c}] \\in \\pofc$, define the component\n\\[\\mathsf{O}_A\\singledbrc \\in \\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}\\]\nas the reflexive coequalizer of the diagram\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\coprod\\limits_{[\\underline{a}] \\in \\pofc} \n\\mathsf{O}\\singledbrabrc \\tensorover{\\Sigma_{\\smallbr{$[\\ua]$}}} (\\mathsf{O} \\circ A)_{\\smallbr{$[\\ua]$}}\n\\ar@<-3pt>[r]_-{d_0} \\ar@<3pt>[r]^-{d_1}\n& \\coprod\\limits_{[\\underline{a}] \\in \\pofc} \n\\mathsf{O}\\singledbrabrc \\tensorover{\\Sigma_{\\smallbr{$[\\ua]$}}} A_{\\smallbr{$[\\ua]$}}\n\\ar@\/_1.5pc\/[l]}\n\\]\nwith\n\\begin{itemize}\n\\item\nthe coequalizer taken in $\\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}$, \n\\item\n$d_0$ induced by the operadic composition on $\\mathsf{O}$,\n\\item\n$d_1$ induced by the $\\mathsf{O}$-algebra action on $A$, and \n\\item\nthe common section induced by $A \\cong \\mathscr{I} \\circ A \\to \\mathsf{O} \\circ A$, where $\\mathscr{I}$ is the unit for the $\\mathfrak{C}$-colored circle product.\n\\end{itemize}\n\\end{definition}\n\n\n\\begin{proposition}[$=$ 5.1.1 in \\cite{white-yau}]\n\\label{o-sub-empty}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad, and $\\varnothing$ is the initial $\\mathsf{O}$-algebra. Then there is an isomorphism\n\\[\\mathsf{O}_{\\varnothing} \\cong \\mathsf{O}\\]\nin $\\symseqc(\\calm)$.\n\\end{proposition}\n\n\\begin{proposition}[$=$ 5.1.3 in \\cite{white-yau}]\n\\label{osuba-empty}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad, $A \\in \\mathsf{Alg}(\\mathsf{O})$, and $d \\in \\mathfrak{C}$. Then there is a natural isomorphism\n\\[\\mathsf{O}_A\\singledempty \\cong A_d\\]\nin $\\mathcal{M}$.\n\\end{proposition}\n\n\\begin{lemma}[$=$ 5.3.1 in \\cite{white-yau}]\n\\label{osubdc}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad, $d \\in \\mathfrak{C}$, and $[\\underline{c}] \\in \\pofc$. Then the functor\n\\[\\mathsf{O}_{(-)}\\singledbrc : \\mathsf{Alg}(\\mathsf{O}) \\to \\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}\\]\npreserves reflexive coequalizers and filtered colimits.\n\\end{lemma}\n\n\n\\begin{definition}\n\\label{def:q-construction}\nSuppose $i : X \\to Y \\in \\calm^{\\fC}$ is concentrated at a single color $c \\in \\mathfrak{C}$ (so $X_b = Y_b = \\varnothing$ whenever $b \\not= c$) and $t \\geq 1$. For $0 \\leq q \\leq t$, define\n\\[\nQ_q^t = Q_q^{[tc]} \\in \\mathcal{M}^{\\Sigma_t}\n\\]\nas follows.\n\\begin{itemize}\n\\item\n$Q^{[tc]}_0 = X^{\\otimes t}$ and $Q^{[tc]}_t = Y^{\\otimes t}$.\n\\item\nFor $0 < q < t$ there is a pushout in $\\mathcal{M}^{\\Sigma_t}$:\n\\[\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\Sigma_t \\dotover{\\Sigma_{t-q} \\times \\Sigma_q} \n\\left[ X^{\\otimes (t-q)} \n\\otimes Q_{q-1}^{[qc]}\\right] \n\\ar[d]_-{(\\id,i_*)} \\ar[r]\n& Q^{[tc]}_{q-1} \\ar[d] \\\\\n\\Sigma_t \\dotover{\\Sigma_{t-q} \\times \\Sigma_q} \n\\left[ X^{\\otimes (t-q)} \\otimes \nY^{\\otimes q}\\right] \\ar[r] & Q^{[tc]}_q.}\n\\]\n\\end{itemize}\n\\end{definition}\n\n\n\n\n\\begin{proposition}[$=$ 4.3.16 in \\cite{white-yau}]\n\\label{free-pushout-filtration}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad, $A \\in \\mathsf{Alg}(\\mathsf{O})$, $i : X \\to Y \\in \\calm^{\\fC}$ is concentrated at a single color $c \\in \\mathfrak{C}$, and\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O} \\circ X \\ar[d]_-{\\id \\circ i} \\ar[r]^-{f} \n& A \\ar[d]^-{j} \\\\\n\\mathsf{O} \\circ Y \\ar[r] & A_{\\infty}}\n\\]\nis a pushout in $\\mathsf{Alg}(\\mathsf{O})$. Then the map $j \\in \\M^{\\mathfrak{C}}$ factors naturally into a countable composition\n\\[\\nicearrow\\xymatrix@R+10pt{A = A_0 \\ar[r]^-{j_1} & A_1 \\ar[r]^-{j_2} & A_2 \\ar[r]^-{j_3} & \\cdots \\ar[r] & A_{\\infty} \\in \\M^{\\mathfrak{C}}}\\]\nsuch that, for each color $d \\in \\mathfrak{C}$ and $t \\geq 1$, the $d$-colored entry of $j_t$ is inductively defined as the pushout in $\\mathcal{M}$\n\\begin{equation}\n\\label{one-colored-jt-pushout}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O}_A \\singledbrtc \\tensorover{\\Sigma_t} Q^{t}_{t-1}\n\\ar[d]_-{\\id \\tensorover{\\Sigma_t} i^{\\mathbin\\square t}} \\ar[r]^-{f^{t-1}_*} \n& (A_{t-1})_d \\ar[d]^-{j_t} \\\\\n\\mathsf{O}_A \\singledbrtc \\tensorover{\\Sigma_t} Y^{\\otimes t} \\ar[r]_-{\\xi_{t}} \n& (A_t)_d}\n\\end{equation}\nwith $f^{t-1}_*$ induced by $f$ and $tc = (c,\\ldots,c)$ with $t$ copies of $c$.\n\\end{proposition}\n\n\n\\begin{proposition}[$=$ 5.3.2 in \\cite{white-yau}]\n\\label{o-ainfinity}\nSuppose $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad, $A \\in \\mathsf{Alg}(\\mathsf{O})$, $i : X \\to Y \\in \\calm^{\\fC}$ is concentrated in one color $b \\in \\mathfrak{C}$, and\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O} \\circ X \\ar[d]_-{\\id \\circ i} \\ar[r]^-{f} & A \\ar[d]^-{j}\n\\\\ \\mathsf{O} \\circ Y \\ar[r] & A_{\\infty}}\n\\]\nis a pushout in $\\mathsf{Alg}(\\mathsf{O})$. Suppose $d \\in \\mathfrak{C}$ and $[\\underline{c}] \\in \\pofc$. Then the map $\\mathsf{O}_{j} \\in \\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}$ factors naturally into a countable composition\n\\[\\nicearrow\\xymatrix@R+10pt{\\mathsf{O}_{A}\\singledbrc = \\mathsf{O}_{A}^0 \\singledbrc \\ar[r]^-{j_1}\n& \\mathsf{O}_{A}^1 \\singledbrc \\ar[r]^-{j_2}\n& \\mathsf{O}_{A}^2 \\singledbrc \\ar[r]^-{j_3}\n& \\cdots \\ar[r] & \\mathsf{O}_{A_{\\infty}} \\singledbrc}\\]\nin $\\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}$ in which each $j_t$ for $t \\geq 1$ is defined inductively as the pushout\n\\begin{equation}\n\\label{osubjt}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\n\\mathsf{O}_{A} \\singledbrtbbrc \\tensorover{\\Sigma_t} Q^{t}_{t-1} \n\\ar[d]_-{\\id \\tensorover{\\Sigma_t} i^{\\mathbin\\square t}} \\ar[r]^-{f_*}\n& \\mathsf{O}_{A}^{t-1}\\singledbrc \\ar[d]^-{j_t}\n\\\\ \\mathsf{O}_{A} \\singledbrtbbrc \\tensorover{\\Sigma_t} Y^{\\otimes t}\n\\ar[r]^-{\\xi_t} & \\mathsf{O}_{A}^{t} \\singledbrc}\n\\end{equation}\nin $ \\mathcal{M}^{\\sigmabrc^{\\smallop} \\times \\{d\\}}$, where $tb = (b,\\ldots,b)$ with $t$ copies of $b$.\n\\end{proposition}\n\n\n\n\\subsection{Model Structure on Algebras over Entrywise Cofibrant Operads}\n\n\\begin{definition}\n\\label{def:club}\nSuppose $\\mathcal{M}$ is a symmetric monoidal category and is a model category. Define the following condition.\n\\begin{quote}\n$(\\clubsuit)$ : For each $n \\geq 1$ and $X \\in \\mathcal{M}^{\\Sigma^{\\smallop}_n}$ that is cofibrant in $\\mathcal{M}$, the function\n\\[X \\tensorover{\\Sigma_n} (-)^{\\mathbin\\square n} : \\mathcal{M} \\to \\mathcal{M}\\]\npreserves cofibrations and trivial cofibrations.\n\\end{quote}\nThe condition $(\\clubsuit)$ for cofibrations will be referred to as $(\\clubsuit)_{\\cof}$, and the condition for trivial cofibrations as $(\\clubsuit)_{\\acof}$. So $(\\clubsuit) = (\\clubsuit)_{\\cof} + (\\clubsuit)_{\\acof}$.\n\\end{definition}\n\n\\begin{example}\nAs in Example \\ref{star.examples}, condition $(\\clubsuit)$ holds whenever cofibrancy in $\\M^{\\Sigma_n}$ coincides with cofibrancy in $\\mathcal{M}$. In particular, it holds in:\n\\begin{itemize}\n\\item simplicial modules over a characteristic $0$ field;\n\\item chain complexes, bounded or unbounded, over a characteristic $0$ field;\n\\item reduced rational simplicial Lie algebras;\n\\item reduced rational dg Lie algebras.\n\\end{itemize}\n\\end{example}\n\n\\begin{theorem}[$=$ 6.2.3 in \\cite{white-yau}]\n\\label{theorem623}\nSuppose \n$\\mathcal{M}$ is a cofibrantly generated monoidal model category satisfying $(\\clubsuit)$. Then for each entrywise cofibrant $\\mathfrak{C}$-colored operad $\\mathsf{O}$ in $\\mathcal{M}$, the category $\\mathsf{Alg}(\\mathsf{O})$ admits a cofibrantly generated \\textbf{semi}-model structure over $\\M^{\\mathfrak{C}}$ such that the weak equivalences and fibrations are created in $\\mathcal{M}$. Moreover:\n\\begin{enumerate}\n\\item\nIf $j : A \\to B \\in \\mathsf{Alg}(\\mathsf{O})$ is a cofibration with $A$ cofibrant in $\\mathsf{Alg}(\\mathsf{O})$, then the underlying map of $j$ is entrywise a cofibration.\n\\item\nEvery cofibrant $\\mathsf{O}$-algebra is entrywise cofibrant in $\\mathcal{M}$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{lemma}[$=$ 6.2.4 in \\cite{white-yau}]\n\\label{middle-row-lemma}\nSuppose $\\mathcal{M}$ is a symmetric monoidal closed category and is a model category satisfying $(\\clubsuit)_{\\cof}$, and $\\mathsf{O}$ is a $\\mathfrak{C}$-colored operad in $\\mathcal{M}$.\n\\begin{enumerate}\n\\item\nSuppose $j : A \\to B \\in \\mathsf{Alg}(\\mathsf{O})$ is a relative $(\\mathsf{O} \\circ \\mathcal{M}_{{\\scalebox{.5}{$\\mathrm{cof}$}}})$-cell complex, i.e., a retract of a transfinite composition of pushouts of maps in $\\mathsf{O} \\circ \\mathcal{M}_{{\\scalebox{.5}{$\\mathrm{cof}$}}}$. Suppose also that $\\mathsf{O}_A$ is entrywise cofibrant in $\\mathcal{M}$. Then $\\mathsf{O}_A \\to \\mathsf{O}_B$ is entrywise a cofibration in $\\mathcal{M}$.\n\\item\nSuppose $\\mathsf{O}$ is entrywise cofibrant in $\\mathcal{M}$, and suppose $A$ is an $(\\mathsf{O} \\circ \\mathcal{M}_{{\\scalebox{.5}{$\\mathrm{cof}$}}})$-cell complex, i.e., $\\varnothing \\to A \\in \\mathsf{Alg}(\\mathsf{O})$ is a relative $(\\mathsf{O} \\circ \\mathcal{M}_{{\\scalebox{.5}{$\\mathrm{cof}$}}})$-cell complex. Then $\\mathsf{O}_A$ is entrywise cofibrant in $\\mathcal{M}$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{definition}\\label{def:nice.qeq}\nA \\emph{nice Quillen equivalence} $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak symmetric monoidal Quillen equivalence (Def. \\ref{def:weak.symmetric.monoidal}) between cofibrantly generated monoidal model categories such that the following conditions hold.\n\\begin{enumerate}\n\\item $(\\filledstar)$ (Def. \\ref{def:star}), $(\\#)$ (Def. \\ref{def:sharp}), and $(\\clubsuit)$ (Def. \\ref{def:club}) hold in $\\mathcal{M}$ and $\\mathcal{N}$. \n\\item $\\mathcal{N}$ satisfies $(\\medstar)$ (Def. \\ref{def:star}).\n\\item Every generating cofibration in $\\mathcal{M}$ has cofibrant domain.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{example}\nThe following are examples of nice Quillen equivalences:\n\\begin{enumerate}\n\\item The Dold-Kan correspondence between simplicial $k$-modules and non-negatively graded chain complexes of $k$-modules for a characteristic $0$ field $k$.\n\\item The adjunction between reduced rational simplicial Lie algebras and reduced rational dg Lie algebras \\cite{quillen} (p.211).\n\\end{enumerate}\n\\end{example}\n\n\\section{Main Result} \\label{sec:main}\n\n\\subsection{Key Step}\n\nBy Proposition \\ref{osuba-empty}, for each color $d \\in \\mathfrak{C}$, there are natural isomorphisms\n\\[L\\mathsf{O}_A\\dnothing \\cong LA_d \\qquad\\text{and}\\qquad \\mathscr{P}_{\\overline{L} A}\\dnothing = (\\overline{L} A)_d.\\]\n A key part of the proof of Theorem \\ref{main.theorem} is the following observation.