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+{"text":"\\section{Introduction}\n\\begin{definition}\\label{def1.0}\nFor $\\lambda\\in \\mathbb{R}$ we denote by $e_\\lambda(x):=e^{2\\pi i\\lambda\\cdot x}$. We say that a finite Borel measure $\\mu$ on $\\mathbb{R}$ is {\\it spectral} if \nthere exists a set $\\Lambda$ such that the family of exponential functions $E(\\Lambda):=\\{e_\\lambda : \\lambda\\in\\Lambda\\}$ is an orthogonal basis for $L^2(\\mu)$. We call $\\Lambda$ a {\\it spectrum} for $\\mu$. If $E(\\Lambda)$ is an orthogonal set then we say that $\\Lambda$ is {\\it orthogonal}. \n\n\nWe say that a bounded Borel subset $\\Omega$ of $\\mathbb{R}$ is {\\it spectral} if the restriction of the Lebesgue measure to $\\Omega$ is a spectral measure.\n\n We say that a finite subset $A$ of $\\mathbb{R}$ is {\\it spectral} if the counting measure on $A$ is a spectral measure. \n\\end{definition}\n\n\nSpectral sets have been introduced in relation to the Fuglede conjecture \\cite{Fug74}:\n\\begin{conjecture}\nA bounded Borel subset $\\Omega$ of $\\mathbb{R}$ is spectral if and only if it tiles $\\mathbb{R}$ by translations, i.e., there exists a set $\\mathcal T$ in $\\mathbb{R}$ such that $\\{\\Omega+t : t\\in\\mathcal T\\}$ is a partition of $\\mathbb{R}$ (up to Lebesgue measure zero). \n\\end{conjecture}\n\nThe conjecture can be formulated in any dimension, but it is known to be false in both directions for dimensions 3 or higher \\cite{Tao04,FaMaMo06}. In dimensions 1 and 2, as far as we know at the moment of writing this paper, the conjecture is open in both directions. \n\n\nIn \\cite{JoPe98}, Jorgensen and Pedersen have constructed a new example of a spectral measure, a fractal one. Their construction is based on a scale 4 Cantor set, where the first and and third intervals are kept and the other two are discarded. The appropriate measure for this set is the Hausdorff measure $\\mu_4$ of dimension $\\frac{\\ln 2}{\\ln 4}=\\frac12$. They proved that this measure is spectral with spectrum \n\\begin{equation}\n\\Lambda:=\\left\\{\\sum_{k=0}^n 4^kl_k: l_k\\in\\{0,1\\}, n\\in\\mathbb{N}\\right\\}.\n\\label{eqspmu4}\n\\end{equation}\n\n\n\nMany other examples of fractal measures have been constructed since, see e.g. \\cite{MR1785282,LaWa02,DJ06,DJ07d}, and many other spectra can be constructed for the same measure, see e.g., \\cite{DHS09}. Among other things, we will show that the spectrum $\\Lambda$ in \\eqref{eqspmu4} tiles $\\mathbb{Z}$ by translations. \n\n\n\n\n\n\n\n\nA large class of examples of spectral measures is based on affine iterated function systems. \n\n\\begin{definition}\\label{def1.1}\nLet $R$ be an integer $R\\geq 2$. We call $R$ the {\\it scaling factor}. Let $B\\subset\\mathbb{Z}$, $0\\in\\mathbb{Z}$, $N:=\\#B$. We define the affine iterated function system\n$$\\tau_b(x)=R^{-1}(x+b),\\quad(x\\in\\mathbb{R},b\\in B).$$\nBy \\cite{Hut81} there exist a unique compact set $X_B$ called {\\it the attractor} of the IFS $\\{\\tau_b\\}$, such that \n$$X_B=\\cup_{b\\in B}\\tau_b(X_B).$$\nThe set $X_B$ can be described using the base $R$ representation of real numbers, with digits in $B$:\n$$X_B=\\left\\{\\sum_{k=1}^\\infty R^{-k}b_k : b_k\\in B\\right\\}.$$\nAlso by \\cite{Hut81}, there exists a unique Borel probability measure $\\mu_B$ on $\\mathbb{R}$ that satisfies the invariance equation \n\\begin{equation}\n\\mu_B(E)=\\frac{1}{N}\\sum_{b\\in B}\\mu_B(\\tau_b^{-1}E)\\mbox{ for all Borel subsets $E$ of $\\mathbb{R}$}.\n\\label{eq1.1.1}\n\\end{equation}\nEquivalently, for all continuous compactly supported functions $f$:\n\\begin{equation}\n\\int f\\,d\\mu_B=\\frac1N\\sum_{b\\in B}\\int f\\circ\\tau_b\\,d\\mu_B.\n\\label{eq1.1.2}\n\\end{equation}\nThe measure $\\mu_B$ is called {\\it the invariant measure} of the IFS $\\{\\tau_b\\}$. \nIn addition the measure $\\mu_B$ is supported on the attractor $X_B$.\n\\end{definition} \n\n\\begin{definition}\\label{def1.1i}\nLet $L\\subset\\mathbb{Z}$, $0\\in L$. We say that $(B,L)$ is a {\\it Hadamard pair with scaling factor $R$} if $\\#L=\\#B=N$ and the matrix \n$$\\frac{1}{\\sqrt N}\\left(e^{2\\pi i R^{-1}b\\cdot l}\\right)_{b\\in B,l\\in L}$$\nis unitary. We call this matrix {\\it the matrix associated with $(B,L)$}.\n\nWe define the function \n\\begin{equation}\nm_B(x)=\\frac{1}{N}\\sum_{b\\in B}e^{2\\pi ib\\cdot x},\\quad (x\\in\\mathbb{R})\n\\label{eq1.1i.1}\n\\end{equation}\n\nGiven a Hadamard pair $(B,L)$ we say that a finite set of points $\\{x_0,\\dots,x_{r-1}\\}$ in $\\mathbb{R}$ is a {\\it cycle} for $L$ if there exist $l_0,\\dots,l_{r-1}$ in $L$ such that \n$$\\frac{x_0+l_0}{R}=x_1,\\dots,\\frac{x_{r-2}+l_{r-2}}{R}=x_{r-1},\\frac{x_{r-1}+l_{r-1}}{R}=x_0.$$\nWe call $l_0,\\dots,l_{r-1}$ {\\it the digits of this cycle.}\nWe say that this cycle is {\\it extreme} for $(B,L)$ if  \n$$|m_B(x_k)|=1\\mbox{ for all }k\\in\\{0,\\dots,r-1\\}.$$\nThe points $\\{x_i\\}$ are called {\\it (extreme) cycle points}.\n\\end{definition}\n\nWhen $(B,L)$ is a Hadamard pair with scaling factor $R$, then the measure $\\mu_B$ is always spectral and a spectrum can be constructed using digits in $L$ and extreme cycles. \n\n\\begin{theorem}\\label{th1.2}\\cite{DJ06}\nIf $(B,L)$ is a Hadamard pair then $\\mu_B$ is a spectral measure with spectrum $\\Lambda(L)$ where $\\Lambda$ is the smallest set which contains $-C$ for all cycles $C$ for $L$ which are extreme for $(B,L)$, and with the property that $R\\Lambda(L)+L\\subset \\Lambda(L)$. \n\\end{theorem}\n\nThis spectrum can be described in terms of base $R$ representations of integers using only digits in $L$. \n\n\\begin{definition}\\label{def1.2}\nLet $L$ be a set of integers. We say that an integer $x$ can be represented in base $R$ using digits in $L$ if there exist integers $x_0,x_1,\\dots$, with $x_0=x$ and digits $l_0,l_1,\\dots$ in $L$ such that \n$$x_k=Rx_{k+1}+l_k\\mbox{ for all $k\\geq0$}.$$\nWe call $l_0l_1\\dots$ a {\\it representation} of $x$ in base $R$. \n\\end{definition}\n\n\n\n\n\\begin{proposition}\\label{pr0.1.6}\nLet $(B,L)$ be a Hadamard pair. Assume in addition that all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then the spectrum $\\Lambda(L)$ defined in Theorem \\ref{th1.2} is the set of integers which can be represented in base $R$ using digits in $L$.\n\\end{proposition}\n\n\n\nNext we turn our attention to finite spectral subsets of $\\mathbb{Z}$. The variant of the Fuglede conjecture for such sets is that a finite subset $A$ of $\\mathbb{Z}$ is spectral if and only if it tiles $\\mathbb{Z}$ by translations. In \\cite{CoMe99}, Coven and Meyerowitz proposed a characterization of sets that tile integers by translations, in terms of cyclotomic polynomials. \n\\begin{definition}\\label{def2.1.1}\nLet $A$ be a finite multiset of nonnegative integers, by multiset we mean that some elements $a\\in A$ might be counted with multiplicity $m_a$. We define the polynomial corresponding to $A$ by \n$$A(x)=\\sum_{a\\in A}m_ax^a.$$\n\nFor $s\\in\\mathbb{N}$, we denote by $\\Phi_s(x)$ the $s$-th cyclotomic polynomial. We denote by $S_A$ the set of all prime powers such that the $s$-th cyclotomic polynomial divides $A(x)$. \n\n\nWe say that the set $A$ (without any multiplicity) {\\it satisfies the Coven-Meyerowitz property (or shortly, $A$ has the CM-property) }if the following two conditions are satsisfied:\n\\begin{enumerate}\n\t\\item[(T1)] $A(1)=\\prod_{s\\in S_A}\\Phi_s(1).$\n\t\\item[(T2)] If $s_1,\\dots,s_m\\in S_A$ are powers of distinct primes then $\\Phi_{s_1\\dots s_m}(x)$ divides $A(x)$.\n\\end{enumerate} \n\\end{definition}\n\nCoven and Meyerowitz proved in \\cite{CoMe99} that a set with the CM-property tiles $\\mathbb{Z}$ by translations and they conjectured that the reverse is also true, and proved the conjecture in some special cases (when the size of the set has at most two prime factors). \\mathcal{L} aba proved in \\cite{Lab02} that the CM-property also implies that the set is spectral. Combining these results we show that the tiling sets and spectra fit together nicely in a {\\it complementary pair}. We are also interested in the extreme cycles due to their importance for the spectra of fractal measures. \n\n\n\\begin{definition}\\label{def2.0}\nLet $A,A'$ be two subsets of $\\mathbb{R}$. We say that $A$ and $A'$ {\\it have disjoint differences} if $(A-A)\\cap(A'-A')=\\{0\\}$. In this case we denote by \n$A\\oplus A'=\\{a+a' : a\\in A, a'\\in A'\\}$; we use the sign $\\oplus$ to indicate that the sets have disjoint differences; equivalently, for any $x\\in A+A'$ there exist unique $a\\in A$ and $a'\\in A'$ such that $x=a+a'$; equivalently, the sets $A+a'$, $a'\\in A'$ are disjoint. \n\\end{definition}\n\n\\begin{definition}\\label{def2.1}\nLet $R\\in\\mathbb{Z}$, $R\\geq 2$. Let $(B,L)$ and $(B',L')$ be two Hadamard pairs with scaling factor $R$, $\\#B=N$, $\\#B'=N'$, not necessarily equal. We say that the two Hadamard pairs are {\\it complementary} if the following conditions are satisfied: \n\\begin{enumerate}\n\t\\item $B\\oplus B'$ and $L\\oplus L'$ are complete sets of representatives $\\mod R$.