\n\n\\begin{proposition}\n\\label{loa.to.pla}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}).\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. The entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise weak equivalence in $\\mathcal{N}$. \n\\end{enumerate}\nSuppose $A$ is a cofibrant $\\mathsf{O}$-algebra. Then the map $f : \\mathsf{O} \\to R\\mathscr{P}$ induces a natural entrywise weak equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O}_A \\ar[r]^-{f_{\\infty}} & \\mathscr{P}_{\\overline{L} A} \\in \\symseqc(\\N)}\\]\nwhose value at $\\dnothing$ is the comparison map $\\chi_A : LA \\to \\overline{L} A$ \\eqref{comparison.map} evaluated at $d$ for each $d \\in \\mathfrak{C}$.\n\\end{proposition}\n\nBelow, we will show that, if $\\mathsf{O}$ and $\\mathscr{P}$ are $\\Sigma$-cofibrant, then we can weaken assumption (1) above to only requiring that $(L,R)$ be a weak symmetric monoidal Quillen equivalence and that the domains of the generating cofibrations in $\\mathcal{M}$ be cofibrant.\n\n\\begin{proof}\nThe generating cofibrations in $\\mathsf{Alg}(\\mathsf{O})$ have the form $\\mathsf{O} \\circ i$ for some generating cofibration $i$ in $\\mathcal{M}$, regarded as a map in $\\M^{\\mathfrak{C}}$ concentrated in a single color. Each cofibration in $\\mathsf{Alg}(\\mathsf{O})$ is a retract of a transfinite composition of pushouts of generating cofibrations. By a retract argument we may assume that the map $\\varnothing_{\\sO} \\to A \\in \\mathsf{Alg}(\\mathsf{O})$ is a transfinite composition\n\\begin{equation}\\label{a.cell.complex}\n\\nicearrow\\xymatrix@R+10pt{\\varnothing_{\\sO} = A^0 \\ar[r] & A^1 \\ar[r] & A^2 \\ar[r] & \\cdots \\ar[r] & A \\in \\mathsf{Alg}(\\mathsf{O})}\n\\end{equation}\nsuch that, for each $t \\geq 1$, the map $A^{t-1} \\to A^t$ is a pushout\n\\begin{equation}\\label{at.pushout}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O} \\circ X \\ar[d]_-{\\Id \\circ i} \\ar[r] & A^{t-1} \\ar[d]\\\\\n\\mathsf{O} \\circ Y \\ar[r] & A^t}\n\\end{equation}\nin $\\mathsf{Alg}(\\mathsf{O})$ for some generating cofibration $i : X \\to Y$ in $\\mathcal{M}$, regarded as a map in $\\M^{\\mathfrak{C}}$ concentrated in a single color, say, $c \\in \\mathfrak{C}$. Both the map $i$ and the color $c$ depend on the index $t$. Note that, since the initial $\\mathsf{O}$-algebra is cofibrant and that $A^{t-1} \\to A^t \\in \\mathsf{Alg}(\\mathsf{O})$ is a cofibration, all the $A^t$ are cofibrant $\\mathsf{O}$-algebras. Moreover, by assumption on $\\mathcal{M}$, the generating cofibration $i$ is a cofibration between cofibrant objects.\n\nWe apply $\\mathsf{O}_{(-)}$ (Def. \\ref{oaalgebra}) to the transfinite composition \\eqref{a.cell.complex} and use Proposition \\ref{o-sub-empty} on $A^0$ and Lemma \\ref{osubdc} on the colimit. We obtain the transfinite composition\n\\begin{equation}\\label{osuba.cell}\n\\nicearrow\\xymatrix@R+10pt{\\mathsf{O} \\cong \\mathsf{O}_{A^0} \\ar[r] & \\mathsf{O}_{A^1} \\ar[r] & \\mathsf{O}_{A^2} \\ar[r] & \\cdots \\ar[r] & \\mathsf{O}_{A} \\in \\symseqc(\\calm).}\n\\end{equation}\nSince $\\mathsf{O}$ is entrywise cofibrant and since all the $A^t$ are cofibrant $\\mathsf{O}$-algebras, by Lemma \\ref{middle-row-lemma}, in \\eqref{osuba.cell} every map is an entrywise cofibration between entrywise cofibrant objects in $\\mathcal{M}$. Applying the left Quillen equivalence \\cite{hirschhorn} (11.6.5(2))\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L : \\symseqc(\\calm) \\ar[r] & \\symseqc(\\N)}\\]\nto \\eqref{osuba.cell}, we obtain the transfinite composition\n\\begin{equation}\\label{losuba.cell}\n\\nicearrow\\xymatrix@R+10pt{L\\mathsf{O} \\cong L\\mathsf{O}_{A^0} \\ar[r] & L\\mathsf{O}_{A^1} \\ar[r] & L\\mathsf{O}_{A^2} \\ar[r] & \\cdots \\ar[r] & L\\mathsf{O}_{A} \\in \\symseqc(\\N)}\n\\end{equation}\nof entrywise cofibrations between entrywise cofibrant objects in $\\mathcal{N}$.\n\nNext we apply the left adjoint $\\overline{L} : \\mathsf{Alg}(\\mathsf{O}) \\to \\mathsf{Alg}(\\sP)$ to the transfinite composition \\eqref{a.cell.complex} and the pushouts \\eqref{at.pushout}. We obtain the transfinite composition\n\\begin{equation}\\label{lbar.star}\n\\nicearrow\\xymatrix@R+10pt{\\varnothing_{\\sP} = \\overline{L}\\varnothing_{\\sO} = \\overline{L} A^0 \\ar[r] & \\overline{L} A^1 \\ar[r] & \\overline{L} A^2 \\ar[r] & \\cdots \\ar[r] & \\overline{L} A \\in \\mathsf{Alg}(\\sP)}\n\\end{equation}\nsuch that, for each $t \\geq 1$, the map $\\overline{L} A^{t-1} \\to \\overline{L} A^t$ is a pushout\n\\begin{equation}\\label{lbar.at.pushout}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathscr{P} \\circ (LX) = \\overline{L}(\\mathsf{O} \\circ X) \\ar[d]_-{\\Id \\circ Li}^-{=\\, \\overline{L}(\\Id \\circ i)} \\ar[r] & \\overline{L} A^{t-1} \\ar[d]\\\\ \\mathscr{P} \\circ (LY) = \\overline{L}(\\mathsf{O} \\circ Y) \\ar[r] & \\overline{L} A^t}\n\\end{equation}\nin $\\mathsf{Alg}(\\sP)$. The equalities on the left come from \\eqref{lbar.ocomp}. Since $\\mathscr{P}$ is also entrywise cofibrant, similar to the paragraph containing \\eqref{osuba.cell}, applying $\\mathscr{P}_{(-)}$ to the transfinite composition \\eqref{lbar.star} yields the transfinite composition\n\\begin{equation}\\label{psub.lbar.star}\n\\nicearrow\\xymatrix@R+10pt{\\mathscr{P} \\cong \\mathscr{P}_{\\overline{L} A^0} \\ar[r] & \\mathscr{P}_{\\overline{L} A^1} \\ar[r] & \\mathscr{P}_{\\overline{L} A^2} \\ar[r] & \\cdots \\ar[r] & \\mathscr{P}_{\\overline{L} A} \\in \\symseqc(\\N)}\n\\end{equation}\nof entrywise cofibrations between entrywise cofibrant objects in $\\mathcal{N}$.\n\nConsider the commutative ladder diagram from \\eqref{losuba.cell} to \\eqref{psub.lbar.star},\n\\begin{equation}\\label{lostar.to.plbarstar}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O} \\cong L\\mathsf{O}_{A^0} \\ar[d]_-{f_0 \\,\\overset{\\mathrm{def}}{=\\joinrel=}\\, \\overline{f}}\\ar[r] & L\\mathsf{O}_{A^1} \\ar[d]_-{f_1} \\ar[r] & L\\mathsf{O}_{A^2} \\ar[d]_-{f_2} \\ar[r] & \\cdots \\ar[r] & L\\mathsf{O}_{A} \\ar[d]_-{\\colim\\, f_t \\,=}^-{ f_{\\infty}}\\\\\n \\mathscr{P} \\cong \\mathscr{P}_{\\overline{L} A^0} \\ar[r] & \\mathscr{P}_{\\overline{L} A^1} \\ar[r] & \\mathscr{P}_{\\overline{L} A^2} \\ar[r] & \\cdots \\ar[r] & \\mathscr{P}_{\\overline{L} A}}\n\\end{equation}\nin $\\symseqc(\\N)$, in which $f_0$ is defined to be $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$. Our goal is to show that the colimit $f_{\\infty}$ is a weak equivalence, i.e., entrywise weak equivalence in $\\mathcal{N}$. By \\cite{hirschhorn} (17.9.1) it suffices to show by induction that each vertical map $f_t$ for $t \\geq 0$ is a weak equivalence. The initial map $f_0 = \\overline{f}$ is a weak equivalence by assumption.\n\nFor the induction step, suppose $t \\geq 1$ and that the map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O}_{A^{t-1}} \\ar[r]^-{f_{t-1}} & \\mathscr{P}_{\\overline{L} A^{t-1}} \\in \\symseqc(\\N)}\\]\nis a weak equivalence. We want to show that $f_t$ is a weak equivalence. The map $f_t$ is inductively defined as follows. Pick $d \\in \\mathfrak{C}$ and $\\underline{b} \\in \\prof(\\fC)$. Applying Proposition \\ref{o-ainfinity} to the pushout \\eqref{at.pushout}, we see that the map\n\\[\\mathsf{O}_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} \\to \\mathsf{O}_{A^t}\\smallprof{$\\binom{d}{\\brb}$}\\]\nis a countable composition\n\\begin{equation}\\label{osubstar.t}\n\\nicearrow\\xymatrix@R+10pt{\\mathsf{O}_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} = \\mathsf{O}_{A^{t-1}}^0\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathsf{O}_{A^{t-1}}^1\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathsf{O}_{A^{t-1}}^2\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\cdots \\ar[r] & \\mathsf{O}_{A^t}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\M^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$ in which, for each $r \\geq 1$, the $r$th map is the pushout\n\\begin{equation}\\label{star.r.t}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(i) \\ar[r] \\ar[d]_{\\Id \\underset{\\Sigma_r}{\\otimes} i^{\\mathbin\\square r}} & \\mathsf{O}_{A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]\\\\\n\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Y^{\\otimes r} \\ar[r] & \\mathsf{O}^r_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\M^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$. In the previous diagram, $rc = (c, \\ldots, c)$ with $r$ copies of the color $c \\in \\mathfrak{C}$, and the top horizontal map is naturally induced by the map $\\mathsf{O} \\circ X \\to A^{t-1}$. We already observed above that every $\\mathsf{O}_{A^k}$ is entrywise cofibrant in $\\mathcal{M}$. Since $i : X \\to Y$ is a cofibration in $\\mathcal{M}$, by $(\\clubsuit)_{\\cof}$ (Def. \\ref{def:club}) the left vertical map in \\eqref{star.r.t} is entrywise a cofibration in $\\mathcal{M}$, hence so is the right vertical map. An induction shows that in \\eqref{osubstar.t} every map is an entrywise cofibration between entrywise cofibrant objects in $\\mathcal{M}$. \n\nFurthermore, since $i : X \\to Y$ is a cofibration between cofibrant objects, the iterated pushout product $i^{\\mathbin\\square r} : Q^r_{r-1}(i) \\to Y^{\\otimes r}$ is also a cofibration between cofibrant objects in $\\mathcal{M}$ by the pushout product axiom. See, for example, \\cite{harper-jpaa} (proof of 7.19) for an explicit iterated construction of $Q^r_{r-1}$. As every $\\mathsf{O}_{A^k}$ is entrywise cofibrant, both objects\n\\[\\mathsf{O}_{A^{t-1}}\\drcub \\otimes Q^r_{r-1}(i) \\qquad\\text{and}\\qquad \\mathsf{O}_{A^{t-1}}\\drcub \\otimes Y^{\\otimes r}\\]\nare entrywise cofibrant in $\\mathcal{M}$ by the pushout product axiom. Condition $(\\filledstar)$ (Def. \\ref{def:star}) in $\\mathcal{M}$ implies that, after taking $\\Sigma_r$-coinvariants, both objects on the left side of \\eqref{star.r.t} are entrywise cofibrant in $\\mathcal{M}$.\n\nNow we apply the left Quillen equivalence \\cite{hirschhorn} (11.6.5(2))\n\\[L : \\M^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}} \\to \\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}\\]\nto the countable composition \\eqref{osubstar.t} and the pushouts \\eqref{star.r.t}. We obtain the countable composition\n\\begin{equation}\\label{losubstar.t}\n\\nicearrow\\xymatrix@R+10pt{L\\mathsf{O}_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} = L\\mathsf{O}_{A^{t-1}}^0\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & L\\mathsf{O}_{A^{t-1}}^1\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & L\\mathsf{O}_{A^{t-1}}^2\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\cdots \\ar[r] & L\\mathsf{O}_{A^t}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$ of entrywise cofibrations between entrywise cofibrant objects. For each $r \\geq 1$, the $r$th map is the pushout\n\\begin{equation}\\label{lstar.r.t}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(i)\\Bigr] \\ar[r] \\ar[d]_{L\\bigl(\\Id \\underset{\\Sigma_r}{\\otimes} i^{\\mathbin\\square r}\\bigr)} & L\\mathsf{O}_{A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]\\\\\nL\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Y^{\\otimes r}\\Bigr] \\ar[r] & L\\mathsf{O}^r_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$ with both vertical maps entrywise cofibrations between entrywise cofibrant objects. \n\nNext, applying Proposition \\ref{o-ainfinity} to the pushout \\eqref{lbar.at.pushout}, we see that the map\n\\[\\mathscr{P}_{\\overline{L} A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} \\to \\mathscr{P}_{\\overline{L} A^t}\\smallprof{$\\binom{d}{\\brb}$}\\]\nis a countable composition\n\\begin{equation}\\label{psub.lbarstar.t}\n\\nicearrow\\xymatrix@R+10pt{\\mathscr{P}_{\\overline{L} A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} = \\mathscr{P}_{\\overline{L} A^{t-1}}^0\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathscr{P}_{\\overline{L} A^{t-1}}^1\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathscr{P}_{\\overline{L} A^{t-1}}^2\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\cdots \\ar[r] & \\mathscr{P}_{\\overline{L} A^t}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$ in which, for each $r \\geq 1$, the $r$th map is the pushout\n\\begin{equation}\\label{lbarstar.r.t}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(Li) \\ar[r] \\ar[d]_{\\Id \\underset{\\Sigma_r}{\\otimes} (Li)^{\\mathbin\\square r}} & \\mathscr{P}_{\\overline{L} A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]\\\\\n\\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} (LY)^{\\otimes r} \\ar[r] & \\mathscr{P}^r_{\\overline{L} A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$. We now argue as in the two paragraphs before \\eqref{losubstar.t} and use conditions $(\\clubsuit)_{\\cof}$ and $(\\filledstar)$ in $\\mathcal{N}$. We then see that in \\eqref{psub.lbarstar.t} every map is an entrywise cofibration between entrywise cofibrant objects. Moreover, both vertical maps in \\eqref{lbarstar.r.t} are entrywise cofibrations between entrywise cofibrant objects.\n\nConsider the commutative ladder diagram from \\eqref{losubstar.t} to \\eqref{psub.lbarstar.t},\n\\begin{equation}\\label{lostart.to.plbarstart}\n\\nicearrow\\xymatrix@R+10pt{L\\mathsf{O}_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} = L\\mathsf{O}_{A^{t-1}}^0\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]_-{f^0_{t-1}}^-{\\,\\overset{\\mathrm{def}}{=\\joinrel=}\\, f_{t-1}}\\ar[r] & L\\mathsf{O}_{A^{t-1}}^1\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]_-{f_{t-1}^1} \\ar[r] & L\\mathsf{O}_{A^{t-1}}^2\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]_-{f_{t-1}^2} \\ar[r] & \\cdots \\ar[r] & L\\mathsf{O}_{A^t}\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]_-{\\colim_r\\, f_{t-1}^r \\,=}^-{f_t}\\\\\n \\mathscr{P}_{\\overline{L} A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} = \\mathscr{P}_{\\overline{L} A^{t-1}}^0\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathscr{P}_{\\overline{L} A^{t-1}}^1\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\mathscr{P}_{\\overline{L} A^{t-1}}^2\\smallprof{$\\binom{d}{\\brb}$} \\ar[r] & \\cdots \\ar[r] & \\mathscr{P}_{\\overline{L} A^t}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$. By \\cite{hirschhorn} (15.10.12(1)), to show that the colimit $f_t$ is a weak equivalence, it suffices to show that each vertical map $f_{t-1}^r$ for $r \\geq 0$ is a weak equivalence. The initial map $f_{t-1}^0$ is defined as $f_{t-1}$, which is a weak equivalence.\n\nFor the induction step, suppose $r \\geq 1$ and that $f_{t-1}^{r-1}$ is a weak equivalence. We want to show that $f_{t-1}^r$ is a weak equivalence. Consider the naturally induced commutative cube from \\eqref{lstar.r.t} (the back face below) to \\eqref{lbarstar.r.t} (the front face),\n\\begin{equation}\\label{lstarrt.to.lbarstarrt}\n\\nicearrow\\xymatrix@C+10pt@R+10pt@C-1.3cm{L\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(i)\\Bigr] \\ar[rr] \\ar[dd]_{L\\bigl(\\Id \\underset{\\Sigma_r}{\\otimes} i^{\\mathbin\\square r}\\bigr)} \\ar@(d,dl)[dr]^-{\\alpha} && L\\mathsf{O}_{A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar'[d][dd] \\ar[dr]^-{f_{t-1}^{r-1}}_-{\\sim} &\\\\\n& \\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(Li) \\ar[rr] \\ar[dd] && \\mathscr{P}_{\\overline{L} A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar[dd]\\\\\nL\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Y^{\\otimes r}\\Bigr] \\ar'[r][rr] \\ar@(d,dl)[dr]_-{\\beta} && L\\mathsf{O}^r_{A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$} \\ar[dr]^-{f_{t-1}^r} &\\\\\n& \\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} (LY)^{\\otimes r} \\ar[rr] && \\mathscr{P}^r_{\\overline{L} A^{t-1}}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nin $\\N^{\\Sigma^{\\smallop}_{[\\underline{b}]} \\times \\{d\\}}$. The map $\\alpha$ factors as the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt@C+.7cm{L\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(i)\\Bigr] \\ar[r]^-{\\alpha} \\ar[d]_-{\\cong} & \\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(Li) \\\\\n\\Bigl[L\\bigl(\\mathsf{O}_{A^{t-1}}\\drcub \\otimes Q^r_{r-1}(i)\\bigr)\\Bigr]_{\\Sigma_r} \\ar[r]^-{\\alpha_1 \\,=\\, (L^2)_{\\Sigma_r}} & \\Bigl[L\\mathsf{O}_{A^{t-1}}\\drcub \\otimes Q^r_{r-1}(Li)\\Bigr]_{\\Sigma_r} \\ar[u]^-{\\alpha_2 \\,=}_-{f_{t-1} \\underset{\\Sigma_r}{\\otimes} \\Id}}\\]\nin which:\n\\begin{itemize}\n\\item $L^2$ is the comonoidal structure map of $L$ \\eqref{comonoidal.map};\n\\item $LQ^r_{r-1}(i) \\cong Q^r_{r-1}(Li)$ because $Q^r_{r-1}$ is a colimit, which is preserved by the left adjoint $L$.\n\\end{itemize}\nObserve that to define the map $\\alpha_2$, the map $f_{t-1}$ must be an equivariant map, instead of merely a map of individual entries. There is a similar factorization for the map $\\beta$. In the top face, the top horizontal map is induced by $L$ applied to the map $\\mathsf{O} \\circ X \\to A^{t-1}$, while the other horizontal map is induced by the map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathscr{P} \\circ (LX) = \\overline{L}(\\mathsf{O} \\circ X) \\ar[r] & \\overline{L} A^{t-1}.}\\]\nWe will prove in Lemma \\ref{top.face.of.cube} below that the top face of the cube \\eqref{lstarrt.to.lbarstarrt} is indeed commutative.\n\nWe already observed above that both vertical maps in the left face are entrywise cofibrations and that all the objects in the cube are entrywise cofibrant. By the Cube Lemma \\cite{hovey} (5.2.6), to show that $f_{t-1}^r$ is a weak equivalence, it suffices to show that both $\\alpha$ and $\\beta$ are entrywise weak equivalences.\n \nTo show that $\\alpha$ is a weak equivalence, it is enough to show that both $\\alpha_1$ and $\\alpha_2$ are weak equivalences. We already observed that $\\mathsf{O}_{A^{t-1}}$ is entrywise cofibrant and that $Q^r_{r-1}(i)$ is cofibrant in $\\mathcal{M}$. So $\\alpha_1$ is a weak equivalence by condition $(\\#)$ (Def. \\ref{def:sharp}). Likewise, since $f_{t-1}$ is a weak equivalence between entrywise cofibrant objects, condition $(\\medstar)$ in $\\mathcal{N}$ (Def. \\ref{def:star}) implies that $\\alpha_2$ is a weak equivalence. This proves that $\\alpha$ is a weak equivalence. A similar argument, with $Y^{\\otimes r}$ in place of $Q^r_{r-1}(i)$, proves that $\\beta$ is a weak equivalence.\n\nTherefore, the map $f^r_{t-1}$ is a weak equivalence. This finishes the induction in the ladder diagram \\eqref{lostart.to.plbarstart}, proving that the map $f_t$ is a weak equivalence. This in turn proves the induction step in the first ladder diagram \\eqref{lostar.to.plbarstar}, so the map $f_\\infty$ is a weak equivalence.\n\nFinally, the assertion $f_\\infty\\dnothing = (\\chi_A)_d$ is a consequence of the naturality of $f_\\infty$ and Example \\ref{comparison.initial}.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{top.face.of.cube}\nThe top face of the cube \\eqref{lstarrt.to.lbarstarrt} is commutative.\n\\end{lemma}\n\n\\begin{proof}\nThe top face of the cube \\eqref{lstarrt.to.lbarstarrt} is the diagram\n\\begin{equation}\\label{top.face}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\Bigl[\\mathsf{O}_{A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(i)\\Bigr] \\ar[r]^-{g_*} \\ar[d]_-{\\alpha} & L\\mathsf{O}_{A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$} \\ar[d]^-{f_{t-1}^{r-1}}\\\\\n\\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub \\underset{\\Sigma_r}{\\otimes} Q^r_{r-1}(Li) \\ar[r]^-{g'_*} & \\mathscr{P}_{\\overline{L} A^{t-1}}^{r-1}\\smallprof{$\\binom{d}{\\brb}$}}\n\\end{equation}\nwith $r,t \\geq 1$ and $i : X \\to Y \\in \\mathcal{M}$ concentrated in a single color $c \\in \\mathfrak{C}$. The top horizontal map is induced by the $\\mathsf{O}$-algebra map $g : \\mathsf{O} \\circ X \\to A^{t-1}$, whose adjoint $X \\to A^{t-1} \\in \\M^{\\mathfrak{C}}$ is also denoted by $g$. The bottom horizontal map is induced by the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{LX \\ar@\/^2pc\/@(ul,ur)[rr]^-{g'} \\ar[r]^-{Lg} & LA^{t-1} \\ar[r]^-{\\chi_A} & \\overline{L} A^{t-1},}\\]\nwhich is adjoint to the map \n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathscr{P} \\circ (LX) = \\overline{L}(\\mathsf{O} \\circ X) \\ar[r]^-{\\overline{L} g} & \\overline{L} A^{t-1}}\\]\nof $\\mathscr{P}$-algebras. As before, we omit writing the forgetful functors.\n\nTo show that \\eqref{top.face} is commutative, observe that $\\mathsf{O}_{A^{t-1}}$ and $\\mathscr{P}_{\\overline{L} A^{t-1}}$ are defined as coequalizers (Def. \\ref{oaalgebra}), that $Q^r_{r-1}$ is a colimit indexed by the punctured $r$-cube $\\{0 < 1\\}^r \\setminus \\{(1,\\ldots,1)\\}$ \\cite{harper-jpaa} (7.19), and that taking $\\Sigma_r$-coinvariants is also a colimit. Therefore, it is enough to check the commutativity of the diagram \\eqref{top.face} when it is restricted to a typical node in the colimiting cone. In other words, it is enough to check the commutativity of the solid-arrow diagram\n\\begin{equation}\\label{top.face.restricted}\n\\begin{footnotesize}\n\\nicearrow\\xymatrix@C+10pt@R+10pt@R-.3cm@C-1.3cm{L\\Bigl[\\mathsf{O}\\duarcub \\otimes A^{t-1}_{[\\underline{a}]} \\otimes X^{p} \\otimes Y^q\\Bigr] \\ar@(r,l)[dr]|-{g_*} \\ar[dd]|-{\\mathrm{comonoidal}} && L\\Bigl[\\mathsf{O}_{A^{t-1}}\\dqcub \\otimes Y^q \\Bigr] \\ar[dd]|-{\\mathrm{comonoidal}}\\\\\n&L\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]} \\otimes Y^q\\Bigr] \\ar@(r,l)[ur]|-{\\mathrm{natural}} \\ar@{.>}[d] &\\\\ \nL\\mathsf{O}(\\vdots) \\otimes (LA^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar[d]_-{(\\chi_{A^{t-1}})_*} & \\bullet \\ar@{.>}[d] & L\\mathsf{O}_{A^{t-1}}\\dqcub \\otimes (LY)^q \\ar[ddd]|-{(f_{t-1}, \\Id)}\\\\\nL\\mathsf{O}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar[dd]|-{(\\overline{f},\\Id)} & \\bullet \\ar@{.