\n\t\\item The extreme cycles for $(B,L)$ and the extreme cycles for $(B',L')$ are contained in $\\mathbb{Z}$.\n\t\\item The greatest common divisor of the points in $B\\oplus B'$ is 1.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\\begin{theorem}\\label{th0.2.1.2}\nLet $B$ a finite set of nonnegative integers with $\\gcd(B)=1$ and which satsifies the Coven-Meyerowitz property. Let $R$ be the lowest common multiple of the elements in $S_B$. Then there exist finite sets $B',L,L'$ of nonnegatve integers such that \n\\begin{enumerate}\n\t\\item $(B,L)$ and $(B',L')$ are complementary Hadamard pairs (relative to the number $R$).\n\t\\item $B'$ satisfies the Coven-Meyerowitz property.\n\\end{enumerate}\n\\end{theorem}\n\nOnce we have two complementary Hadamard pairs $(B,L)$, $(B',L')$ with scaling factor $R$, we can construct the two fractal measures $\\mu_B$ and $\\mu_{B'}$ with spectra $\\Lambda(L)$ and $\\Lambda(L')$ respectively. The next theorem shows that the convolution of the two measures $\\mu_B$ and $\\mu_{B'}$ is the Lebesgue measure on a tile of $\\mathbb{R}$, it is also the invariant measure $\\mu_{B\\oplus B'}$ for the affine IFS associated to scaling by $R$ and digits $B\\oplus B'$. The two spectra always have disjoint differences and moreover, under some restrictions on the encodings of the extreme cycles for $(B\\oplus B',L\\oplus L')$, $(B,L)$ and $(B',L')$, the two sets complement each other, in the sense that $\\Lambda(L)$ tiles $\\mathbb{Z}$ with $\\Lambda(L')$.\n\n\n\n\\begin{definition}\\label{def2.2}\nLet $(B,L)$ and $(B',L')$ be complementary Hadamard pairs with scaling factor $R$. We define the maps $p:L\\oplus L'\\rightarrow L$ and $p':L\\oplus L'\\rightarrow L'$ by \n$$p(l+l')=l,\\quad p'(l+l')=l'\\mbox{ for all }l\\in L ,l'\\in L'.$$\nFor a sequence $a_0a_1\\dots$ of digits in $L\\oplus L'$ we define \n$$p(a_0a_1\\dots)=p(a_0)p(a_1)\\dots,\\quad p'(a_0a_1\\dots)=p'(a_0)p'(a_1)\\dots.$$\n\\end{definition}\n\n\\begin{theorem}\\label{th2.3}\nLet $(B,L)$ and $(B',L')$ be complementary Hadamard pairs with scaling factor $R$. Let $\\Lambda(L)$ be the set of integers that can be represented in base $R$ using digits from $L$, and similarly for $\\Lambda(L')$.\n\\begin{enumerate}\n\t\\item The measure $\\mu_{B\\oplus B'}$ is the Lebesgue measure on the attractor $X_{B\\oplus B'}$ and has spectrum $\\mathbb{Z}$. Moreover $X_{B\\oplus B'}$ is translation congruent to $[0,1]$, i.e., there exists a measurable partition $\\{A_n\\}_{n\\in\\mathbb{Z}}$ of $[0,1]$ such that $\\{A_n+n\\}_{n\\in\\mathbb{Z}}$ is a partition of $X_{B\\oplus B'}$. \n\t\\item The measure $\\mu_{B\\oplus B'}$ is the convolution of the measures $\\mu_B$ and $\\mu_{B'}$. \n\t\\item The set $\\Lambda(L)$ is a spectrum for $\\mu_B$ and the set $\\Lambda(L')$ is a spectrum for $\\mu_{B'}$.\n\t\\item The sets $\\Lambda(L)$ and $\\Lambda(L')$ have disjoint differences. \t\n\t\\item The set $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$ if and only if for any digits $a_0\\dots a_{r-1}$ of an extreme cycle for $(B\\oplus B',L\\oplus L')$, the sequence $p(a_0\\dots a_{r-1})$ consists of the digits of an extreme cycle for $(B,L)$ and the sequence $p'(a_0\\dots a_{r-1})$ consists of the digits of an extreme cycle for $(B',L')$. \n\tThe equality $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$ means that $\\Lambda(L)$ tiles $\\mathbb{Z}$ by $\\Lambda(L')$.\n\\end{enumerate}\n\n\\end{theorem}\n\n\n\n\n\n\nNext, we focus on sets $B$ of small size: 2,3,4,5 and investigate when such a set is spectral and when a Hadamard pair with scaling factor $R$ can be complemented. We base our results on the classification of Hadamard matrices of size 2,3,4,5. For size $\\#B=2,3,4$ this is fairly simple, see \\cite{TaZy06}. For size 5, the problem becomes more complicated but it was solved by Haagerup \\cite{Haa97}.\n\n\\begin{definition}\nA $N\\times N$ matrix $H$ is called a {\\it Hadamard matrix} if it is unitary and all its entries have the same absolute value $\\frac{1}{\\sqrt N}$. Two Hadamard matrices $H$, $H$' are said to be {\\it equivalent} if one can be obtained from the other after permutations of row and columns and multiplication of rows and columns by complex numbers of absolute value 1; formally: there exist permutation $\\pi$ and $\\rho$ of the set $\\{1,\\dots,N\\}$ and complex numbers $c_1,\\dots,c_N,d_1,\\dots,d_N$ on the unit circle $\\mathbb{T}=\\{z: |z|=1\\}$ such that \n$$H_{ij}'=c_id_jH_{\\pi(i)\\rho(j)},\\quad(i,j\\in\\{1,\\dots,N\\}).$$\nThe matrix of the Fourier transform on $\\mathbb{Z}_N$, $\\frac{1}{\\sqrt{N}}(e^{2\\pi i\\frac{jk}{N}})_{j,k=0}^{N-1}$ is called the {\\it standard Hadamard matrix}.\n\n\\end{definition}\n\n\n\\begin{theorem}\\label{th1.15}(See \\cite{TaZy06,Haa97})\nLet $N=2,3$ or $5$. Any Hadamard matrix of size $N$ is equivalent to the standard Hadamard matrix. If $N=4$, any $4\\times 4$ Hadamard matrix is equivalent to one of the following form:\n\\begin{equation}\\frac12\n\\begin{pmatrix}\n1&1&1&1\\\\\n1&-1&\\rho&-\\rho\\\\\n1&-1&-\\rho&\\rho\\\\\n1&1&-1&-1\n\\end{pmatrix}\n\\label{eqmat4}\n\\end{equation}\nfor some $\\rho\\in\\mathbb{T}$.\n\\end{theorem}\n\nAs far as we know, there is no classification for Hadamard matrices of size 6 or higher. Beauchamp and Nicoar\\u a gave a classification of {\\it self-adjoint} $6\\times 6$ Hadamard matrices in \\cite{MR2398121}.\n\n\n\nA Hadamard matrix is said to be in de-phased form if its first row and column contain only the number $1$.\n\n\n\\begin{corollary} \\label{perm}\nLet $N=2$, $3$, $4$, or $5$. Any two Hadamard matrices $A$ and $B$ of size $N$ in de-phased form which are equivalent are also equivalent via permutations only, that is, there are permutation matrices $P_1$ and $P_2$ such that $A=P_1 B P_2$.\n\n\\end{corollary}\n\\begin{definition}\\label{def0.3.1}\nLet $B$ be a finite spectral subset of $\\mathbb{R}$ with spectrum $\\Lambda$, $\\#B=\\#\\Lambda=:N$. The matrix \n$$\\frac{1}{\\sqrt{N}}(e^{2\\pi ib\\cdot\\lambda})_{b\\in B,\\lambda\\in\\Lambda}$$\nis a Hadamard matrix and we called it {\\it the Hadamard matrix associated to $B$ and $\\Lambda$}.\n\\end{definition}\n\nThis enables us to describe the spectral sets of size $2,3,4,5$.\n\n\\begin{theorem} \\label{standard}\nLet $B \\subset \\mathbb{Z}$ have $N$ elements and spectrum $\\Lambda$. Assume $0$ is in $B$ and $\\Lambda$. Suppose the Hadamard matrix associated to $(B,\\Lambda)$ is equivalent to the standard $N$ by $N$ Hadamard matrix. Then $B$ has the form $B=d B_0$ where $d$ is an integer and $B$ is a complete set of residues modulo $N$ with $\\gcd(B)=1$. In this case any such spectrum $\\Lambda$ has the form $\\Lambda = \\frac{1}{R} f L_0$ where $f$ and $R$ are integers, $L_0$ is a complete set of residues modulo $N$ with greatest common divisor one, and $R=NS$ where $S$ divides $df$ and $\\frac{df}{S}$ is mutually prime with $N$. The converse also holds.\n\n\\end{theorem}\n\n\n\n\n\n\n\\begin{corollary}\\label{pr0.1}A set $B \\subset \\mathbb{Z}$ with $|B|=N=2$, $3$, or $5$, where $0 \\in B$ is spectral if and only if $B= N^k B_0$ where $k$ is a positive integer and $B_0$ is a complete set of residues modulo $N$.\n\\end{corollary}\n\n\n\nWe can also describe all possible Hadamard pairs of size 2,3,4,5.\n\n\n\\begin{theorem}\\label{thha4}\nLet $B$ be spectral with spectrum $\\Lambda$ and size $N=4$. Assume $0$ is in both sets. Then there exists a set of integers $L$, containing $0$, and an integer scaling factor $R$ so that $\\Lambda= \\frac{1}{R} L$.\n\n$(B,L)$ is a Hadamard pair (each containing $0$) of integers of size $N=4$, with scaling factor $R$, if and only if $R=2^{C+M+a+1} d$, $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$, and $L=2^M \\{0, n_1, n_1 + 2^a n_2, 2^a n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ is a positive integer, $C$ and $M$ are non-negative integers, and $d$ divides $c_1 n$, $c_3 n$, $n_2 c$, and $n_3 c$, where $c$ is the greatest common divisor of the $c_k$'s and similarly for $n$.\n\\end{theorem}\n\nUsing the classification of Hadamard matrices of small dimension we can also show that Hadamard pairs of size 2,3,4,5 can always be complemented. We can give a more general result:\n\n\\begin{theorem}\\label{th0.4a}\nLet $(B,L)$ be a Hadamard pair of integers of size $N$ (containing zero as their first element), with scaling factor an integer $R$, where the matrix associated with $(B,L)$ is equivalent to the $N\\times N$ standard Hadamard matrix. Assume that all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then $(B,L)$ has a complementary Hadamard pair of integers.\n\\end{theorem}\n\n\n \n\n\n\n\n\\begin{theorem}\\label{th0.5a}\nLet $(B,L)$ be a Hadamard pair of size $|B|=|L|=2,3,4$ or $5$, with scaling factor $R$, and assume all extreme cycles for $(B,L)$ are contained in $\\mathbb{Z}$. Then $(B,L)$ has a complementary Hadamard pair.