>}[d] &\\\\\n& \\mathscr{P}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar@(r,l)[dr]|-{\\mathrm{natural}}&\\\\\n\\mathscr{P}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar@(r,l)[ur]|-{g'_*} && \\mathscr{P}_{\\overline{L} A^{t-1}}\\dqcub \\otimes (LY)^q}\n\\end{footnotesize}\n\\end{equation}\nin which $\\underline{a} \\in \\Sigma_{\\frakC}$ is arbitrary, $(\\vdots) = \\duarcub$, and $p+q = r$ with $p>0$ (hence $0 \\leq q < r$). We will show that this diagram is commutative by factoring it into two commutative diagrams as indicated by the dotted arrows.\n\nThe left half of the diagram \\eqref{top.face.restricted} is the commutative diagram\n\\begin{equation}\\label{top.face.left}\n\\begin{small}\n\\nicearrow\\xymatrix@C+10pt@R+10pt@C-.3cm{L\\Bigl[\\mathsf{O}\\duarcub \\otimes A^{t-1}_{[\\underline{a}]} \\otimes X^{p} \\otimes Y^q\\Bigr] \\ar[r]^-{g_*} \\ar[d]|-{\\mathrm{comonoidal}} & L\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]} \\otimes Y^q\\Bigr] \\ar[d]|-{\\mathrm{comonoidal}}\\\\ \nL\\mathsf{O}(\\vdots) \\otimes (LA^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar[d]_-{(\\chi_{A^{t-1}})_*} \\ar[r]^-{g_*} & L\\mathsf{O}(\\vdots) \\otimes (LA^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[d]^-{(\\chi_{A^{t-1}})_*}\\\\\nL\\mathsf{O}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar[d]_-{(\\overline{f},\\Id)} \\ar[r]^-{g'_*} & L\\mathsf{O}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[d]^-{(\\overline{f},\\Id)} &\\\\\n\\mathscr{P}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a}]} \\otimes (LX)^p \\otimes (LY)^q \\ar[r]^-{g'_*}\n& \\mathscr{P}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q}\n\\end{small}\n\\end{equation}\nin which the top square is commutative by naturality. The middle square is commutative by the definition of the map $g'$, and the bottom square is commutative by definition.\n\nIt remains to show that the right half of the diagram \\eqref{top.face.restricted} is commutative. First observe that the upper right region of the diagram \\eqref{top.face.restricted} can be rewritten as in the commutative diagram:\n\\begin{equation}\\label{upper.right}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]} \\otimes Y^q\\Bigr] \\ar[r]^-{\\mathrm{natural}} \\ar[d]|-{\\mathrm{comonoidal}} & L\\Bigl[\\mathsf{O}_{A^{t-1}}\\dqcub \\otimes Y^q \\Bigr] \\ar[d]|-{\\mathrm{comonoidal}}\\\\ \nL\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]}\\Bigr] \\otimes (LY)^q \\ar[r]^-{\\mathrm{natural}} & L\\mathsf{O}_{A^{t-1}}\\dqcub \\otimes (LY)^q}\n\\end{equation}\nTherefore, to show that the right half of the diagram \\eqref{top.face.restricted} is commutative, it is enough to show that the diagram\n\\begin{equation}\\label{right.half.1}\n\\begin{footnotesize}\n\\nicearrow\\xymatrix@C+10pt@R+10pt@R-.3cm@C-1.3cm{L\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]} \\otimes Y^q\\Bigr] \\ar[dr]|-{\\mathrm{comonoidal}} \\ar[dd]|-{\\mathrm{comonoidal}} && L\\mathsf{O}_{A^{t-1}}\\dqcub \\otimes (LY)^q \\ar[dddddd]|-{(f_{t-1},\\Id)}\\\\\n& L\\Bigl[\\mathsf{O}(\\vdots) \\otimes A^{t-1}_{[\\underline{a},pc]}\\Bigr] \\otimes (LY)^q \\ar[ur]|-{\\mathrm{natural}} \\ar[dl]|-{\\mathrm{comonoidal}} \\ar[dd]|-{L(f,\\mathrm{unit})} &\\\\\nL\\mathsf{O}(\\vdots) \\otimes (LA^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[dd]_-{(\\chi_{A^{t-1}})_*} \\ar[dddr]|-{\\bigl(Lf,L(\\mathrm{unit})\\bigr)} &&\\\\ \n& L\\Bigl[R\\mathscr{P}(\\vdots) \\otimes (R\\overline{L} A^{t-1})_{[\\underline{a},pc]}\\Bigr] \\otimes (LY)^q \\ar[dd]|-{\\mathrm{comonoidal}} &\\\\\nL\\mathsf{O}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[dd]|-{(\\overline{f},\\Id)}&&\\\\\n& LR\\mathscr{P}(\\vdots) \\otimes (LR\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[dl]|-{\\mathrm{counit}}&\\\\\n\\mathscr{P}(\\vdots) \\otimes (\\overline{L} A^{t-1})_{[\\underline{a},pc]} \\otimes (LY)^q \\ar[rr]^-{\\mathrm{natural}} && \\mathscr{P}_{\\overline{L} A^{t-1}}\\dqcub \\otimes (LY)^q}\n\\end{footnotesize}\n\\end{equation}\nis commutative. The left column in \\eqref{right.half.1} is equal to the right column in \\eqref{top.face.left} (i.e. the middle column in \\ref{top.face.restricted}). The path from the top left to the lower right, across the top then down the right side, is isomorphic to the right column in \\ref{top.face.restricted}. The right sub-diagram is commutative by the definition of $f_{t-1}$. The top left triangle (with three comonoidal maps) and the middle left triangle (with two comonoidal maps) are commutative by naturality. The lower left triangle is commutative because, by adjunction, $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is equal to the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O} \\ar[r]^-{Lf} & LR\\mathscr{P} \\ar[r]^-{\\mathrm{counit}} & \\mathscr{P}.}\\]\nLikewise, the comparison map $\\chi_{A^{t-1}} : LA^{t-1} \\to \\overline{L} A^{t-1}$ is equal to the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{LA^{t-1} \\ar[r]^-{L(\\mathrm{unit})} & LR\\overline{L} A^{t-1} \\ar[r]^-{\\mathrm{counit}} & \\overline{L} A^{t-1}.}\\]\nTherefore, the diagram \\eqref{right.half.1} is commutative. As discussed above, together with the commutative diagrams \\eqref{top.face.left} and \\eqref{upper.right}, we conclude that the diagrams \\eqref{top.face.restricted} and, therefore, \\eqref{top.face} are commutative.\n\\end{proof}\n\n\n\\subsection{Main Theorem}\n\nThe following theorem is our main result. Roughly speaking, it says that operadic algebras are homotopically well-behaved with respect to Quillen equivalences.\n\n\\begin{theorem}[Lifting Quillen Equivalences]\n\\label{main.theorem}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}).\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. The entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise weak equivalence in $\\mathcal{N}$. \n\\end{enumerate}\nThen the lifted adjunction \\eqref{lbar.ocomp.diagram}\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{R}}\\]\nis a Quillen equivalence between the semi-model categories of $\\mathsf{O}$-algebras in $\\mathcal{M}$ and of $\\mathscr{P}$-algebras in $\\mathcal{N}$ (Theorem \\ref{theorem623}).\n\\end{theorem}\n\nBelow, we will show that, if $\\mathsf{O}$ and $\\mathscr{P}$ are $\\Sigma$-cofibrant, then we can weaken assumption (1) above to only requiring that $(L,R)$ be a weak symmetric monoidal Quillen equivalence and that the domains of the generating cofibrations in $\\mathcal{M}$ be cofibrant.\n\n\\begin{proof}\nRecall that weak equivalences and fibrations in $\\mathsf{Alg}(\\mathsf{O})$ and $\\mathsf{Alg}(\\sP)$ are defined entrywise in $\\mathcal{M}$ and $\\mathcal{N}$, respectively. The lifted adjunction $\\overline{L} \\dashv R$ is a Quillen adjunction--i.e., the right adjoint $R$ preserves fibrations and trivial fibrations--because $UR = RU$ in the diagram \\eqref{lbar.ocomp.diagram}.\n\nTo see that $\\overline{L} \\dashv R$ is a Quillen equivalence between semi-model categories, suppose $A$ is a cofibrant $\\mathsf{O}$-algebra, $B$ is a fibrant $\\mathscr{P}$-algebra, and $\\varphi : \\overline{L} A \\to B \\in \\mathsf{Alg}(\\sP)$. We want to show that $\\varphi$ is a weak equivalence if and only if its adjoint $\\overline{\\varphi} : A \\to RB$ is a weak equivalence. By Proposition \\ref{loa.to.pla} the comparison map $\\chi_A : LA \\to \\overline{L} A \\in \\N^{\\mathfrak{C}}$ is an entrywise weak equivalence. By the $2$-out-of-$3$ property, $\\varphi$ is a weak equivalence if and only if the composite\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{LA \\ar[r]^-{\\chi_A}_-{\\sim} & \\overline{L} A \\ar[r]^-{\\varphi} & B \\in \\N^{\\mathfrak{C}}}\\]\nis a weak equivalence. Note that $B \\in \\N^{\\mathfrak{C}}$ is fibrant and that $A \\in \\M^{\\mathfrak{C}}$ is cofibrant by Theorem \\ref{theorem623}(2). Since the entrywise prolongation\n\\[L : \\M^{\\mathfrak{C}} \\hspace{-.1cm \\N^{\\mathfrak{C}} : R\\]\nis a Quillen equivalence \\cite{hirschhorn} (11.6.5(2)), the map $\\varphi\\chi_A \\in \\N^{\\mathfrak{C}}$ is a weak equivalence if and only if its adjoint $\\overline{\\varphi} : A \\to RB \\in \\M^{\\mathfrak{C}}$ is a weak equivalence (Example \\ref{comparison.adjoint}). This proves that $\\overline{L} \\dashv R$ is a Quillen equivalence.\n\\end{proof}\n\nNote that in Theorem \\ref{main.theorem}, we only ask that the colored operads $\\mathsf{O}$ and $\\mathscr{P}$ be \\emph{entrywise} cofibrant, instead of the much stronger conditions of being $\\Sigma$-cofibrant and admissible \\cite{bm03} (section 4). In particular, our operads will almost never be admissible. \n\n\\subsection{$\\Sigma$-cofibrant colored operads}\n\nIf we require $\\mathsf{O}$ and $\\mathscr{P}$ to be $\\Sigma$-cofibrant, then we can weaken our conditions on the adjunction $(L,R)$. The results that follow provide an extension of Proposition 12.3.4 in \\cite{fresse-book}, which considers one operad acting in two different model categories.\n\n\\begin{proposition}\n\\label{loa.to.pla.Sigma.cof}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak symmetric monoidal Quillen equivalence (Def. \\ref{def:weak.symmetric.monoidal}).\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. The entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise weak equivalence in $\\mathcal{N}$. \n\\item Every generating cofibration in $\\mathcal{M}$ has cofibrant domain.\n\\end{enumerate}\nSuppose $A$ is a cofibrant $\\mathsf{O}$-algebra. Then the map $f : \\mathsf{O} \\to R\\mathscr{P}$ induces a natural entrywise weak equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O}_A \\ar[r]^-{f_{\\infty}} & \\mathscr{P}_{\\overline{L} A} \\in \\symseqc(\\N)}\\]\nwhose value at $\\dnothing$ is the comparison map $\\chi_A : LA \\to \\overline{L} A$ \\eqref{comparison.map} evaluated at $d$ for each $d \\in \\mathfrak{C}$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof proceeds exactly as in Proposition \\ref{loa.to.pla}. Instead of Lemma \\ref{middle-row-lemma}, we use the colored version of Proposition 5.17 in \\cite{harper-gnt}, which implies $\\mathsf{O}_{A^t}$ and $\\mathscr{P}_{\\overline{L}A^t}$ are $\\Sigma$-cofibrant for all $t$.\nInstead of $(\\clubsuit)_{\\cof}$ in \\ref{star.r.t}, we use the observation that, for a $\\Sigma_r$-projectively cofibrant object $X$, the functor $X \\otimes_{\\Sigma_r} -$ is left Quillen. In the proof, $X$ is first $\\mathsf{O}_{A^{t-1}}\\drcub$ in \\eqref{star.r.t} and is later $\\mathscr{P}_{\\overline{L} A^{t-1}}\\drcub$ in \\eqref{lbarstar.r.t}. Rather than condition $(\\filledstar)$, observe that $X \\otimes i^{\\mathbin\\square r}$ is a cofibration in the projective model structure on $\\mathcal{M}^{\\Sigma_r}$, and the domain and codomain are projectively cofibrant. Thus, after taking $\\Sigma_r$-coinvariants, we are left with a cofibration between cofibrant objects in $\\mathcal{M}$. Similarly, $X\\otimes_{\\Sigma_r} (Li)^{\\mathbin\\square r}$ is a cofibration between cofibrant objects in $\\mathcal{N}$.\n \nFinally, when proving the maps $\\alpha$ and $\\beta$ are weak equivalences, instead of using conditions $(\\#)$ and $(\\medstar)$, we use that $(L,R)$ is a weak monoidal Quillen pair.\nThis implies $L^2$ is a weak equivalence between $\\Sigma_r$-projectively cofibrant objects (since $L$ induces a left Quillen functor on $\\mathcal{M}^{\\Sigma_r}$). It follows from Ken Brown's Lemma that $\\alpha_1$ is a weak equivalence. The situation for $\\alpha_2$ is similar, since $f_{t-1}\\otimes \\Id$ is a weak equivalence between $\\Sigma_r$-projectively cofibrant objects. It follows that $\\alpha$ is a weak equivalence, a similar argument shows that $\\beta$ is a weak equivalence, and then the double induction demonstrates that $f_\\infty$ is a weak equivalence as required.\n\\end{proof}\n\nSimilarly, we have a version of Theorem \\ref{main.theorem} for $\\Sigma$-cofibrant colored operads:\n\n\\begin{theorem}[Lifting Quillen Equivalences for $\\Sigma$-Cofibrant Operads]\n\\label{main.theorem.Sigma}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak symmetric monoidal Quillen equivalence (Def. \\ref{def:weak.symmetric.monoidal}).\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$ with $\\mathfrak{C}$ a set, $\\mathsf{O}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, and $\\mathscr{P}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$. The entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise weak equivalence in $\\mathcal{N}$. \n\\item Every generating cofibration in $\\mathcal{M}$ has a cofibrant domain.\n\\end{enumerate}\nThen the lifted adjunction \\eqref{lbar.ocomp.diagram}\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{R}}\\]\nis a Quillen equivalence between the semi-model categories of $\\mathsf{O}$-algebras in $\\mathcal{M}$ and of $\\mathscr{P}$-algebras in $\\mathcal{N}$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof proceeds exactly as in Theorem \\ref{main.theorem}, but uses Proposition \\ref{loa.to.pla.Sigma.cof} instead of Proposition \\ref{loa.to.pla}. The existence of the semi-model structures on $\\mathsf{Alg}(\\mathsf{O})$ and $\\mathsf{Alg}(\\sP)$ is now due to Theorem 6.3.1 in \\cite{white-yau}, which also proves that cofibrant $\\mathsf{O}$-algebras are cofibrant in $\\M^{\\mathfrak{C}}$, avoiding the need for Theorem \\ref{theorem623} and $(\\clubsuit)$. \n\\end{proof}\n\n\n\\section{Special Cases: Rectification and Derived Change of Category} \\label{sec:rect-and-change}\n\nIn this section, we discuss special cases of our main results, Theorems \\ref{main.theorem} and \\ref{main.theorem.Sigma}. We begin with the strongest possible condition on $L$ (that it is the identity), and successively weaken our conditions on $L$. We see that rectification, change of rings, and change of underlying model category (i.e., lifting Quillen equivalences) are all special cases of the same general framework.\n\n\\subsection{Rectification}\n\nRestricting Theorem \\ref{main.theorem} to the special case $L = R = \\Id$ (so condition $(\\#)$ (Def. \\ref{def:sharp}) holds automatically), we obtain the following rectification result for entrywise cofibrant operads.\n\n\\begin{corollary}[Rectification of Operadic Algebras]\n\\label{cor.rectification}\nSuppose $\\mathcal{M}$ is a cofibrantly generated monoidal model category satisfying the conditions $(\\medstar)$, $(\\filledstar)$ (Def. \\ref{def:star}), and $(\\clubsuit)$ (Def. \\ref{def:club}), in which every generating cofibration has a cofibrant domain. Suppose $\\mathfrak{C}$ is a set, and $f : \\mathsf{O} \\to \\mathscr{P}$ is a map of entrywise cofibrant $\\mathfrak{C}$-colored operads that is an entrywise weak equivalence in $\\mathcal{M}$. Then the induced adjunction\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O}) \\ar@<2.5pt>[r]^-{f_!} & \\mathsf{Alg}(\\sP) \\ar@<2.5pt>[l]^-{f^*}}\\]\nis a Quillen equivalence between semi-model categories.\n\\end{corollary}\n\nThe right adjoint $f^* : \\mathsf{Alg}(\\sP) \\to \\mathsf{Alg}(\\mathsf{O})$ is given by restriction along the map $f$. The left adjoint $f_!$ may be constructed as a certain coequalizer \\cite{bm07} (section 4).\n\n\n\\begin{example}\nCorollary \\ref{cor.rectification} applies when $\\mathcal{M} = \\Chk_{\\geq 0}$, the category of non-negatively graded chain complexes of $k$-modules for a characteristic $0$ field $k$, and $f : \\mathsf{O} \\to \\mathscr{P}$ is the cofibrant replacement:\n\\begin{enumerate}\n\\item $\\mathsf{A}_{\\infty} \\to \\mathsf{As}$, where $\\mathsf{A}_{\\infty}$ is the operad for $A_{\\infty}$-algebras \\cite{stasheff} and $\\mathsf{As}$ is the operad for differential graded algebras.\n\\item $\\End_{\\infty} \\to \\mathsf{Com}$, where $\\End_{\\infty}$ is an $E_{\\infty}$ operad \\cite{may72} and $\\mathsf{Com}$ is the operad for commutative dg algebras.\n\\item $\\mathsf{L}_{\\infty} \\to \\mathsf{Lie}$, where $\\mathsf{L}_{\\infty}$ is the operad for $L_{\\infty}$-algebras and $\\mathsf{Lie}$ is the operad for dg Lie algebras \\cite{fmy,lada-markl}.\n\\end{enumerate}\n\\end{example}\n\n\\begin{remark}\nSimilar rectification results for \\emph{admissible $\\Sigma$-cofibrant} operads--as opposed to entrywise cofibrant operads--have been obtained by Berger and Moerdijk \\cite{bm03,bm07}. Admissibility means that the category of algebras over the operad in question has a model category structure in which the fibrations and weak equivalences are defined entrywise in the underlying category. Another rectification result for admissible operads is in \\cite{dmitri}. Concrete examples (e.g. \\cite{batanin-white-eilenberg}) demonstrate that admissibility is a strong condition, and one should instead expect only semi-model structures. A rectification result for $1$-colored, entrywise cofibrant, non-symmetric operads is \\cite{muro11} (1.3). For operads in symmetric spectra, a rectification result is \\cite{em06} (1.4); a $1$-colored version is \\cite{agt1} (1.4).\n\\end{remark}\n\n\\begin{example} \\label{ex:change-of-rings}\nRectification results specialize to change of rings results. Let $\\mathcal{M}$ be a cofibrantly generated monoidal model category whose generating cofibrations have cofibrant domains. Let $f:R\\to T$ be a weak equivalence of monoids in $\\mathcal{M}$ with $R$ and $T$ cofibrant as objects in $\\mathcal{M}$.\n\\begin{enumerate}\n\\item Then $f$ induces a Quillen equivalence between the semi-model categories of $R$-modules and $T$-modules. \n\\item If $f$ is a map of commutative monoids, then it induces a Quillen equivalence between the semi-model categories of $R$-algebras and $T$-algebras.\n\\item Suppose $f$ is a map of commutative monoids and $\\mathcal{M}$ satisfies the conditions in Corollary \\ref{cor.rectification}. Then $f$ induces a Quillen equivalence between the semi-model categories of commutative $R$-algebras and commutative $T$-algebras.\n\\end{enumerate}\nThe first two of these examples date back to \\cite{ss}, and can be viewed as special cases of rectification for $\\Sigma$-cofibrant operads. As discussed in Theorem \\ref{main.theorem.Sigma}, when the operads involved are $\\Sigma$-cofibrant, the conditions $(\\medstar)$, $(\\filledstar)$ (Def. \\ref{def:star}), and $(\\clubsuit)$ (Def. \\ref{def:club}) are not needed. The last example for commutative algebras requires the theory of entrywise cofibrant operads as in Theorem \\ref{main.theorem}.\n\\end{example}\n\n\\subsection{Modules}\n\nEach (commutative) monoid $T$ in $\\mathcal{M}$ admits a category $\\mathbf{Mod}(T)$ of left $T$-modules \\cite{maclane} (VII.4). If $L : \\mathcal{M} \\to \\mathcal{N}$ is a lax (symmetric) monoidal functor, then $LT$ is a (commutative) monoid in $\\mathcal{N}$, so it admits a category $\\mathbf{Mod}(LT)$ of left $LT$-modules. Using the respective operads for left $T$-modules and left $LT$-modules, Theorem \\ref{main.theorem.Sigma} yields the following result. It has both a commutative version and an associative version, the latter of which is closely related to \\cite{ss03} (3.12(1)).\n\n\\begin{corollary}[Modules]\n\\label{cor.com.module}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak (symmetric) monoidal Quillen equivalence with $L$ lax (symmetric) monoidal, and $T$ is a (commutative) monoid that is cofibrant as an object in $\\mathcal{M}$. Assume that the domains of the generating cofibrations in $\\mathcal{M}$ are cofibrant. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathbf{Mod}(T) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathbf{Mod}(LT) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of left $T$-modules in $\\mathcal{M}$ and of left $LT$-modules in $\\mathcal{N}$.\n\\end{corollary}\n\n\\begin{proof}\nThe proof is the same in the symmetric and non-symmetric contexts. We need to check condition (2) in Theorem \\ref{main.theorem.Sigma}. The $1$-colored operad $\\mathsf{O}$ for left $T$-modules has\n\\[\\mathsf{O}(n) = \\begin{cases} T & \\text{ if $n=1$};\\\\ \\varnothing & \\text{ if $n\\not= 1$.}\\end{cases}\\]\nLikewise, the only non-$\\varnothing$ entry in the operad for $LT$-modules is $\\mathscr{P}(1) = LT$. Both $\\mathsf{O}$ and $\\mathscr{P}$ are $\\Sigma$-cofibrant. The map $f : \\mathsf{O} \\to R\\mathscr{P}$ is determined by the unit of the adjunction $T \\to RLT$, and $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is the identity map. \n\\end{proof}\n\n\\subsection{Monoids and Algebras}\n\nConsider Theorem \\ref{main.theorem} restricted to the special case with $\\mathsf{O}$ the operad for associative monoids \\cite{maclane}(VII.3) in $\\mathcal{M}$ and $\\mathscr{P}$ the operad for associative monoids in $\\mathcal{N}$. In this setting, we recover the following result from \\cite{ss03} (3.12(3)) with slightly different assumptions. The slight difference in assumptions is due to the generality of our result. \n\n\\begin{corollary}[Monoids]\n\\label{cor.schwede.shipley}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak monoidal Quillen equivalence, that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant, and that the domains of the generating cofibrations in $\\mathcal{M}$ are cofibrant. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Monoid}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Monoid}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of monoids in $\\mathcal{M}$ and in $\\mathcal{N}$.