\n\\end{theorem}\n\nThe cases 2,3,5 follow imediately from Theorem \\ref{th0.4a} since the Hadamard matrix associated to the pair $(B,L)$ has to be equivalent to the standard one. For size 4, the situation is different. \n\nA useful tool for our construction of Hadamard pairs is the following proposition, which is closely relation to Di\\c{t}\\u a's construction of Hadamard matrices (see e.g. \\cite{TaZy06}):\n\n\\begin{proposition} \\label{prHP}\n Let $(B,L)$ and $(F,G)$ be Hadamard pairs of integers with the same scaling factor $R$ and such that $b\\cdot g$ is a multiple of $R$ for every $g\\in G$ and $b\\in B$. Then $(B\\oplus F, L \\oplus G)$ is a Hadamard pair with scaling factor $R$.\n \\end{proposition}\n \n\n\nFinally, we study spectral sets with Lebesgue measure as part of the original Fuglede conjecture. A wonderful result due to Iosevich and Kolountzakis \\cite{IoKo12} states that the spectrum $\\Lambda$ of a bounded spectral subset $\\Omega$ of $\\mathbb{R}$ has to be periodic. More precisely\n\\begin{theorem}(\\cite{IoKo12}) Let $\\Omega$ be a bounded Borel subset of $\\mathbb{R}$ with Lebesgue measure $|\\Omega|=1$. If $\\Omega$ is spectral with spectrum $\\Lambda$ then $\\Lambda$ is periodic, i.e., there exists $p>0$ such that $\\Lambda+p=\\Lambda$; moreover the period $p$ is an integer. \n\\end{theorem}\n\n\n\\begin{definition}\\label{def0.7}\nLet $p\\in\\mathbb{N}$, we say that {\\it spectral implies tile for period $p$} if every bounded Borel subset $\\Omega$ of $\\mathbb{R}$, with Lebesgue measure $|\\Omega|=1$ and which has a spectrum $\\Lambda$ of period $p$, is also a tile. \n\\end{definition}\n\nIn the original paper \\cite{Fug74}, Fuglede proved that his conjecture is true in the case when the spectrum or the tiling set is a lattice. This corresponds to the case of period equal to 1. We prove that spectral implies tile for periods 2,3,4,5. \n\n\n\\begin{theorem}\\label{th0.9}\nSpectral implies tile for period $2$, $3$, $4$, or $5$. \n\\end{theorem}\n\nWe end the paper with some examples to illustrate our results. Example \\ref{ex4.1} shows that in the well known Jorgensen Pedersen example, of a scale 4 Cantor set, the spectrum $\\Lambda$ described in \\eqref{eqspmu4} tiles $\\mathbb{Z}$ with translations and the tiling set is the spectrum of a complementary fractal measure. \n\n\n\\section{Proofs and other results}\n\n\n\\subsection{Spectra of fractals and base $R$ representations of integers}\n\n\\begin{proposition}\\label{pr1.2}\nLet $d$ be the greatest common divisor of the points in $B$. Let $M=\\max\\{ l: l\\in L\\}$, $m=\\min\\{ l: l\\in L\\}$. Then for every extreme cycle point $x$ for $(B,L)$ we have $x\\in\\frac1d\\mathbb{Z}$ and $\\frac m{R-1}\\leq x\\leq \\frac{M}{R-1}$. \n\\end{proposition}\n\n\\begin{proof}\nSince $|m_B(x)|=1$ and $0\\in B$, using the triangle inequality we obtain that $e^{2\\pi i bx}=1$ for all $b\\in B$. Therefore $bx\\in\\mathbb{Z}$ for all $b\\in B$. \nThis implies that $dx\\in\\mathbb{Z}$ so $x\\in\\frac1d\\mathbb{Z}$. \n\nLet $x=x_0, x_1,\\dots, x_{r-1}$ be a cycle for $L$, with digits $l_0,\\dots, l_{r-1}$. Then we have \n$$\\frac{x_0+l_0+Rl_1+\\dots+R^{r-1}l_{r-1}}{R^r}=x_0\\mbox{ so }x_0=\\frac{l_0+Rl_1+\\dots+R^{r-1}l_{r-1}}{R^r-1}$$\nwhich implies\n$$x_0\\leq \\frac{M\\frac{R^{r}-1}{R-1}}{R^r-1}=\\frac{M}{R-1},$$\nand similarly for the lower bound. \n\\end{proof}\n\n\\begin{proposition}\\label{pr1.3}\nLet $L$ be a complete set of representatives $\\mod R$. Then every integer $x$ has a unique representation in base $R$ using digits in $L$. Moreover any such representation $l_0l_1\\dots$ is eventually periodic, i.e., there exists $n_0\\geq0$ and $r\\geq 1$ such that $l_{n+r}=l_n$ for all $n\\geq n_0$.\n\\end{proposition}\n\n\\begin{proof}\nLet $x\\in\\mathbb{Z}$ and $x_0=x$. Since $L$ is a complete set of representatives $\\mod R$, there is a unique $l_0\\in L$ and some $x_1\\in\\mathbb{Z}$ such that $x_0=Rx_1+l_0$. By induction, we obtain the sequence $\\{x_n\\}$ and $\\{l_n\\}$ in a unique way. We have to show that the sequence $\\{l_n\\}$ is eventually periodic. \nLet $M=\\max\\{|l| : l\\in L\\}$. By induction we can show that, for all $n\\in\\mathbb{N}$,\n$$x_n=\\frac{x_0-l_0-Rl_1-\\dots-R^{n-1}l_{n-1}}{R^n}.$$\nThis implies that for $n$ large enough such that $|x_0\/R^n|\\leq 1$ we have \n$$| x_n|\\leq 1+M\\left(\\frac{1}R+\\dots+\\frac{1}{R^{n-1}}\\right)\\leq1+ \\frac M{R-1}.$$\nSo $x_n$ lies in a compact interval from some point on. But $x_n$ is also an integer so the numbers $x_n$ take finitely many values. Therefore there exists $n_0\\geq 0$ and $r\\geq 1$ such that $x_{n_0}=x_{n_0+r}$. This implies that $l_{n_0}=l_{n_0+r}$ and $x_{n_0+1}=x_{n_0+r+1}$. By induction $l_{n}=l_{n+r}$ for all $n\\geq  n_0$.  \n\\end{proof}\n\n\\begin{definition}\\label{def1.4}\nIf $L$ is a complete set of representatives $\\mod R$, we write $x=l_0l_1\\dots$ if $l_0l_1\\dots$ is the base $R$ representation of $x$ using digits in $L$. For $l_0,\\dots, l_{r-1}$ in $L$ we denote by $\\uln{l_0\\dots l_{r-1}}$ the periodic sequence $l_0\\dots l_{r-1}l_0\\dots l_{r-1}\\dots$. If $x=\\uln{l_0\\dots l_{r-1}}$ for some $l_0,\\dots,l_{r-1}\\in L$, we say that $x$ has a {\\it periodic} representation in base $R$ using digits in $L$.\n\nWe say that $l_0\\dots l_{r-1}$ is a {\\it cycle} for $L$ if there exists an integer that has base $R$ representation equal to $\\uln{l_0\\dots l_{r-1}}$.\n\\end{definition}\n\n\n\\begin{proposition}\\label{pr1.5}\nIf $\\{x_0,\\dots,x_{r-1}\\}$ is a cycle for $L$ with digits $l_0,\\dots, l_{r-1}$, and if $x_0\\in\\mathbb{Z}$ then $x_1,\\dots, x_{r-1}\\in\\mathbb{Z}$ and the points $-x_0,\\dots,-x_{r-1}$ have periodic expansions in base $R$ using digits in $L$:\n\\begin{equation}\n-x_0=\\uln{l_0\\dots l_{r-1}},\\quad -x_1=\\uln{l_1\\dots l_{r-1}l_0},\\quad\\dots\\quad, -x_{r-1}=\\uln{l_{r-1}l_0\\dots l_{r-2}}.\n\\label{eq1.5.1}\n\\end{equation}\nConversely, if $-x_0\\in\\mathbb{Z}$ has a periodic expansion in base $R$ using digits in $L$ , $-x_0=\\uln{l_0\\dots l_{r-1}}$, and we define \n$$x_1=-\\,\\uln{l_1\\dots l_{r-1}l_0},\\quad\\dots\\quad, x_{r-1}=-\\,\\uln{l_{r-1}l_0\\dots l_{r-2}},$$\nthen $\\{x_0,\\dots, x_{r-1}\\}$ is a cycle for $L$ contained in $\\mathbb{Z}$. \n\n\\end{proposition}\n\n\\begin{proof}\nIf $\\{x_0,x_1\\dots,x_{r-1}\\}$ is a cycle for $L$ with digits $l_0,\\dots, l_{r-1}$ then $-x_{r-1}=R(-x_0)+l_{r-1}$ so $x_{r-1}\\in\\mathbb{Z}$. By induction, all points in this cycle are in $\\mathbb{Z}$. We have also $-x_0=R(-x_1)+l_0$, $-x_1=R(-x_2)+l_1,\\dots$. This shows that $-x_0=\\uln{l_0\\dots,l_{r-1}}$, $-x_1=\\uln{l_1\\dots l_{r-1}l_0}$, etc.\n\nFor the converse, if $-x_0=\\uln{l_0\\dots l_{r-1}}$ then $-x_0=R(-x_1)+l_0$ and the number $-x_1$ will have the representation $\\uln{l_1\\dots l_{r-1}l_0}$. \nThis implies also $(x_0+l_0)\/R=x_1$. The rest follows by induction.\n\\end{proof}\n\n\n\\begin{myproof}[Proof of Proposition \\ref{pr0.1.6}]\nFirst, note that the points in $L$ are incongruent $\\mod R$. Indeed if $l-l'=Rk$ for some $k\\in\\mathbb{Z}$, then from the unitarity of the matrix in Definition \\ref{def1.1i} we have\n$$0=\\frac1N\\sum_{b\\in B}e^{2\\pi i R^{-1}b\\cdot(l-l')}=\\frac{1}{N}\\sum_{b\\in B}e^{2\\pi i b\\cdot k}=1,$$\na contradiction.\nFrom Proposition \\ref{pr1.5} we see that for any extreme cycle point $x$, the point $-x$ has a periodic representation using digits in $L$. Also, if $x$ is an integer that has a periodic representation using digits in $L$, then $-x$ is an cycle point in $\\mathbb{Z}$. Since $-x$ is in $\\mathbb{Z}$ it follows that $m_B(-x)=1$ so $-x$ is an extreme cycle point for $(B,L)$. \n\nThis implies that the set $\\Lambda'$ of integers that can be represented in base $R$ using digits in $L$, contains $-C$ for all extreme cycles $C$. We show that $R\\Lambda'+L\\subset \\Lambda'$. \nIf $x\\in\\Lambda'$, $x=l_0l_1\\dots$ then $Rx+l_{-1}=l_{-1}l_0l_1\\dots$ so $Rx+l_{-1}\\in \\Lambda'$ for any $l_{-1}\\in L$. \n\nThe minimality of $\\Lambda(L)$ implies that $\\Lambda(L)\\subset\\Lambda'$. To obtain the converse inclusion, take \n$x\\in\\Lambda'$. With Proposition \\ref{pr1.3}, $x$ has an eventually periodic expansion \n$x=k_0\\dots k_{n-1}\\uln{l_0\\dots l_{r-1}}$. If $c=\\uln{l_0\\dots l_{r-1}}$ then $x=k_0+R k_1+\\dots+R^{n-1} k_{n-1}+R^n c$. We have that $-c$ is an extreme cycle point so $c$ is in $\\Lambda(L)$. By the invariance of $\\Lambda(L)$ we get that $x\\in\\Lambda(L)$. So $\\Lambda'\\subset\\Lambda$.  \n\\end{myproof}\n\n\n\n\n\n\n\n\n\\subsection{The Coven-Meyerowitz property and complementary Hadamard pairs}\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.2.1.2}]\nThe hard part of this theorem was covered in \\cite[Theorem A]{CoMe99} where the tiling property is proved, i.e., the existence of the set $B'$ such that $B\\oplus B'=\\mathbb{Z}_R$, and in \\cite[Theorem 1.5]{Lab02} where it is shown the spectral property, i.e., the existence of the set $L$. We will include parts of their proofs here to be able to get some more information. \n\n\nThe set $B'$ is defined as follows: first, define the polynomial $B'(x)=\\prod \\Phi_s(x^{t(s)})$ where the product is take over all the prime power factors of $R$ which are not in $S_A$ and $t(s)$ is the largest factor of $R$ which is prime to $s$. It is shown in \\cite{CoMe99} that this polynomial has coefficients $0$ or $1$, therefore it corresponds to a set $B'$, and $B\\oplus B'=\\mathbb{Z}_R$ (addition modulo $R$). \n\nTake a number $s$ that appears in the product that defines $B'(x)$. Since $s$ is a prime power, say $s=p^\\alpha$, the cyclotomic polynomial is of the form $\\Phi_s(x)=1+x^{p^{\\alpha-1}}+x^{2p^{\\alpha-1}}+\\dots+x^{(p-1)p^{\\alpha-1}}$ (see e.g. \\cite[Lemma 1.1]{CoMe99}). Hence\n$$\\Phi_s(x^{t(s)})=1+x^{p^{\\alpha-1}t(s)}+x^{2p^{\\alpha-1}t(s)}+\\dots+x^{(p-1)p^{\\alpha-1}t(s)}.$$\n\n So all the coefficients are nonnegative for all the factors that appear in this product. Therefore, there are no cancelations. This implies that $x^{p^{\\alpha-1}t(s)}$ appears with a positive coefficient in $B'(x)$. So $p^{\\alpha-1}t(s)$ is in $B'$. The greatest common divisor of the elements in $B'$ must divide $p^{\\alpha-1}t(s)$ which divides $st(s)$, and by the definition of $t(s)$ this will divide $R$. \n \n Therefore we have that $\\gcd(B')$ divides $R$. We will use this property to show that all the extreme cycles for the Hadamard pair $(B',L')$ are in $\\mathbb{Z}$.\n \n \n Since $B\\oplus B'=\\mathbb{Z}_R$, we have by \\cite[Lemma1.3]{CoMe99} that for any prime power $s$ that divides $R$, the cyclotomic polynomial $\\Phi_s(x)$ divides $B(x)$ or $B'(x)$. Then, with \\cite[Lemma 2.1]{CoMe99}, we obtain that $S_B$ and $S_{B'}$ are disjoint sets whose union is the set of all prime power factors of $R$, and also \n $B'(1)=\\prod_{s\\in S_{B'}}\\Phi_s(1)$, so the (T1) property is satsisfied by $B'$.\n \n To see that the (T2) property is satsisfied by $B'$ we follow again the proof of \\cite[Theorem A]{CoMe99}: it is shown there that if $s=s_1\\dots s_m$ is a product of distinct prime power factors of $R$ and $s_i$ is not in $S_B$, then $\\Phi_s(x)$ divides $\\Phi_{s_i}(t(x))$ (\\cite[Lemma 1.1.(6)]{CoMe99}) so it divides $B'(x)$. So, if all $s_1,\\dots,s_m$ are in $S_{B'}$, then they will not be in $S_B$ so $\\Phi_s(x)$ will divide $B'(x)$, which proves (T2). Hence $B'$ has the CM-property.\n \n Since $\\gcd(B)=1$, we have also $\\gcd(B\\oplus B')=1$.\n \n \n Now we take care of the spectral part. We use the proof of \\cite[Theorem 1.5]{Lab02}. The set $L$ will contain all sums of the form $R\\cdot \\sum_{s\\in S_B}\\frac{k_s}{s}$ where $s\\in S_B$, $s=p^\\alpha$ for some $\\alpha>0$, $p$ prime and $k_s\\in\\{0,\\dots,p-1\\}$. Since $B$ satsifies the CM-property, it is shown in \\cite{Lab02} that $(B,L)$ is a Hadamard pair. Obviously $L$ is a subset of $\\mathbb{Z}$, since the elements of $S_B$ divide $R$. Similarly we can construct $L'$ for $B'$, since we showed that $B'$ has the CM-property. \n \n Next we show that $L\\oplus L'=\\mathbb{Z}_R$. We have $|L|\\cdot |L'|=|B|\\cdot|B'|=R$. \n We prove that $(L-L)\\cap (L'-L')=\\{0\\}$ (in $\\mathbb{Z}_R$). Suppose we have \n \\begin{equation}\\label{eqdisj2.1}\nR\\cdot\\sum_{s\\in S_B}\\frac{k_s-l_s}{s}=R\\cdot \\sum_{s\\in S_{B'}}\\frac{k_s-l_s}{s}\\quad\\mod R,\n\\end{equation}\nwhere $k_s$, $l_s$ for $s$ in either $S_B$ or $S_{B'}$ are as above. We proved above that $S_B$ and $S_{B'}$ are disjoint and their union consists of all prime power factors of $R$. \n \n  Take some prime $p$ that divides $R$ and let $s=p^\\alpha$ be the largest power that divides $R$. Then $s$ appears in one of the sums in \\eqref{eqdisj2.1}, and $R\\cdot\\frac{k_s-l_s}{s}$ is not divisible by $p$ unless $k_s=l_s$. For all the other elements $s'\\in S_B\\cup S_{B'}$ the numbers $R\\cdot\\frac{k_{s'}-l_{s'}}{s'}$ are divisible by $p$. Therefore, the equality \\eqref{eqdisj2.1} implies that $k_s=l_s$. By induction we assume that $k_s=l_s$ for all $s$ in $S_B$ or $S_{B'}$ of the form $s=p^\\beta$ with $1\\leq \\gamma\\leq \\beta\\leq\\alpha$. Then consider $s=p^{\\gamma-1}$, which is in either $S_B$ or $S_{B'}$. Then $R\\frac{k_s-l_s}{s}$ is not divisible by $p^{\\alpha-\\gamma+2}$ unless $k_s=l_s$ and for the other $s'\\in S_B\\cup S_{B'}$ for which $k_s\\neq l_s$, the number $R\\frac{k_{s'}-l_{s'}}{s'}$ is divisible by $p^{\\alpha-\\gamma+2}$. Using equation \\eqref{eqdisj2.1} we obtain that $k_s=l_s$. Therefore for all the powers $s$ of $p$ that appear in either $S_B$ or $S_{B'}$ we have $k_s=l_s$. Since the prime $p$ was an arbitrary prime factor of $R$, we get that $k_s=l_s$ for all $s\\in S_B\\cup S_{B'}$. Hence $(L-L)\\cap(L'-L')=\\{0\\}$ in $\\mathbb{Z}_R$. This means that the map from $L\\times L'$ to $\\mathbb{Z}_R$, $(l,l')\\mapsto l+l'\\mod R$ is injective. But since $|L\\times L'|=R$ the map will be also surjective so $L\\oplus L'=\\mathbb{Z}_R$.\n  \n  It remains to deal with the extreme cycles. By Proposition \\ref{pr1.2}, since $\\gcd(B)=1$, we have that the extreme cycles for $(B,L)$ are in $\\mathbb{Z}$. \n  We also proved above that $d':=\\gcd(B')$ divides $R$. By Proposition \\ref{pr1.2} any extreme cycle for $(B',L')$ is contained in $\\frac1{d'}\\mathbb{Z}$. Take the first two points in such a cycle $x_0=\\frac{k_0}{d'}$, $x_1=\\frac{k_1}{d'}$, and for some $l_0'\\in L'$:\n  $$\\frac{\\frac{k_0}{d'}+l_0'}{R}=\\frac{k_1}{d'}.$$\n  Then $$\\frac{k_0}{d'}+l_0'=R\\cdot\\frac{k_1}{d'}.$$\n  But since $R$ is divisble by $d'$, it follows that $R\\cdot\\frac{k_1}{d'}$ is in $\\mathbb{Z}$; also $l_0'$ is in $\\mathbb{Z}$, so $x_0=\\frac{k_0}{d'}$ is in $\\mathbb{Z}$. Thus all extreme cycles for $(B',L')$ are contained in $\\mathbb{Z}$.\n \n \n \n \n \n\\end{myproof}\n\n\\begin{myproof}[Proof of Theorem \\ref{th2.3}]\nSince $L\\oplus L'$ and $B\\oplus B'$ are complete sets of representatives $\\mod R$, they form a Hadamard pair. With Proposition \\ref{pr1.2}, we have that all extreme cycles for $(B\\oplus B', L\\oplus L')$ are contained in $\\mathbb{Z}$. With Propositions \\ref{pr1.3} and \\ref{pr0.1.6} we see that $\\mathbb{Z}$ is the spectrum for $\\mu_{B\\oplus B'}$. Using the results from \\cite{DJ11} we obtain that $\\mu_{B\\oplus B'}$ is the Lebesgue measure on its support $X_{B\\oplus B'}$ and $X_{B\\oplus B'}$ is translation congruent to $[0,1]$. \n\n\nFor (ii), note that \n$m_{B\\oplus B'}=m_B\\cdot m_{B'}$. According to \\cite{DJ06}, the Fourier transforms of the measures are \n$$\\widehat\\mu_{B\\oplus B'}(x)=\\prod_{k=1}^\\infty m_{B\\oplus B'}(R^{-k}x)$$\nand similarly for $\\mu_B$ and $\\mu_{B'}$ and the products are uniformly and absolutely convergent on compact sets. Therefore $\\widehat\\mu_{B\\oplus B'}=\\widehat\\mu_B\\cdot\\widehat\\mu_{B'}$ which implies that $\\mu_{B\\oplus B'}=\\mu_B*\\mu_{B'}$.\n\n\n(iii) follows from Proposition \\ref{pr0.1.6}.\n\nFor (iv), let $\\lambda=l_0l_1\\dots$, $\\lambda'=l_0'l_1'\\dots$, $\\gamma=k_0k_1\\dots$, $\\gamma'=k_0'k_1'\\dots$ such that $\\lambda-\\gamma=\\lambda'-\\gamma'$. Reducing $\\mod R$, we obtain $l_0-k_0\\equiv l_0'-k_0'\\mod R$. This implies $l_0+k_0'\\equiv l_0'+k_0\\mod R$, but since $L\\oplus L'$ is a complete set of representatives $\\mod R$, we get $l_0+k_0'=l_0'+k_0$ so $l_0-k_0=l_0'-k_0'$. Since $L$ and $L'$ have disjoint differences, it follows that $l_0=k_0$ and $l_0'=k_0'$. Then, by induction $l_n=k_n$ and $l_n'=k_n'$ for all $n$, so $\\lambda=\\gamma$ and $\\lambda'=\\gamma'$. \n\n\nFor (v), take an extreme cycle for $(B\\oplus B',L\\oplus L')$ with digits $a_0\\dots a_{r-1}$. Then $x=\\uln{a_0\\dots a_{r-1}}$ is a point in $\\mathbb{Z}=\\Lambda(L)\\oplus \\Lambda(L')$. Thus $x=l_0l_1\\dots + l_0'l_1'\\dots$. This implies that $a_0\\equiv l_0+l_0'\\mod R$. Since $L\\oplus L'$ is a complete set of representatives $\\mod R$, this means that $a_0=l_0+l_0'$ and $l_0=p(a_0)$, $l_0'=p'(a_0)$. By induction $l_n=p(a_n)$ and $l_n'=p'(a_n)$ for all $n$. So $l_0l_1\\dots=p(\\uln{a_0\\dots a_{r-1}})$ and $l_0'l_1'\\dots=p'(\\uln{a_0\\dots a_{r-1}})$, so $p(a_0\\dots a_{r-1})$ contains the digits of an extreme cycle for $(B,L)$ and similarly for $p'$.\n\nFor the converse, note that $R(\\Lambda(L)\\oplus\\Lambda(L'))+L\\oplus L'\\subset \\Lambda(L)\\oplus \\Lambda(L')$. So, by Proposition \\ref{pr0.1.6} and Proposition \\ref{pr1.5}, it is enough to show that $\\Lambda(L)\\oplus\\Lambda(L')$ contains all points $\\uln{a_0\\dots a_{r-1}}$ where $a_0\\dots a_{r-1}$ are the digits of an extreme cycle for $(B\\oplus B',L\\oplus L')$. But the hypothesis implies that $p(\\uln{a_0\\dots a_{r-1}})$ represents a point in $\\Lambda(L)$ and $p'(\\uln{a_0\\dots a_{r-1}})$ represents a point in $\\Lambda(L')$. One can easily see that \n$$\\uln{a_0\\dots a_{r-1}}=p(\\uln{a_0\\dots a_{r-1}})+p'(\\uln{a_0\\dots a_{r-1}})$$\nbecause the two sides are congruent $\\mod R^n$ for all $n$. This implies that $\\uln{a_0\\dots a_{r-1}}\\in\\Lambda(L)\\oplus\\Lambda(L')$.\n\n\\end{myproof}\n\n\n\\begin{proposition}\\label{pr3.4}\nLet $R$ be an integer $R\\geq 2$. Let $B,B'$ finite sets of integers such that $B\\oplus B'=\\{0,1,\\dots,R-1\\}$. Then $\\mu_B*\\mu_{B'}$ is the Lebesgue measure on $[0,1]$. If $\\Lambda$ is an orthogonal set for $\\mu_B$ and $\\Lambda'$ is an orthogonal set for $\\mu_{B'}$ then $\\Lambda$ and $\\Lambda'$ have disjoint differences.