\n\\end{corollary}\n\n\\begin{proof}\nAs above, we need to check condition (2) in Theorem \\ref{main.theorem.Sigma}. The $1$-colored operad $\\mathsf{O}$ for monoids in $\\mathcal{M}$ has\n\\[\\mathsf{O}(n) = \\coprodover{\\Sigma_n}\\, \\tensorunit^{\\M}\\] \nand similarly the $1$-colored operad $\\mathscr{P}$ for monoids in $\\mathcal{N}$ has $\\mathscr{P}(n) = \\coprod_{\\Sigma_n} \\tensorunit^{\\N}$. In fact, $\\mathsf{O}$ is the image of the associative operad in the category of sets under the strong symmetric monoidal functor\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Set} \\ar[r] & \\mathcal{M},}\\quad \\nicearrow\\xymatrix@C+10pt@R+10pt{S \\ar@{|->}[r] & \\coprodover{S}\\, \\tensorunit^{\\M}}\\]\nand similarly for $\\mathscr{P}$. Both $\\mathsf{O}$ and $\\mathscr{P}$ are $\\Sigma$-cofibrant. The map $\\overline{R}^0 : L\\tensorunit^{\\M} \\to \\tensorunit^{\\N}$ \\eqref{unit.adjoint} is a weak equivalence between cofibrant objects in $\\mathcal{N}$. So the coproduct map\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{L\\mathsf{O}(n) = L\\Bigl(\\coprodover{\\Sigma_n}\\, \\tensorunit^{\\M}\\Bigr) \\cong \\coprodover{\\Sigma_n}\\, L\\tensorunit^{\\M} \\ar[r]^-{\\coprod \\overline{R}^0} & \\coprodover{\\Sigma_n}\\, \\tensorunit^{\\N} = \\mathscr{P}(n)}\\]\nis also a weak equivalence.\n\\end{proof}\n\nA commutative monoid $T$ also admits categories $\\mathsf{Alg}(T)$ (and $\\mathsf{CAlg}(T)$) of (commutative) $T$-algebras, which are (commutative) monoids in the category of $T$-modules \\cite{ss} (p.499). An analogous proof to Corollary \\ref{cor.schwede.shipley} demonstrates:\n\n\\begin{corollary}[Algebras]\n\\label{cor.algebras}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak monoidal Quillen equivalence with $L$ lax monoidal, and $T$ is a commutative monoid that is cofibrant as an object in $\\mathcal{M}$. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(T) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(LT) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of $T$-algebras in $\\mathcal{M}$ and of $LT$-algebras in $\\mathcal{N}$.\n\\end{corollary}\n\n\\begin{proof}\nAs above, we check condition (2) in Theorem \\ref{main.theorem.Sigma}. The $1$-colored operad for $T$-algebras is the enveloping operad $\\mathsf{O}_T$, where $\\mathsf{O}$ is the operad for monoids. In this operad, $T$ replaces the unit $\\tensorunit^{\\M}$. Since $T$ is cofibrant and $\\mathsf{O}$ is $\\Sigma$-cofibrant, $\\mathsf{O}_T$ is $\\Sigma$-cofibrant by Proposition 5.17 in \\cite{harper-gnt}. Similarly, $\\mathsf{O}_{LT}$ is $\\Sigma$-cofibrant and $LT$ replaces $\\tensorunit^{\\N}$. The proof now proceeds precisely as above.\n\\end{proof}\n\n\\subsection{Commutative Monoids, Commutative Algebras, and Non-Symmetric Operads}\nThe following two special cases of Theorem \\ref{main.theorem} are the commutative versions of the previous two results.\n\n\\begin{corollary}[Commutative Monoids]\n\\label{cor.com.monoids}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}) and that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{CMonoid}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{CMonoid}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of commutative monoids in $\\mathcal{M}$ and in $\\mathcal{N}$.\n\\end{corollary}\n\n\\begin{proof}\nWe simply reuse the proof of Corollary \\ref{cor.schwede.shipley} with $\\mathsf{O}$ the commutative monoid operad in $\\mathcal{M}$, which has $\\mathsf{O}(n) = \\tensorunit^{\\M}$ for all $n \\geq 0$, and $\\mathscr{P}$ the commutative monoid operad in $\\mathcal{N}$. As the commutative monoid operad is not $\\Sigma$-cofibrant, we need to assume $(L,R)$ is a nice Quillen equivalence, and we need to use Theorem \\ref{main.theorem}.\n\\end{proof}\n\nUsing essentially the same proof as in the previous corollaries with the respective operads for commutative $T$-algebras and commutative $LT$-algebras, Theorem \\ref{main.theorem} yields the following result.\n\n\\begin{corollary}[Commutative Algebras]\n\\label{cor.com.algebra}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}) with $L$ lax symmetric monoidal, and $T$ is a commutative monoid that is cofibrant as an object in $\\mathcal{M}$. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{CAlg}(T) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{CAlg}(LT) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of commutative $T$-algebras in $\\mathcal{M}$ and of commutative $LT$-algebras in $\\mathcal{N}$.\n\\end{corollary}\n\nThis result improves on Theorem 4.19 in \\cite{white-commutative}, which required $L$ to be strong symmetric monoidal.\n\nUsing instead the operads for $1$-colored non-symmetric operads in $\\mathcal{M}$ and in $\\mathcal{N}$, essentially the same proof yields the following special case of Theorem \\ref{main.theorem.Sigma}. A similar result for full model categories is \\cite{muro14} (1.1). \n\n\\begin{corollary}[Non-Symmetric Operads]\n\\label{cor.muro}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a weak monoidal Quillen equivalence, that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant, and that the generating cofibrations in $\\mathcal{M}$ have cofibrant domains. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\Omega}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\Omega}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of $1$-colored non-symmetric operads in $\\mathcal{M}$ and in $\\mathcal{N}$.\n\\end{corollary}\n\n\n\\subsection{Generalized Props}\n\nThe previous five corollaries can be vastly extended to \\emph{generalized props} associated to any pasting scheme in the sense of \\cite{jy2} (10.39). We refer the reader to \\cite{jy2} for detailed discussion of pasting schemes and their associated generalized props. For any fixed set $\\mathfrak{C}$ of colors, generalized props over a pasting scheme include: enriched $\\mathfrak{C}$-categories ($=$ enriched categories with object set $\\mathfrak{C}$ and object-preserving functors), $\\mathfrak{C}$-colored operads, $\\mathfrak{C}$-colored half-props, $\\mathfrak{C}$-colored dioperads, $\\mathfrak{C}$-colored prop(erad)s, $\\mathfrak{C}$-colored wheeled operads, and $\\mathfrak{C}$-colored wheeled prop(erad)s. See \\cite{jy2} (Chapter 11) for detailed discussion of these objects. \n\nWhen Theorem \\ref{main.theorem} is restricted to the special case with $\\mathsf{O}$ and $\\mathscr{P}$ the operads for the generalized props under discussion in $\\mathcal{M}$ and in $\\mathcal{N}$, which are explicitly described in \\cite{jy2} (14.1), we obtain the following result. The proof is once again basically the same as that of Corollary \\ref{cor.schwede.shipley}, but uses Theorem \\ref{main.theorem} because the colored operads in question are in general not $\\Sigma$-cofibrant.\n\n\\begin{corollary}\n\\label{cor.gprop}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}) and that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then for each pasting scheme $\\mathfrak{G}$ in the sense of \\cite{jy2} (Def. 8.2), there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Prop}^{\\fG}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Prop}^{\\fG}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of $\\mathfrak{G}$-props \\cite{jy2} (10.39) in $\\mathcal{M}$ and in $\\mathcal{N}$. In particular, for each color set $\\mathfrak{C}$, there are induced Quillen equivalences between semi-model categories:\n\\begin{tabular}{rl}\nenriched $\\mathfrak{C}$-categories & $\\mathsf{Cat}^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\mathsf{Cat}^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored operads & $\\operad^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\operad^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored half-props & $\\frac{1}{2}\\mathsf{Prop}^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\frac{1}{2}\\mathsf{Prop}^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored dioperads & $\\dioperad^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\dioperad^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored properads & $\\mathsf{Properad}^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\mathsf{Properad}^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored props & $\\mathsf{Prop}^{\\fC}(\\mathcal{M}) \\hspace{-.1cm \\mathsf{Prop}^{\\fC}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored wheeled operads & $\\operad^{\\fC \\rcirclearrowdown}(\\mathcal{M}) \\hspace{-.1cm \\operad^{\\fC \\rcirclearrowdown}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored wheeled properads & $\\mathsf{Properad}^{\\fC \\rcirclearrowdown}(\\mathcal{M}) \\hspace{-.1cm \\mathsf{Properad}^{\\fC \\rcirclearrowdown}(\\mathcal{N})$\\\\\n$\\mathfrak{C}$-colored wheeled props & $\\mathsf{Prop}^{\\fC \\rcirclearrowdown}(\\mathcal{M}) \\hspace{-.1cm \\mathsf{Prop}^{\\fC \\rcirclearrowdown}(\\mathcal{N})$\n\\end{tabular}\n\\end{corollary}\n\nThis result extends a result from \\cite{hry15} to non-shrinkable contexts. In particular, the application to properads, colored props, and colored wheeled props, is new.\n\n\\subsection{Cyclic and Modular Operads}\n\nSimilarly, using the operads for $\\mathfrak{C}$-colored cyclic operads or $\\mathfrak{C}$-colored modular operads for a fixed color set $\\mathfrak{C}$ in $\\mathcal{M}$ and in $\\mathcal{N}$ \\cite{gk95,gk98,mss}, Theorem \\ref{main.theorem} yields the following result.\n\n\\begin{corollary}\n\\label{cor.cyclic}\nSuppose $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}) and that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then for each color set $\\mathfrak{C}$, there are induced Quillen equivalences\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\fC}_{\\mathsf{cyc}}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\fC}_{\\mathsf{cyc}}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\n\\quad\n\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\fC}_{\\mathsf{mod}}(\\mathcal{M}) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\fC}_{\\mathsf{mod}}(\\mathcal{N}) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of $\\mathfrak{C}$-colored cyclic (resp., modular) operads in $\\mathcal{M}$ and in $\\mathcal{N}$.\n\\end{corollary}\n\n\n\\section{Applications to Left Bousfield Localization} \\label{sec:bous-loc}\n\nLeft Bousfield localization is a general framework that starts with a (nice) model category $\\mathcal{M}$ and a set of morphisms $\\mathcal{C}$, and produces a new model structure $L_\\mathcal{C}(\\mathcal{M})$ on the same category in which maps in $\\mathcal{C}$ are now weak equivalences (along with all the old weak equivalences). When we say Bousfield localization we will always mean \\emph{left} Bousfield localization, so cofibrations in $L_\\mathcal{C}(\\mathcal{M})$ will be the same as the cofibrations in $\\mathcal{M}$. The model category $L_\\C(\\M)$ satisfies a universal property (Theorem 3.3.20, \\cite{hirschhorn}): for any left Quillen functor $F:\\mathcal{M} \\to \\mathcal{N}$ taking the maps in $\\mathcal{C}$ to weak equivalences, there is an induced left Quillen functor $\\tilde{F}:L_\\C(\\M) \\to \\mathcal{N}$. \n\nApplications of left Bousfield localization abound: it has been used to study generalized homology theories, to create stable model structures for spectra (including equivariant and motivic spectra), for spectral sequence computations, and to give models for presentable $\\infty$-categories, just to name a few. We refer the interested reader to \\cite{hirschhorn} to learn more. Recently, it has become advantageous to study the interplay between Bousfield localization and operad algebra structure. A lengthy list of applications in this vein can be found in \\cite{white-localization} and \\cite{white-yau}.\n\nIn this section we will specialize the machinery of Theorems \\ref{main.theorem} and \\ref{main.theorem.Sigma} to the local setting, and prove results relating $\\mathsf{Alg}(\\mathsf{O};\\lcm)$ and $\\mathsf{Alg}(\\sP;\\ldn)$, the categories of algebras valued in local model categories. We begin with an adjunction $L:\\mathcal{M} \\hspace{-.1cm \\mathcal{N}:R$ and a class of maps $\\mathcal{C}$ in $\\mathcal{M}$. We define a class of maps $\\mathcal{D} = \\underline{L} \\mathcal{C}$ in $\\mathcal{N}$, where $\\underline{L}$ is the left derived functor of the left adjoint $L$.\n\n\\subsection{Local Quillen Equivalences}\n\nIn order for operad algebras to have a well-behaved local homotopy theory, we will need $L_\\C(\\M)$ and $L_\\D(\\N)$ to be monoidal model categories. Such localizations are studied in \\cite{white-thesis}, where they are called \\textit{monoidal left Bousfield localizations}. In particular, the following characterization is given. We say that \\textit{cofibrant objects are flat} when, for every cofibrant $X$, $X\\otimes -$ preserves weak equivalences.\n\n\\begin{theorem}[Monoidal Bousfield Localization] \\label{thm:PPAxiom-nontractable}\nSuppose $\\mathcal{M}$ is a cofibrantly generated monoidal model category in which cofibrant objects are flat. Then the following are equivalent:\n\\begin{enumerate}\n\\item $L_\\mathcal{C}(\\mathcal{M})$ satisfies the pushout product axiom and has cofibrant objects flat.\n\\item Every map of the form $f \\otimes \\Id_K$, where $f$ is in $\\mathcal{C}$ and $K$ is cofibrant, is a $\\mathcal{C}$-local equivalence. \n\\end{enumerate}\nIf the domains of the generating cofibrations are cofibrant, then it suffices to check this condition for (co)domains $K$ of the generating cofibrations.\n\\end{theorem}\n\nThroughout this section, we will assume that $\\mathcal{M}$, $\\mathcal{N}$, $L_\\C(\\M)$, and $L_\\D(\\N)$ are cofibrantly generated monoidal model categories. We state our main result:\n\n\\begin{theorem}[Lifting Local Quillen Equivalences]\n\\label{main.theorem.local}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a Quillen equivalence where $R$ is lax symmetric monoidal. Suppose\n\\begin{enumerate}\n\\item For all $X,Y$ cofibrant in $\\mathcal{M}$, the comonoidal map $L(X\\otimes Y)\\to LX \\otimes LY$ is a local weak equivalence in $\\mathcal{N}$.\n\\item For some cofibrant replacement $Q \\tensorunit^{\\M}$ of the unit $\\tensorunit^{\\M}$, the composition \\\\\n$LQ\\tensorunit^{\\M} \\to L \\tensorunit^{\\M} \\to \\tensorunit^{\\N}$ is a local weak equivalence in $\\mathcal{N}$.\n\\end{enumerate} \nRecall that $\\mathcal{D} = \\underline{L}\\mathcal{C}$.\n\\item $(\\filledstar)$ holds in $\\mathcal{M}$ and $\\mathcal{N}$. The generating cofibrations in $\\mathcal{M}$ have cofibrant domain.\n\\item $(\\#)$ and $(\\clubsuit)$ hold in $L_\\C(\\M)$ and $L_\\D(\\N)$.\n\\item $\\mathcal{N}$ satisfies the local version of $(\\medstar)$; i.e. for any local weak equivalence $g$ between objects of $\\mathcal{N}^{\\Sigma^{\\smallop}_n}$ that are cofibrant in $\\mathcal{N}$ (same as being cofibrant in $L_\\D(\\N)$), and for any $X$ in $\\mathcal{N}^{\\Sigma_n}$ that is cofibrant in $\\mathcal{N}$, then $g\\otimes_{\\Sigma_n} X$ is a local weak equivalence.\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$, $\\mathsf{O}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, $\\mathscr{P}$ an entrywise cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$, and the entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise local weak equivalence in $\\mathcal{N}$. \n\\end{enumerate}\nThen the lifted adjunction \\eqref{lbar.ocomp.diagram}\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O};\\lcm) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(\\sP;\\ldn) \\ar@<2.5pt>[l]^-{R}}\\]\nis a Quillen equivalence between the semi-model categories of $\\mathsf{O}$-algebras in $L_\\C(\\M)$ and of $\\mathscr{P}$-algebras in $L_\\D(\\N)$ (Theorem \\ref{theorem623}).\n\\end{theorem}\n\nWe also have a streamlined version for $\\Sigma$-cofibrant colored operads, that we state after the proof.\n\n\\begin{proof}\nThe definition of $\\mathcal{D}$ as $\\underline{L}\\mathcal{C}$ guarantees that the adjunction $(L,R)$ descends to an adjuction $L: L_\\C(\\M) \\hspace{-.1cm L_\\D(\\N) : R$, by Theorem 3.3.20 in \\cite{hirschhorn}. We will apply Theorem \\ref{main.theorem} to this adjunction. Condition (5) is the local version of condition (2) of Theorem \\ref{main.theorem}, since $\\mathsf{O}$ (resp., $\\mathscr{P}$) is entrywise cofibrant locally if and only if it is entrywise cofibrant in $\\mathcal{M}$ (resp., $\\mathcal{N}$). We are left to prove $L: L_\\C(\\M) \\hspace{-.1cm L_\\D(\\N) : R$ is a nice Quillen equivalence (Def. \\ref{def:nice.qeq}). It is a weak symmetric monoidal Quillen equivalence by condition (1) of the theorem. Note that this is a weaker condition than simply assuming $(L,R)$ is a weak symmetric monoidal Quillen equivalence relative to $\\mathcal{M}$ and $\\mathcal{N}$. \n\nNext, $(\\filledstar)$ only references the cofibrations, and so holds in $\\mathcal{M}$ if and only if it holds in $L_\\C(\\M)$, because whenever an object $X$ is $\\Sigma_n$-cofibrant in $\\mathcal{M}$, it is $\\Sigma_n$-cofibrant in $L_\\C(\\M)$. The same holds for $\\mathcal{N}$. The same argument shows that the domains of the generating cofibrations in $L_\\C(\\M)$ are cofibrant.\n\nWe have assumed $(\\#)$ and $(\\medstar)$ for $L_\\C(\\M)$ and $L_\\D(\\N)$, but we note that condition (3) above is weaker than simply assuming $(\\#)$ for $\\mathcal{M}$ and $\\mathcal{N}$, since every weak equivalence is a local weak equivalence. Similarly, assuming $(\\medstar)$ is weaker than the usual method of getting a functor to preserve local weak equivalences (namely, Theorem 3.3.18 in \\cite{hirschhorn}), because we do not need the functor $-\\otimes_{\\Sigma_n} X$ to be left Quillen.\n\nLastly, we have assumed $(\\clubsuit)$ in $L_\\C(\\M)$ and $L_\\D(\\N)$, and this implies that \\\\\n$L:L_\\C(\\M) \\hspace{-.1cm L_\\D(\\N):R$ is a nice Quillen equivalence. Note that $(\\clubsuit)_{\\cof}$ is the same in $\\mathcal{M}$ and in $L_\\C(\\M)$. Conditions guaranteeing $(\\clubsuit)_{\\tcof}$ to hold in any left Bousfield localization are given in \\cite{white-yau}. Examples include spaces, spectra, and chain complexes over a field of characteristic zero.\n\\end{proof}\n\nWe now state the version of this result for $\\Sigma$-cofibrant colored operads. The proof involves applying Theorem \\ref{main.theorem.Sigma} to the induced adjunction $L:L_\\C(\\M) \\hspace{-.1cm L_\\D(\\N):R$, as above.\n\n\\begin{theorem}[Lifting Local Quillen Equivalences for $\\Sigma$-Cofibrant Operads]\n\\label{main.theorem.local.Sigma}\nSuppose:\n\\begin{enumerate}\n\\item $L : \\mathcal{M} \\hspace{-.1cm \\mathcal{N} : R$ is a Quillen equivalence where $R$ is lax symmetric monoidal. Suppose\n\\begin{enumerate}\n\\item For all $X,Y$ cofibrant in $\\mathcal{M}$, the comonoidal map $L(X\\otimes Y)\\to LX \\otimes LY$ is a local weak equivalence in $\\mathcal{N}$.\n\\item For some cofibrant replacement $Q \\tensorunit^{\\M}$ of the unit $\\tensorunit^{\\M}$, the composition \\\\\n$LQ\\tensorunit^{\\M} \\to L \\tensorunit^{\\M} \\to \\tensorunit^{\\N}$ is a local weak equivalence in $\\mathcal{N}$.\n\\end{enumerate} \nRecall that $\\mathcal{D} = \\underline{L}\\mathcal{C}$.\n\\item The generating cofibrations in $\\mathcal{M}$ have cofibrant domain.\n\\item $f : \\mathsf{O} \\to R\\mathscr{P}$ is a map of $\\mathfrak{C}$-colored operads in $\\mathcal{M}$, $\\mathsf{O}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{M}$, $\\mathscr{P}$ a $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operad in $\\mathcal{N}$, and the entrywise adjoint $\\overline{f} : L\\mathsf{O} \\to \\mathscr{P}$ is an entrywise local weak equivalence in $\\mathcal{N}$. \n\\end{enumerate}\nThen the lifted adjunction \\eqref{lbar.ocomp.diagram}\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O};\\lcm) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(\\sP;\\ldn) \\ar@<2.5pt>[l]^-{R}}\\]\nis a Quillen equivalence between the semi-model categories of $\\mathsf{O}$-algebras in $L_\\C(\\M)$ and of $\\mathscr{P}$-algebras in $L_\\D(\\N)$.\n\\end{theorem}\n\n\\begin{remark}\nRecently, the first author and Michael Batanin have investigated local operadic algebras. In Theorem 3.4 of \\cite{batanin-white-eilenberg}, they prove that, whenever $\\mathsf{Alg}(\\mathsf{O};\\lcm)$ has a transferred semi-model structure, the Bousfield localization $L_{\\mathsf{O}(\\mathcal{C})} \\mathsf{Alg}(\\mathsf{O};\\M)$ exists and coincides with $\\mathsf{Alg}(\\mathsf{O};\\lcm)$, where $\\mathsf{O}(\\mathcal{C})$ denotes the free $\\mathsf{O}$-algebra maps on $\\mathcal{C}$. Combining this with Theorem \\ref{main.theorem.local} provides a Quillen equivalence between $L_{\\mathsf{O}(\\mathcal{C})} \\mathsf{Alg}(\\mathsf{O};\\M)$ and $L_{\\mathscr{P}(\\mathcal{D})} \\mathsf{Alg}(\\sP;\\N)$. This can be viewed as an enhancement to Theorem 3.3.20 in \\cite{hirschhorn}, as it allows for simultaneously changing the model category and the operad.\n\\end{remark}\n\n\\subsection{Special Cases}\n\nFollowing Section \\ref{sec:rect-and-change}, we provide several applications of the results above. We begin with rectification.\n\n\\begin{corollary}[Local Rectification]\n\\label{cor.rectification.local}\nSuppose $\\mathcal{M}$ is a cofibrantly generated monoidal model category, and $L_\\C(\\M)$ is a monoidal left Bousfield localization. Suppose $\\mathcal{M}$ satisfies $(\\filledstar)$ (Def. \\ref{def:star}), $(\\clubsuit)_{\\cof}$ (Def. \\ref{def:club}), and every generating cofibration has a cofibrant domain. Suppose $\\mathcal{M}$ satisfies local versions of $(\\medstar)$ and $(\\clubsuit)_{\\tcof}$ as in Theorem \\ref{main.theorem.local}. Suppose $\\mathfrak{C}$ is a set, and $f : \\mathsf{O} \\to \\mathscr{P}$ is a map of entrywise cofibrant $\\mathfrak{C}$-colored operads that is an entrywise $\\mathcal{C}$-local weak equivalence in $\\mathcal{M}$. Then the induced adjunction\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O};\\lcm) \\ar@<2.5pt>[r]^-{f_!} & \\mathsf{Alg}(\\sP;\\lcm) \\ar@<2.5pt>[l]^-{f^*}}\\]\nis a Quillen equivalence between semi-model categories.\n\\end{corollary}\n\n\\begin{corollary}[Local Rectification for $\\Sigma$-Cofibrant Operads]\n\\label{cor.rectification.local.sigma}\nSuppose $\\mathcal{M}$ is a cofibrantly generated monoidal model category, and $L_\\C(\\M)$ is a monoidal left Bousfield localization. Suppose that in $\\mathcal{M}$ every generating cofibration has a cofibrant domain. Suppose $\\mathfrak{C}$ is a set, and $f : \\mathsf{O} \\to \\mathscr{P}$ is a map of $\\Sigma$-cofibrant $\\mathfrak{C}$-colored operads that is an entrywise $\\mathcal{C}$-local weak equivalence in $\\mathcal{M}$. Then the induced adjunction\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(\\mathsf{O};\\lcm) \\ar@<2.5pt>[r]^-{f_!