\n\\end{proposition}\n\n\\begin{proof}\nThe proof that $\\mu_B*\\mu_{B'}$ is the Lebesgue measure on $[0,1]$ is the same as the proof of Theorem \\ref{th2.3}(i) and (ii), the attractor of the IFS associated to $B\\oplus B'=\\{0,\\dots,R-1\\}$ is $[0,1]$. \nTake $\\lambda\\neq\\gamma$ in $\\Lambda$ and $\\lambda'\\neq\\gamma'$ in $\\Lambda'$ such that $\\lambda-\\gamma=\\lambda'-\\gamma'$. Since $e_\\lambda$ is orthogonal to $e_\\gamma$, we have that $\\widehat\\mu_B(\\lambda-\\gamma)=0$. Also $\\widehat\\mu_{B'}(\\lambda'-\\gamma')=0$. But $\\widehat\\mu_B$ and $\\widehat\\mu_{B'}$ can be extended to entire functions \n$$\\widehat\\mu_B(z)=\\int e^{-2\\pi itz}\\,\\mu_B(t),\\quad(z\\in\\mathbb{C})$$\nand similarly for $\\mu_B'$. Their product is the Fourier transform of the Lebesgue measure on $[0,1]$ which is \n$$\\widehat\\mu_{B\\oplus B'}(z)=e^{-\\pi i z}\\frac{\\sin\\pi z}{\\pi z}.$$\nThe zeros of $\\widehat\\mu_{B\\oplus B'}$ on $\\mathbb{R}$ have multiplicity one. But the relations above shows that $\\lambda-\\gamma=\\lambda'-\\gamma'$ is zero of multiplicity at least 2 for $\\widehat\\mu_B\\cdot\\widehat\\mu_{B'}=\\widehat\\mu_{B\\oplus B'}$. This contradiction proves the conclusion.\n\\end{proof}\n\n\n\n\\subsection{Finite sets of size 2,3,4,5}\n\n\n\\begin{remark}\\label{remf1}\nWe will often ignore the multiplicative constant $\\frac{1}{\\sqrt{N}}$ for Hadamard matrices. So, when we say that some number $z$ with $|z|=1$ is an entry in a Hadamard matrix, we actually mean that $\\frac{1}{\\sqrt{N}}z$ is.\n\n\n\nWe also note here that many times the study of a spectral set $B$ in $\\mathbb{Z}$ with spectrum $\\Lambda$ in $\\mathbb{R}$ can be reduced to the study of Hadamard pairs, so we can assume in addition that $\\Lambda$ has the form $\\Lambda=\\frac{1}{R}L$ for some $R$ integer and $L$ in $\\mathbb{Z}$. First, we examine what happens if $\\Lambda$ has only rational numbers. \n\n\nIf $\\beta$ is a finite subset of the rational numbers, and $\\Lambda$ is a spectrum of rational numbers for $\\beta$, then $B,L$ is a Hadamard pair with scaling factor $RQ$, where $R$ is the least common multiple of the denominators of the numbers in $\\Lambda$ and $Q$ the least common multiple of the the denominators of the numbers in $\\beta$, and $L=R \\Lambda$, $B=Q \\beta$.\n\n\n\nIndeed, the matrix associated with $(\\beta,\\Lambda)$ is unitary, and therefore so is the matrix associated with $(Q \\beta, R \\Lambda)$ with scaling factor $QR$.\n\n\n\nNow assume $\\beta$ is a finite subset of the rational numbers and $\\Lambda$ is a spectrum of real numbers for $\\beta$. If the unitary matrix associated with $(\\beta, \\Lambda)$ has at least one column which contains only roots of unity, then $\\Lambda$ must contain only rational numbers, because the entries of that column are $e^{2 \\pi i b \\lambda_j}$ for all $\\lambda_j \\in \\Lambda$. Thus, whenever we know such a thing about the columns of the Hadamard matrices for a certain size $N=\\#B$, we know from the above theorem that when considering spectra (for finite sets of integers), it is sufficient to consider Hadamard pairs. For example, this property holds true of $N=2$, $3$, $4$, and $5$. \n\nFor the remainder of the section we assume Hadamard pairs $(B,L)$ are such that $B$ and $L$ each contain $0$ as their first element, and due to the above notions we restrict our attention to Hadamard pairs which are subsets of $\\mathbb{Z}$.\n\nIt is clear that the first row and first column of a Hadamard matrix associated to such a Hadamard pair must contain only $1$'s (ignoring the multiplicative constant $\\frac1{\\sqrt N}$). Therefore, when we consider Hadamard matrices in this section, we consider only the ones which are in \"de-phased\" form, i.e. their first row and column contains only $1$'s. For any Hadamard matrix $H$ there are diagonal matrices $D_1$ and $D_2$ so that $D_1 H D_2$ is de-phased (see e.g. \\cite{TaZy06}), so we lose no generality in dealing with matrices in this way. In addition, we only consider one ordering of the rows and the columns of a Hadamard matrix, for changing the ordering of the rows and the columns corresponds to changing the ordering of the elements in $B$ and $L$. Since we know the equivalence classes of the Hadamard matrices for $N=2$, $3$, $4$, and $5$, and we know by Corollary \\ref{perm} that everything in those equivalence classes are permutation equivalent, we lose no generality in dealing with Hadamard matrices in this way.\n\n\n\n\\end{remark}\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{standard}]\n\nFirst, we need some lemmas.\n\n\\begin{lemma}\nLet $H$, $H'$ be two equivalent Hadamard matrices whose first rows and columns are constant $\\frac{1}{\\sqrt{N}}$. Then there exist permutations $\\pi$, $\\psi$ of $\\{1,...,N\\}$ such that\n\\begin{equation}\n{H'}_{j,k} = \\frac{H_{\\pi (1) \\psi (1)} H_{\\pi (j) \\psi (k)} }{\\sqrt{N} H_{\\pi (j) \\psi (1)} H_{\\pi (1) \\psi (k)} } .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $H$ and $H'$ are equivalent, there are permutations $\\pi$ and $\\psi$ and constants $c_1,...,c_N$, $d_1,...,d_N \\in \\mathbb{T}$ (the unit circle) such that\n\\begin{equation}\nH'_{j,k} = c_j d_k H_{\\pi (j) \\psi (k)} .\n\\end{equation}\nSince $H'_{j,1} = \\frac{1}{\\sqrt{N}} = c_j d_1 H_{\\pi (j) \\psi (1)} $, we obtain $c_j = \\frac{1}{\\sqrt{N} d_1 H_{\\pi (j) \\psi (1)} } $. Similarly, $d_k = \\frac{1}{\\sqrt{N} c_1 H_{\\pi (1) \\psi (k)} } $. Since $H'_{1,1} = \\frac{1}{\\sqrt{N}} = c_1 d_1 H_{\\pi (1) \\psi (1)}$, we obtain\n\\begin{equation}\n{H'}_{j,k} = \\frac{ H_{\\pi (j) \\psi (k)} }{{N} c_1 d_1 H_{\\pi (j) \\psi (1)} H_{\\pi (1) \\psi (k)} } ,\n\\end{equation}\nand the result follows.\n\n\n\\end{proof}\n\n\\begin{lemma}\\label{lem2.8}\nLet $H$ be a Hadamard matrix whose first row and columns are constant $\\frac{1}{\\sqrt{N}}$. Suppose $H$ is equivalent to the standard Hadamard matrix of size $N$. Then this matrix is permutation equivalent to the standard Hadamard matrix.\n\n\\end{lemma}\n\n\\begin{proof}\n Using the previous lemma, we find permutations $\\tau,\\psi$ of $\\{0,1,2,\\dots,N-1\\}$ such that\n\\begin{equation}\nH_{j,k} = \\frac{e^\\frac{2 \\pi i \\tau (j) \\psi (k) }{N} e^\\frac{2 \\pi i \\tau (1) \\psi (1) }{N} }{\\sqrt{N} e^\\frac{2 \\pi i \\tau (1) \\psi (k) }{N} e^\\frac{2 \\pi i \\tau (j) \\psi (1) }{N} } = \\frac{1}{\\sqrt{N}} e^\\frac{2 \\pi i ( \\tau (j) - \\tau (1) )( \\psi (k) - \\psi (1) }{N} .\n\\end{equation}\nNow notice that modulo $N$, the functions $\\tau'(j)= \\tau(j)-\\tau(1)$ and $\\psi'(k)=\\psi(k)-\\psi(1)$ are permutations of $\\{0,1,2,\\dots,N-1\\}$. Thus $H$ is permutation equivalent to the standard Hadamard matrix.\n\n\n\\end{proof}\n\nNow assume that $B \\subset \\mathbb{Z}$ with spectrum $\\Lambda$ has $N$ elements, and $0$ is in both sets, and the matrix associated with $B$ and $\\Lambda$ is equivalent to the standard Hadamard matrix of size $N$. If the greatest common divisor $d$ of $B$ is $1$, we may perform our calculations on the sets $\\frac{1}{d} B$ and $d \\Lambda$, which have the same associated matrix. Therefore, we assume without loss of generality that the greatest common divisor of $B$ is $1$. \n\nWe apply the lemma above, and relabel the elements in $B$, so that $C$ is a permutation of $B$ and $\\Gamma$ is a permutation of $\\Lambda$, with elements $c_0=0,c_1,\\dots,c_{N-1}$ and $\\gamma_0=0,\\gamma_1,\\dots,\\gamma_{N-1}$ respectively, and the matrix associated to $C$ and $\\Gamma$ is the standard Hadamard matrix of size $N$. From the second row of this matrix, we obtain (here $i$ is the complex number, not an index)\n\\begin{equation}\ne^{2 \\pi i  c_j \\gamma_1 } = e^{ 2 \\pi i  j\/N  } ,\n\\end{equation}\nso $c_j \\gamma_1 = \\frac{j}{N} + m_j$ for some integers $m_j$. Then we write $\\gamma_1 = \\frac{z_1}{z_2}$ in lowest terms, as it is a rational number. Thus $c_j \\frac{z_1}{z_2} = \\frac{j+N m_j}{N} $. Taking $j=1$, we find that $z_2$ is divisible by $N$, so we let $z_2 = N z_3$. Thus $c_j z_1 = (j+N m_j) z_3$. Thus, since $z_1$ and $z_3$ are mutually prime, $z_3$ divides $c_j$ for all $j$. Since we know the greatest common divisor of $C$ is one, $z_3 = 1$. Thus $c_j z_1 \\equiv j$ modulo $N$, so $C$ is a complete set of residues modulo $N$. Therefore, so is $B$.\n\nTo prove that we can take $\\gcd(B_0)=1$, suppose $\\gcd(B_0)=e$. Then $\\frac{1}{e}B_0$ is again a complete set of representatives modulo $N$; indeed, if $\\frac{b_1}{e}\\equiv\\frac{b_2}{e}\\mod N$ then $b_1\\equiv b_2\\mod N$ so $b_1=b_2$. Also $\\gcd(B_0)=1$ and we can write $B=dB_0=de\\frac{1}{e}B_0$. \n\nNow we consider $\\Lambda$. Examining the second column of the standard matrix, we find that $e^{2 \\pi i c_1 \\gamma_k} = e^{2 \\pi i k\/N}$. Therefore $\\gamma_k$ is rational for all $k$, so $\\Lambda$ is a set of rational numbers. Let $R$ be their lowest common denominator. Then $\\Lambda = \\frac{1}{R} L$ where $L$ is a set of integers containing zero. Thus $L$ is spectral with spectrum $\\frac{1}{R} B$. So $L = f L_0$ where $L_0$ is a complete set of residues modulo $N$ with greatest common divisor one.