} & \\mathsf{Alg}(\\sP;\\lcm) \\ar@<2.5pt>[l]^-{f^*}}\\]\nis a Quillen equivalence between semi-model categories.\n\\end{corollary}\n\nWe turn now to modules, (commutative) monoids, (commutative) algebras, non-symmetric operads, generalized props, cyclic operads, and modular operads. \n\n\\begin{corollary}\nAssume that the conditions of Theorem \\ref{main.theorem.local} are satisfied (for $\\Sigma$-cofibrant situations, Theorem \\ref{main.theorem.local.Sigma} suffices).\n\\begin{enumerate}\n\\item Suppose $T$ is a (commutative) monoid that is cofibrant as an object in $\\mathcal{M}$, and $L$ is lax (symmetric) monoidal. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathbf{Mod}(T;L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathbf{Mod}(LT;L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of left $T$-modules in $L_\\C(\\M)$ and of left $LT$-modules in $L_\\D(\\N)$.\n\n\\item Suppose $T$ is a commutative monoid that is cofibrant as an object in $\\mathcal{M}$, and $L$ lax symmetric monoidal. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Alg}(T;L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Alg}(LT;L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}.\\]\n\n\\item Suppose $T$ is a commutative monoid that is cofibrant as an object in $\\mathcal{M}$, and $L$ lax symmetric monoidal. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{CAlg}(T;L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{CAlg}(LT;L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}.\\]\n\n\\item Suppose the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\Omega}(L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\Omega}(L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}.\\]\n\n\\item Suppose that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then for each pasting scheme $\\mathfrak{G}$ in the sense of \\cite{jy2} (Def. 8.2), there is an induced Quillen equivalence\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\mathsf{Prop}^{\\fG}(L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\mathsf{Prop}^{\\fG}(L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}\\]\nbetween the semi-model categories of $\\mathfrak{G}$-props \\cite{jy2} (10.39) in $L_\\C(\\M)$ and in $L_\\D(\\N)$.\n\n\\item Suppose that the tensor units in $\\mathcal{M}$ and $\\mathcal{N}$ are cofibrant. Then for each set $\\mathfrak{C}$, there are induced Quillen equivalences\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\fC}_{\\mathsf{cyc}}(L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\fC}_{\\mathsf{cyc}}(L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}\\]\nand\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{\\operad^{\\fC}_{\\mathsf{mod}}(L_\\C(\\M)) \\ar@<2.5pt>[r]^-{\\overline{L}} & \\operad^{\\fC}_{\\mathsf{mod}}(L_\\D(\\N)) \\ar@<2.5pt>[l]^-{R}}.\\]\n\\end{enumerate}\n\n\\end{corollary}\n\nTaking $T$ to be the tensor unit in (2) and (3) implies a Quillen equivalence for local (commutative) monoids.\n\n\n\\section{Applications to (Commutative) $HR$-Algebras, (C)DGAs, $E_\\infty$-Algebras, and Motivic Ring Spectra} \\label{sec:applications}\n\n\\subsection{Dold-Kan Equivalence}\n\nThe main application of \\cite{ss03} proves that the Dold-Kan equivalence lifts to categories of modules and algebras. This can be viewed as a special case of Theorem \\ref{main.theorem.Sigma}, since these are categories of algebras over $\\Sigma$-cofibrant operads (as explained in Section \\ref{sec:rect-and-change}), since the model categories involved have generating cofibrations with cofibrant domains, and since the Dold-Kan equivalence satisfies the conditions of Theorem \\ref{main.theorem.Sigma} (see Example \\ref{example:ss03}). \n\n\\subsection{(Commutative) DGAs and $HR$-Algebra Spectra}\n\nThe main theorem of \\cite{shipley-hz-spectra} proves that the model categories of $HR$-algebra spectra and unbounded differential graded $R$-algebras are Quillen equivalent, where $R$ is a discrete commutative ring. Shipley lifts the chain of Quillen equivalences (with left adjoints on top)\n\\[\\nicearrow\\xymatrix@C+10pt@R+10pt{ \nHR-Mod \\quad \n\\ar@<.4ex>^-{Z}[r] &\n\\quad \\mathsf{Sp}^\\Sigma (\\mathsf{sAb}) \\quad \n\\ar@<.4ex>^-{U}[l]\n\\ar@<-.4ex>_-{\\phi^*N}[r] &\n\\quad \\mathsf{Sp}^\\Sigma(\\mathsf{ch}) \\quad \n\\ar@<-.4ex>_-{L}[l]\n\\ar@<.4ex>^-{D}[r] &\n\\quad \\mathsf{Ch} \\quad \\ar@<.4ex>^-{R}[l]}\\]\nto the level of monoids. Here $HR$ is the Eilenberg-Maclane spectrum, $\\mathsf{sAb}$ is the category of simplicial $R$-modules, $\\mathsf{ch}$ is the category of non-negatively graded chain complexes of $R$-modules, $\\mathsf{Ch}$ is the category of unbounded chain complexes of $R$-modules with the projective model structure, and $\\mathsf{Sp}^{\\Sigma}(-)$ is a suitable category of symmetric spectra. The Quillen equivalences $(Z,U)$ and $(D,R)$ are strong symmetric monoidal, and the middle one $(L,\\phi^*N)$ is weak symmetric monoidal.\n\nShipley's main result may be viewed as a special case of Theorem \\ref{main.theorem.Sigma}, applied to each adjunction. Proposition 2.10 of \\cite{shipley-hz-spectra} demonstrates that the adjunctions all satisfy the conditions of Theorem \\ref{main.theorem.Sigma}. The domains of the generating cofibrations in all settings are cofibrant. To recover Shipley's result, in each of the three Quillen equivalences, $\\mathsf{O}$ and $\\mathscr{P}$ are both the operads whose algebras are monoids as in Corollary \\ref{cor.schwede.shipley}. A key point here is that the associative operad is $\\Sigma$-cofibrant, so Theorem \\ref{main.theorem.Sigma} is applicable. Similarly, Shipley's extension to modules (Theorem 2.13 of \\cite{shipley-hz-spectra}) can be viewed as a special case of Theorem \\ref{main.theorem.Sigma}.\n\nFurthermore, when $R$ has characteristic $0$ (which implies that $(\\filledstar)$, $(\\medstar)$ (Def. \\ref{def:star}), $(\\#)$ (Def. \\ref{def:sharp}), and $(\\clubsuit)$ (Def. \\ref{def:club}) are satisfied), Theorem \\ref{main.theorem} can be applied to each of Shipley's three Quillen equivalences with $\\mathsf{O}$ and $\\mathscr{P}$ the operads whose algebras are \\emph{commutative} monoids as in Corollary \\ref{cor.com.monoids}. This yields, in the characteristic $0$ setting, a zig-zag of three Quillen equivalences \n\\begin{equation}\\label{shipley.com}\n\\nicearrow\\xymatrix@C+10pt@R+10pt{ C(HR-Mod)\n\\ar@<.4ex>^-{Z}[r] &\nC\\Bigl(\\mathsf{Sp}^\\Sigma (\\mathsf{sAb})\\Bigr) \n\\ar@<.4ex>^-{U}[l]\n\\ar@<-.4ex>_-{\\phi^*N}[r] &\nC\\Bigl(\\mathsf{Sp}^\\Sigma(\\mathsf{ch})\\Bigr)\n\\ar@<-.4ex>_-{\\overline{L}}[l]\n\\ar@<.4ex>^-{D}[r] &\nC(\\mathsf{Ch}) \\ar@<.4ex>^-{R}[l]}\n\\end{equation}\nbetween the categories of commutative $HR$-algebra spectra and of commutative differential graded $R$-algebras. This confirms a belief expressed in \\cite{shipley-hz-spectra}. As we will discuss below, a zig-zag of Quillen equivalences between the same end categories is also achieved in \\cite{richter-shipley} (Corollary 8.4) using \\emph{six} Quillen equivalences instead of three here.\n\n\\subsection{Commutative $HR$-Algebra Spectra, CDGAs, and $E_\\infty$-Algebras}\n\nThe main theorem of \\cite{richter-shipley} proves a result analogous to the above, but for commutative $HR$-algebra spectra and $E_\\infty$-algebras in $\\mathsf{Ch}$ for a discrete commutative ring $R$. The chain of Quillen equivalences produced is now:\n\n$$ \\nicearrow\\xymatrix@C+10pt@R+10pt{\n{C(HR\\text{-mod})} \\ar@<0.5ex>[r]^Z &\n\\ar@<0.5ex>[l]^U {C(\\mathsf{Sp}^\\Sigma(\\mathsf{sAb}))}\n\\ar@<-0.5ex>[r]_N &\n\\ar@<-0.5ex>[l]_{L_N} {C(\\mathsf{Sp}^\\Sigma(\\mathsf{ch}))} \\ar@<0.5ex>[r]^i &\n\\ar@<0.5ex>[l]^{C_0} {C(\\mathsf{Sp}^\\Sigma(\\mathsf{Ch}))}\n\\ar@<-0.5ex>[d]_{\\varepsilon_*}\\\\\n & &\n{E_\\infty \\mathsf{Ch}} \\ar@<-0.5ex>[r]_-{\\ev_0}\n& \\ar@<-0.5ex>[l]_>>>>>>>{F_0} {E_\\infty(\\mathsf{Sp}^\\Sigma(\\mathsf{Ch}))}\n\\ar@<-0.5ex>[u]_{\\varepsilon^*}\n}$$\nThe Quillen equivalence in the bottom row is a special case of Theorem \\ref{main.theorem.Sigma}, because $E_\\infty$ operads are $\\Sigma$-cofibrant. The vertical Quillen equivalence is a special case of rectification. As the commutative operad is not $\\Sigma$-cofibrant, we need Theorem \\ref{main.theorem} in this setting. Unfortunately, we do not know if the conditions of this theorem are satisfied for symmetric spectra in $\\mathsf{Ch}$ for general $R$. We do, however, know that the conditions are satisfied if $R$ is replaced by a field $k$ of characteristic zero. Once this replacement is made, the vertical Quillen equivalence is a special case of Theorem \\ref{main.theorem}. Of the remaining three Quillen equivalences, the outer ones are induced by strong symmetric monoidal Quillen equivalences, while the inner one $(L_N, N)$ is induced by a weak symmetric monoidal Quillen equivalence. However, in the characteristic $0$ setting, this is enough to deduce $(\\#)$, $(\\filledstar), (\\medstar),$ and $(\\clubsuit)$. Thus, in the characteristic $0$ case, all five Quillen equivalences are special cases of Theorem \\ref{main.theorem}. \n\nFurthermore, in the characteristic $0$ setting, there is a rectification Quillen equivalence between $E_\\infty$-algebras in differential graded modules $E_\\infty\\mathsf{Ch}$ and commutative differential graded algebras $C(\\mathsf{Ch})$. In this case, the above zig-zag is prolonged to a zig-zag of six Quillen equivalences between commutative $Hk$-algebra spectra and commutative differential graded $k$-algebras, which is Corollary 8.4 in \\cite{richter-shipley}. However, as discussed in the previous section \\eqref{shipley.com}, using the main results of this paper, we actually have a zig-zag with the same end categories involving only three Quillen equivalences, which are Shipley's original adjunctions.\n\n\\subsection{Motivic Applications}\n\nThe final application of \\cite{dmitri-motivic} constructs a strictly commutative ring spectrum for Deligne cohomology. This is done by pushing a commutative differential graded algebra through a chain of Quillen equivalences terminating with strictly commutative motivic ring spectra. However, since both the starting category and the ending category admit rectification, one could instead view the CDGA as an $E_\\infty$-algebra in chain complexes, then use Theorem \\ref{main.theorem.Sigma} to push this object through a chain of lifted Quillen equivalences, and then use rectification in the positive stable model structure on motivic symmetric spectra (Theorem 3.10, \\cite{hornbostel}) to strictify the resulting $E_\\infty$-algebra into a strictly commutative ring spectrum.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}