\n\nWe now have that $(B,L)$ is a Hadamard pair with scaling factor $R$, whose matrix $H$ is equivalent (and thus permutation equivalent) to the standard Hadamard matrix. We assume without loss of generality that $H$ is the standard Hadamard matrix (after changing the order of the elements in $B$ and $L$). We let the elements in $B_0$ and $L_0$ be $b_j$ and $l_k$ respectively, $b_0=l_0=0$. Then\n\\begin{equation}\ne^{\\frac{2 \\pi i d f b_j l_k}{R}} = e^{\\frac{2 \\pi i j k}{N}}.\n\\end{equation}\nThus there are integers $m_{j,k}$ such that\n\\begin{equation}\nNdfb_j l_k = R(jk+ Nm_{j,k}) .\n\\end{equation}\nLetting $j=k=1$, we have that $N$ divides $R$ and thus $R=NS$. Thus\n\\begin{equation}\ndfb_j l_k = S(jk+ Nm_{j,k}) .\n\\end{equation}\nThus $S$ divides $dfW$ where $W$ is the product of the greatest common divisors of $B_0$ and $L_0$, and thus $W=1$. Therefore $S$ divides $df$, so $df=St$. Thus\n\\begin{equation}\ntb_j l_k = jk+ Nm_{j,k} .\n\\end{equation}\nThus $tb_1 l_1 = 1+ Nm_{j,k}$ so $t=\\frac{df}{S}$ is mutually prime with $N$.\n\nConversely, let $B= d B_0$, $L= f L_0$ and $R = NS$ where $S$ divides $df$ and that quotient $t$ is mutually prime with $N$. Assume $B_0$ and $L_0$ are complete sets of residues modulo $N$. Since $t$ is mutually prime with $N$, $t B_0$ is a complete set of residues modulo $N$. Reorder $t B_0$ and $L_0$ from least to greatest modulo $N$. Then the matrix associated with $B$ and $L$ with scaling factor $R$ has entries\n\\begin{equation}\ne^{\\frac{2 \\pi i d f b_j l_k}{R} } = e^{\\frac{2 \\pi i t b_j k}{N} } = e^{\\frac{2 \\pi i  j k}{N} }.\n\\end{equation}\nThus the matrix associated with $B,L$ with scaling factor $R$ is equivalent to the standard Hadamard matrix of size $N$, so $B,L$ is a Hadamard pair with scaling factor $R$. The same reasoning applies to any spectrum of rational numbers which meets the criteria.\n\n\\end{myproof}\n\n\nThe above classifies the Hadamard pairs for a certain class which contains all Hadamard pairs of size $N=2$, $3$, and $5$. More specifically, we have the next item.\n\n\\begin{myproof}[Proof of Corollary \\ref{pr0.1}]\nAll Hadamard matrices of size $2$, $3$ and $5$ are equivalent to the respective standard Hadamard matrices of those sizes, so by Theorem \\ref{standard} the spectral sets are in the form $B=H_0 B_0$. We know $B_0$ contains $0$, and that $B_0$ is a complete set of residues modulo $N$. We let $H_0 = q N^k$ with $q$ not divisible by $N$. Then, since $N$ is prime in this special case, $q$ is an automorphism of the integers modulo $N$, so $N^k q B_0 = N^k B_1$ where $B_1=qB_0$ is a complete set of residues modulo $N$ which contains $0$.\n\n\\end{myproof}\n\n\n\n\n\n\n\nWe now move on to $N=4$, a case where there are other types of Hadamard matrices.\n\n\n\\begin{myproof}[Proof of Corollary \\ref{perm}]\nFor $N$ equal to $2$, $3$, or $5$, all dephased Hadamard matrices are equivalent to the standard one, so by Lemma \\ref{lem2.8} they are permutation equivalent to it.\n\nLet $N=4$. Let $A$ and $B$ be equivalent de-phased Hadamard matrices, where\n$$A=\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&q&-q\\\\\n  1&-1&-q&q\\\\\n  1&1&-1&-1\n\\end{pmatrix}.\n$$\nWe shall prove that $B$ is permutation equivalent to $A$.\nBefore we proceed, let us prove a lemma:\n\\begin{lemma}\\label{lem3i.1}\nIf the numbers $\\alpha,\\beta,\\gamma\\in\\mathbb{T}=\\{z\\in\\mathbb{C} : |z|=1\\}$ satisfy the relation\n\\begin{equation}\n1+\\alpha+\\beta+\\gamma=0,\n\\label{eq3i.1.1}\n\\end{equation}\nthen one of them must be $-1$.\n\\end{lemma}\n\n\\begin{proof}\nTake the conjugate in \\eqref{eq3i.1.1} and multiply by $\\alpha\\beta\\gamma$: $\\alpha\\beta\\gamma+\\alpha\\beta+\\alpha\\gamma+\\beta\\gamma=0$. Multiply \\eqref{eq3i.1.1} by $\\alpha\\beta$: $\\alpha\\beta+\\alpha^2\\beta+\\alpha\\beta^2+\\alpha\\beta\\gamma=0$. Now substract these two reltations to obtain\n$0=\\alpha\\beta^2-\\beta\\gamma+\\alpha^2\\beta-\\alpha\\gamma=(\\beta+\\alpha)(\\alpha\\beta-\\gamma)$ so\n$\\alpha+\\beta=0$ or $\\alpha\\beta=\\gamma$. Similarly, by symmetry, we obtain $\\alpha+\\gamma=0$ or $\\alpha\\gamma=\\beta$. Also $\\beta+\\gamma=0$ or $\\beta\\gamma=\\alpha$.\n\n\nIf $\\alpha+\\beta=0$ then, using \\eqref{eq3i.1.1}, we get that $\\gamma=-1$. Therefore, if one of these relations is true, then the lemma is proved. If none is true, then we must have $\\alpha\\beta=\\gamma$ and $\\alpha\\gamma=\\beta$ and $\\beta\\gamma=\\alpha$. Multiply them: $\\alpha^2\\beta^2\\gamma^2=\\alpha\\beta\\gamma$, so $\\alpha\\beta\\gamma=1$. Multiply the first relation by $\\gamma$, we obtain that $\\gamma^2=1$. Similarly $\\alpha^2=1$ and $\\beta^2=1$. So $\\alpha,\\beta,\\gamma=\\pm 1$. We cannot have all of them $1$, because of \\eqref{eq3i.1.1}, therefore one has to be -1.\n\\end{proof}\n\nTherefore the matrix $B$ has the number negative one in every row and every column. \nThus each row and column has a 1 and -1. Consider now the other entries of the matrix which are not $\\pm 1$. If we fix one, denote it by $t$ then the other non $\\pm1$ entries which lie on the same row or column will have to be $-t$, because of \\eqref{eq3i.1.1}. Using the same procedure we can fill out some more entries by $t$ and all the entries of the matrix are completely determined in this way. Now suppose we have two rows such that the entries 1 and $-1$ do not match, for example $(1, -1, \\ast,\\ast)$ and $(1,\\ast, -1,\\ast)$. Then\nthe two rows will be of the form $(1,-1, t,-t)$ and $(1,-t,-1,t)$. By orthogonality, we get $0=1+\\cj t-t-t\\cj t=\\cj t-t$. So $t$ has to be real so $t=\\pm1$.\n\nIf we have two rows such that the $\\pm1$ entries match, for example $(1,-1,t,-t)$ and $(1,-1, -t, t)$, then the last row is forced to be $(1,1,-1,-1)$.\nThus, in both cases, one of the rows has to have two ones and two $-1$. Similarly, one of the columns has the same property. Thus $B$ is permutation equivalent to a matrix of the form:\n\n\n$$C=\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&t&-t\\\\\n  1&-1&-t&t\\\\\n  1&1&-1&-1\n\\end{pmatrix}.\n$$\nTherefore $A$ is equivalent to $C$. Now let us prove another lemma. This lemma is found in \\cite{TaZy06}, where it is attributed to Haagerup \\cite{Haa97}, though it does not appear in its present form in \\cite{Haa97}.\n\n\\begin{lemma}\\label{haag}\nLet $H$ be a Hadamard matrix and consider the set $T(H) = \\{ H_{j,k} \\overline{H_{n,k}} H_{n,m} \\overline{H_{j,m}} \\}$. If $A$ and $B$ are equivalent Hadamard matrices, $T(A) = T(B)$. The set $T$ is called the invariants of a Hadamard matrix. \n\n\n\\end{lemma}\n\n\\begin{proof}\nAssume $A$ and $B$ are permutation equivalent. Then, since they have the same entries, $T(A) = T(B)$. Then it is sufficient to prove that if $A$ and $C$ are equivalent via diagonal matrices $X$ and $Y$, so that $A = X C Y$, then they have the same invariants. Note that $A_{j,k} = X_{j,j} C_{j,k} Y_{k,k}$. We compute the elements of $T(A)=T(XCY)$:\n\\begin{equation}\nA_{j,k} \\overline{A_{n,k}} A_{n,m} \\overline{A_{j,m}} = X_{j,j} C_{j,k} Y_{k,k} \\overline{X_{n,n} C_{n,k} Y_{k,k}} X_{n,n} C_{n,m} Y_{m,m} \\overline{X_{j,j} C_{j,m} Y_{m,m}}\n\\end{equation}\nSimplifying the right hand side, we then obtain the desired result as $X_{q,q} \\overline{X_{q,q}} =1$.\n\n\\end{proof}\n\nNow notice that the above lemma implies that if two Hadamard matrices are equivalent and de-phased, they have the same elements (we let $j$ and $k$ be any numbers and the rest of the indices be $1$). Therefore, examining $A$ and $C$, we can see that $t = \\pm q$, so $A$ and $C$ are permutation equivalent. But $B$ and $C$ are permutation equivalent, so $A$ and $B$ are permutation equivalent. \n\n\n\n\n\\end{myproof}\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{thha4}]\n\nWe first prove that the spectra can be decomposed as $\\frac1R L$. We know from Theorem \\ref{th1.15} and Corollary \\ref{perm} that the matrix associated with $B$ and $\\Lambda$ is permutation equivalent with a matrix that has a $-1$ in every row except the first. Therefore for each non-zero element $\\lambda_k$ of $\\Lambda$ there is a $j$ such that $b_j \\in B$ and $e^{2 \\pi i b_j \\lambda_k} = -1$. Therefore since $b_j$ are integers, $\\Lambda$ is a set of rational numbers. So we let $\\Lambda = \\frac{1}{R} L$ where $L$ is a set of integers containing $0$, and we have the result.\n\n Using Theorem \\ref{th1.15} and Corollary \\ref{perm} we have that the matrix $H:=\\frac1{\\sqrt{4}}\\left(e^{2\\pi ib\\lambda}\\right)_{b\\in B,\\lambda\\in\\Lambda}$ is of the form given in \\eqref{eqmat4}, after some pemutations of $B$ and $\\Lambda$. This means, upon some relabelling, that we have for some $\\lambda\\in\\Lambda$ and $B=\\{0,b_1,b_2,b_3\\}$: $e^{2\\pi ib_1\\lambda}=1$, $e^{2\\pi i b_2\\lambda}=-1$, $e^{2\\pi ib_3\\lambda}=-1$. Therefore $b_1\\lambda=k_1, b_2\\lambda=\\frac{2k_2+1}2,b_3\\lambda=\\frac{2k_3+1}2$ for some $k_1,k_2,k_3\\in\\mathbb{Z}$.\n\nWe can write $b_1=2^{a_1}c_1$, $b_2=2^{a_2}c_2$, $b_3=2^{a_3}c_3$ with $a_1,a_2,a_3\\geq0$ in $\\mathbb{Z}$ and $c_1,c_2,c_3$ odd. We get that $\\frac{2^{a_1}c_1}{2^{a_2}c_2}=\\frac{2k_1}{2k_2+1}$ so $2^{a_1}c_1(2k_2+1)=2^{a_2+1}k_1c_2$. This implies that $a_1\\geq a_2+1$.\n\nAlso $\\frac{2^{a_2}c_2}{2^{a_3}c_3}=\\frac{2k_2+1}{2k_3+1}$ so $2^{a_2}c_2(2k_3+1)=2^{a_3}c_3(2k_2+1)$, which implies that $a_2=a_3$.\n\nSince $B$ is spectral iff $\\frac{1}{2^{a_2}}B$ is spectral, we can assume, without loss of generality, dividing by $\\frac{1}{2^{a_1}}$, that $B$ is of the form\n$$B=\\{0,2^ac_1,c_2,c_3\\},$$\nwith $a\\geq 1$, $c_1,c_2,c_3$ odd.\n\nSince every row has a $-1$, there is a $\\lambda_2\\in\\Lambda$ such that $e^{2\\pi i 2^a c_1\\lambda_2}=-1$. Therefore $2^{a+1}c_1\\lambda_2=2m+1$ for some $m\\in\\mathbb{Z}$. So $\\lambda_2=\\frac{2m+1}{2^{a+1}c_1}$. The other two entries on the column of $\\lambda_2$ must be opposite:\n$$e^{2\\pi i c_2\\frac{2m+1}{2^{a+1}c_1}}=-e^{2\\pi i c_3\\frac{2m+1}{2^{a+1}c_1}},$$\nwhich means that\n$$\\frac{2m+1}{2^ac_1}c_2=\\frac{2m+1}{2^ac_1}c_3+2q+1,$$\nfor some $q\\in\\mathbb{Z}$. Then $(2m+1)c_2=(2m+1)c_3+2^ac_1(2q+1)$. This implies that $c_3-c_2=2^ad$ for some odd number $d$. This proves that $B$ has the given form.\n\n\n\nWe have proved that $B$ (containing $0$) is a set of integers with $N=4$ elements is spectral if and only if it is of the form given in the theorem.\n\nAssume now $(B,L)$ is a Hadamard pair with scaling factor $R$. Then, since $B$ and $L$ are both spectral sets of integers, we must have  $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$ and $L=2^M \\{0, n_1, n_1 + 2^K n_2, 2^K n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ and $K$ are positive integers, and $C$ and $M$ are non-negative integers.\n\nRecall the Hadamard matrix for $N=4$,\n$$\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&e^{\\pi i q}&-e^{\\pi i q}\\\\\n  1&-1&-e^{\\pi i q}&e^{\\pi i q}\\\\\n  1&1&-1&-1\n\\end{pmatrix}.\n$$\nHere $q$ is any rational number, though we will see that not all rational numbers correspond to a Hadamard pair. We do not yet know which elements (other than $0$) in $B$ and $L$ are associated with which rows and columns. \n\nFirst we shall prove that the odd elements in $\\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$ can not be associated with the entry $+1$ in the matrix above. Let us assume for contradiction's sake that the elements $2^C g \\in B$ and $2^M f \\in L$ are associated with the matrix entry $1$, where $g$ is odd. Then $\\text{exp} \\left( \\frac{2 \\pi i 2^{C+M} g f}{R} \\right) = 1$, so $\\frac{2 \\pi i 2^{C+M} g f}{R} = 2 \\pi  i Z $ for some integer $Z$. Then, the matrix entry associated with $2^{C+a} c_1 \\in B$ and  $2^M f \\in L$ must be $-1$, as $0$ is associated with $1$ and $-1$ are the only other entries in that column. Then $\\text{exp} \\left( 2 \\pi  i \\frac{2^{a+M+C} c_1 f}{R} \\right) = -1 = \\text{exp} \\left( 2 \\pi  i \\frac{2^{a+M+C} c_1 f g}{R g} \\right)$. Substituting, $-1 = \\text{exp} \\left( 2 \\pi  i \\frac{2^{a} c_1 Z}{ g} \\right)$. Since $g$ is odd, this is impossible. Therefore, the first non-zero element of $B$ (in our current ordering) must be associated with the matrix element $1$ that is not in the first row or column. By similar reasoning, so must the last element of $L$.\n\nTherefore, the first non-zero element of $B$ is associated with the second column of the matrix, as depicted above, and the last element of $L$ is associated with the last row of the matrix. In making these statements we make use of the fact that changing the order of the elements in a set which is part of a Hadamard pair permutes the columns or rows of the associated matrix and vice versa, and that therefore it is sufficient to consider the order of the rows and columns of $A$ as depicted above.\n\nNow we shall show $K=a$. We have, from the second column and last row: $\\text{exp} (\\pi i) = \\text{exp} \\left( \\frac{2 \\pi i 2^{C+M+a} c_1 g}{R} \\right) = \\text{exp} \\left( \\frac{2 \\pi i 2^{C+M+K} n_3 f}{R} \\right)$, where $g$ and $f$ are odd. Thus $R$ has a power of $2$ exactly equal to both $1+C+M+a$ and $1+C+M+K$, so $K=a$.\n\nWe now also know that $R= 2^{C+M+a+1} d$, where $d$ is odd. Let $c$ be the greatest common divisor of the $c_k$'s and $n$ that of the $n_k$'s. Examining column two in the matrix above, we can see that for every $2^Mg$ in $L$, we have $\\exp\\left(\\frac{\\pi i c_1g}{d}\\right)=\\pm1$. Therefore, $d$ must divide $c_1 g$. Thus, since $d$ is odd, $d$ divides $c_1 n_1$, then it divides $c_1 n_2$ and $c_1n_3$. Therefore $d$ divides $c_1 n$. Similarly, from the last row we have that $d$ divides $n_3 c$. From the third column and the last column, since the corresponding entries are equal or opposite, we get that $\\text{exp} \\left( 2 \\pi i \\frac{2^{a+C} c_3 l_j}{R} \\right) = \\pm 1$ for all $l_j \\in L$. Therefore, since $d$ is odd, $d$ must divide $c_3 l_j$ for every $l_j \\in L$, so as before $d$ must divide $c_3 n$. Similarly, comparing the second and third rows, we have that $d$ divides $n_2 c$.\nThus, we have that $B$, $L$, and $R$ are as stated.\n\nConversely, it is easy to check that such a $B$, $L$, and $R$ lead to the Hadamard matrix above.\n\n\n\n\\end{myproof}\n\nThis gives a complete classification of Hadamard pairs of integers when $N=4$. \n\n\\begin{remark}\nThere are Hadamard matrices that do not correspond to Hadamard pairs.\n\n\nConsider the case when $q=0$ in the construction above for Hadamard matrices where $N=4$, which corresponds to the matrix\n$$\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&1&-1\\\\\n  1&-1&-1&1\\\\\n  1&1&-1&-1\n\\end{pmatrix} .\n$$\nAssume this matrix has a Hadamard pair, so it can be written as above. Consider the matrix element associated with $c_2$ and $n_1$. We have from the proof of Theorem \\ref{thha4} that $k=\\frac{c_2 n_1}{2^a d}$, where $k$ is an integer (so that the matrix entry is $-1$ or $1$), but $c_2$ and $n_1$ are odd and the denominator of the right hand side is even, so no Hadamard pair of integers has this as the associated matrix. Therefore no Hadamard pair of rational numbers has this as the associated matrix, and therefore, since every column contains an $R$th root of unity for some integer $R$, no set of integers $B$ and set of real numbers $\\Lambda$ has this as the associated matrix.\n\nAt this point we recall that the Hadamard matrices for $N=6$ are not completely classified. The above example suggests a question: what are the Hadamard matrices for $N=6$ that arise from Hadamard pairs? We do not yet know how to answer this question.\n\\end{remark}\n\n\n\n\\subsection{Spectral sets in $\\mathbb{R}$}\n\n\n\\begin{lemma}\\label{lem0.8}\nLet $p\\in \\mathbb{N}$. Assume the following statement is true: for every set $\\Gamma=\\{\\lambda_0=0,\\lambda_1,\\dots,\\lambda_{p-1}\\}$ in $\\mathbb{R}$, which has a spectrum of the form $\\frac1p A$ with $A\\subset\\mathbb{Z}$, there exists a subset $\\mathcal T$ of $\\mathbb{Z}$ such that for any spectrum of $\\Gamma$ of the form $\\frac1pA'$ with $A'\\subset\\mathbb{Z}$, the set $A'$ tiles $\\mathbb{Z}$ by $\\mathcal T$. \n\nThen spectral implies tile for period $p$. \n\n\\end{lemma}\n\n\\begin{proof}\nThe result follows from \\cite{DJ12}.\n\n\\end{proof}\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.9}] We use Lemma \\ref{lem0.8}.\n\nFor $p=2$, take a set $\\Gamma=\\{0,\\lambda\\}$ which has a spectrum of the form $\\frac12A$ with $A\\subset\\mathbb{Z}$. Using a translation we can assume $0\\in A$, so $A=\\{0,b\\}$ with $b\\in\\mathbb{Z}$. Write $b=2^ac$ with $a\\geq 0$, $c$ odd. Since $\\frac12 A$ is a spectrum for $\\Gamma$, the matrix $\\frac1{\\sqrt2}(e^{2\\pi i\\lambda a})_{\\lambda\\in\\Lambda,a\\in A}$ is unitary and the first row is $\\frac{1}{\\sqrt2}(1,1)$ and the second is $\\frac{1}{\\sqrt2}(1, e^{2\\pi i\\lambda\\frac12 2^a c})$. Therefore $e^{2\\pi i\\lambda\\frac12 2^ac}=-1$, hence $\\frac12 2^ac\\lambda=\\frac12+k$ for some $k\\in\\mathbb{Z}$. Thus $\\lambda=\\frac{1+2k}{2^ac}$. \n\nNow take another spectrum of the same form $\\frac12A'$ with $A'=\\{0,2^{a'}c'\\}$. Then $\\lambda=\\frac{1+2k'}{2^{a'}c}$ with $k'\\in\\mathbb{Z}$. This implies that \n$2^{a'}c'(1+2k')=2^ac(1+2k)$. Since $c$ and $c'$ are odd this means that $a=a'$. So the number $a$ depends only on $\\Gamma$, not on the choice of the spectrum $\\frac12A$. \n\nIf a set $A$ is of the form $\\{0,2^ac\\}$ with $a\\geq0$, $c$ odd then $A$ tiles $\\mathbb{Z}$ by $\\mathcal T:=\\{0,1,\\dots, 2^a-1\\}\\oplus 2^{a+1}\\mathbb{Z}$. Indeed $\\{0,c\\}\\oplus 2\\mathbb{Z}=\\mathbb{Z}$ so $2^a\\{0,c\\}\\oplus 2^{a+1}\\mathbb{Z}=2^a\\mathbb{Z}$ so $A\\oplus 2^a\\mathbb{Z}\\oplus \\{0,1,\\dots, 2^a-1\\}=\\mathbb{Z}$. Since $\\mathcal T$ depends only on $a$ and not on $c$, hence it depends only on $\\Gamma$ and not on the choice of the spectrum $\\frac12A$, it follows that the hypothesis of Lemma \\ref{lem0.8} are satisfied for $p=2$ and therefore spectral implies tile for period 2. \n\n\nFor $p=3$, $\\Gamma=\\{0,\\lambda_1,\\lambda_2\\}$ which has a spectrum  of the form $\\frac13A$ with $A\\subset\\mathbb{Z}$. Again, we can assume all the spectra contain $0$. \nThen $A$ is also spectral with spectrum $\\frac13\\Gamma$. From Corollary \\ref{pr0.1} we see that $A=3^aB$ with $a\\geq0$ and $B$ a complete set of representatives modulo 3. We claim that the number $a$ depends only on $\\Gamma$, not on the choice of the spectrum $\\frac13A$. As we see from the proof of Corollary \\ref{pr0.1}, the first row of the matrix $(e^{2\\pi i\\lambda b})_{\\lambda\\in\\Gamma,b\\in\\frac13A}$ is $(1,1,1)$ and the other two have the entries $\\{1,e^{2\\pi i\/3},e^{4\\pi i\/3}\\}$. This means that there is a $b_1\\in B$ such that $e^{2\\pi i \\lambda_1\\frac13 3^ab_1}=e^{2\\pi i\/3}$ and $b_1\\not\\equiv 0\\mod 3$.\nThen $\\frac13 3^ab_1\\lambda_1=\\frac13+k$ for some $k\\in\\mathbb{Z}$, so $\\lambda_1=\\frac{1+3k}{3^ab_1}$. \n\nNow take anothe spectrum $\\frac13A'$ with $A'=3^{a'}B'$. We get $\\lambda_1=\\frac{1+3k'}{3^{a'}b_1'}$ for some $k'\\in\\mathbb{Z}$, $b_1'\\in B'$, $b_1'\\not\\equiv0\\mod 3$. Then $3^{a'}b_1'(1+3k)=3^ab_1(1+3k')$. Since $b_1',b_1$ are not divisible by 3, it follows that $a=a'$. \n\nA set of the form $3^aB$ where $a\\geq 0$ and $B$ is a complete set of representatives modulo 3 tiles $\\mathbb{Z}$ by $\\mathcal T:=\\{0,1,\\dots,3^a-1\\}\\oplus 3^{a+1}\\mathbb{Z}$. Indeed \n$B\\oplus 3\\mathbb{Z}=\\mathbb{Z}$, which implies that $3^aB\\oplus 3^{a+1}\\mathbb{Z}=3^a\\mathbb{Z}$, so $3^aB\\oplus 3^{a+1}\\mathbb{Z}\\oplus\\{0,1,\\dots,3^a-1\\}=\\mathbb{Z}$.\n\nSince $\\mathcal T$ depends only on $\\Gamma$, Lemma \\ref{lem0.8} shows that spectral implies tile for period 3. \n\n\n\nFor $p=4$, $\\Gamma=\\{0,\\lambda_1,\\lambda_2, \\lambda_3 \\}$ with spectrum  of the form $\\frac14A$ with $A\\subset\\mathbb{Z}$. We assume all the spectra contain $0$.\nThen $A$ is also spectral with spectrum $\\frac14\\Gamma$. From Theorem \\ref{thha4} we have $A=2^m \\{0, 2^a c_1, c_2, c_2 +2^a c_3 \\}$ where all $c_i$ are odd, $m$ and $a$ are integers, and $a$ is positive.\nIn the present case, up to permutation of rows and columns, all Hadamard matricies are equivalent to the following (we omit the constant $\\frac{1}{2}$):\n$$\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&e^{\\pi i q}&-e^{\\pi i q}\\\\\n  1&-1&-e^{\\pi i q}&e^{\\pi i q}\\\\\n  1&1&-1&-1\n\\end{pmatrix},\n$$\nwhere $q$ is a real (even rational) number.\nWithout loss of generality, we assume that $\\lambda_1$ is associated to the first column of the matrix. Then we deduce that $2^{m+a} c_1$ is associated to the last row of the matrix (see the proof of Theorem \\ref{thha4}). Hence, $e^{2 \\pi i \\frac14 \\lambda_1 2^{m+a} c_1}=1$. Thus, from the second column and either (or both) the second or third row, $e^{2 \\pi i \\frac14 \\lambda_1 2^{m} c_2}=-1$. Thus $\\lambda_1 2^{m-1}c_2$ is odd. Thus, since $c_2$ is odd, $\\lambda_1$ determines $m$. From the third column and last row, we obtain $e^{2 \\pi i \\frac14 \\lambda_2 2^{m+a} c_1}=-1$. Then $\\lambda_2 2^{m+a-1}c_1$ is odd. Thus, $\\lambda_2$ determines $m+a$. This means that $\\Gamma$ determines $m$ and $a$.\n\nTherefore, in our calculations we take $A$ as above with $a$ and $m$ fixed. It remains to show the existence of a tile $\\mathcal T$ for $A$ that depends only on $a$ and $m$, which will show by Lemma \\ref{lem0.8} that spectral implies tile for period 4.\n\nWe shall turn our attention to the simpler problem of finding a tile dependent only on $a$ for $A_0 = \\{0, 2^a c_1, c_2, c_2 +2^a c_3\\}$. We consider this set, modulo $2^{a+1}$. We have representatives for $0$, $2^a$, an odd number, and $2^a$ plus that odd number. We consider $T_0 = \\{0,2,4,6,\\dots,2^a -2\\}$. We notice that $T_0 \\oplus A_0 = \\mathbb{Z} (\\mod  2^{a+1})$. Hence, $T_0 \\oplus A_0 \\oplus 2^{a+1} \\mathbb{Z} = \\mathbb{Z}$, so we have a tile for $A_0$. We notice that $2^m T_0 \\oplus 2^m A_0 \\oplus 2^m 2^{a+1} \\mathbb{Z} = 2^m \\mathbb{Z}$, so $\\{0,1,\\dots,2^m -1\\} \\oplus 2^m T_0 \\oplus 2^{m+a+1} \\mathbb{Z} \\oplus A = \\mathbb{Z}$. Therefore, $\\mathcal T = \\{0,1,\\dots,2^m -1\\} \\oplus 2^m T_0 \\oplus 2^{m+a+1} \\mathbb{Z}$ is a tile for $A$ which depends only on $a$ and $m$, so spectral implies tile for period 4.\n\n\n\n\n\n For $p=5$, $\\Gamma = \\{0, \\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4\n \\}$, with spectrum $\\frac{1}{5} B$ where $B$ is a set of integers\n containing $0$. Then $B$ is spectral with spectrum $\\frac{1}{5}\n \\Gamma$. Then, by Corollary \\ref{pr0.1}, $B=5^a \\{0,b_1,b_2,b_3,b_4\\}$ where\n $\\{0,b_1,b_2,b_3,b_4\\}$ is a complete set of residues modulo $5$ and\n $a$ is a non-negative integer. We shall show that the number $a$\n depends only on $\\Gamma$. \\\\\n With Lemma \\ref{lem2.8} the matrix associated with $(B,\\frac15\\Gamma)$ is (after some relabeling of $B,\\Gamma$):\n $$\\begin{pmatrix}\n  1&1&1&1&1\\\\\n  1&w&w^2&w^3&w^4\\\\\n  1&w^2&w^4&w&w^3\\\\\n  1&w^3&w&w^4&w^2 \\\\\n  1&w^4&w^3&w^2&w \n\\end{pmatrix},\n$$\nwhere $w=e^{\\frac{2 \\pi i}{5}}$. Select $j$ so that $b_j \\equiv 1 (\\mod 5)$. Now select $k$ so that $e^{\\frac{2 \\pi i}{5}} = e^{\\frac{2 \\pi i b_j \\lambda_k}5}$. Then $e^{\\frac{2 \\pi i}{5}} = e^{\\frac{2 \\pi i 5^a \\lambda_k}5}$. Thus, $a$ depends only on $\\Gamma$. Thus, it remains to show the existence of a tile $\\mathcal T$ for $B$ that depends only on $a$. \\\\\nSince $B_0=\\{0,b_1,b_2,b_3,b_4\\}$ is a complete set of residues modulo $5$, a tile for $B_0$ is $T_0 = 5\\mathbb{Z}$. Therefore, a tile for $B$ is $\\mathcal T = \\{0,1,\\dots,5^a -1\\} \\oplus 5^{a+1} \\mathbb{Z}$. This tile depends only on $a$, so spectral implies tile for period $5$.\n\n\\end{myproof}\n\n\n\\subsection{Complementing Hadamard pairs}\nNow we would like to find complementary Hadamard pairs whenever possible for the cases $N=2,3,4,5$ that we have been exploring.\n\\begin{myproof}[Proof of Proposition \\ref{prHP}] One can check this directly, by verifying the orthogonality of the rows, but we show that we are in a particular case of a more general construction of Hadamard matrices. \n\n We shall prove that the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$ can be obtained by Di\\c t\\u a's construction (see e.g. \\cite{TaZy06}), and is therefore a Hadamard matrix. Di\\c ta's construction is a generalization of the fact that the tensor product of Hadamard matrices is a Hadamard matrix: Let $A$ be a Hadamard matrix and $\\{Q_1, \\dots, Q_k\\}$ be (possibly different) Hadamard matrices. Let $\\{E_1, E_2, \\dots, E_k\\}$ be unitary diagonal matrices whose first element is $1$, and where $E_1$ is the identity. Then the following is a Hadamard matrix:\n $$D = \\begin{pmatrix}\n  A_{1,1} E_1 Q_1 & A_{1,2} E_2 Q_2 & \\dots & A_{1,k} E_k Q_k \\\\\n  . & . & . & . \\\\\n  A_{k,1} E_1 Q_1 & A_{k,2} E_2 Q_2 & \\dots & A_{k,k} E_k Q_k \\\\\n\\end{pmatrix} .\n$$\nConsider one way to write the matrix elements of the tensor product of matrices of size $N$:\n\\begin{equation}\n(A\\otimes B)_{\\alpha, \\beta} = A_{j,l} B_{m,n},\n\\end{equation}\nwhere $\\alpha = N(j-1) +m $ and $\\beta = N(l-1) +n$. As one varies $n$, $m$, $j$, and $l$, one obtains the elements of $(A\\otimes B)$. We generalize this formula to fit Di\\c{t}\\u a's construction, and assume $A$ is size $J$ and the $Q$s and $E$s are size $N$:\n\\begin{equation}\nD_{\\alpha, \\beta} = A_{j,l} \\left( E_l Q_l \\right)_{m,n} ,\n\\end{equation}\nwhere $\\alpha = J(j-1) +m $ and $\\beta = J(l-1) +n$. As before one varies the indexes on the right to obtain the entries in $D$. We notice that the $E$s are diagonal matrices, and therefore  \n\\begin{equation} \\label{dita}\nD_{\\alpha, \\beta} = A_{j,l} ( E_l )_{m,m} (Q_l )_{m,n} .\n\\end{equation}\n\n\nNow consider the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$. Let $B_j \\in B$, $L_l \\in L$, $F_m \\in F$, $G_n \\in G$. Thus $j$ and $l$ range from $1$ to, say, $J$, and $n$ and $m$ range from $1$ to, say, $N$. We have\n\\begin{equation}\nX_{\\alpha, \\beta} =\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j+F_m)(L_l+G_n)\\right) \\right)_{j,l,m,n}.\n\\end{equation}\nThe interaction of the indexes on the left and right depends on the way we organize $B\\oplus F$ and $ L \\oplus G$. We shall choose to organize $B\\oplus F$ in such a way that the first $N$ elements of the set are given by $B_1 + F_m$, for $1\\leq m \\leq N$, and so on. We shall do the same things with $L \\oplus G$, fix $L$ first and vary $G$. In this way, we have determined that, as in the constructions above, $\\alpha = J(j-1) +m $ and $\\beta = J(l-1) +n$, and thus by varying $j$, $l$, $m$, and $n$, we obtain $X$. \n\n\nFrom the hypothesis, $\\texttt{exp}\\left( \\frac{2 \\pi i}{R}B_j G_n\\right)=1$ for $B_j\\in B$, $G_n\\in G$. Thus we have\n\n\\begin{equation}\nX_{\\alpha, \\beta}=\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{j,l,m,n}.\n\\end{equation}\nWe arrange the indices:\n\\begin{equation} \nX_{\\alpha, \\beta}=\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\right)_{j,l} \\left(\\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\right)_{l,m} \\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{m,n}.\n\\end{equation}\nThis is exactly like Di\\c{t}\\u{a}'s construction \\eqref{dita}: the role of the constants $(E_l)_{m,m}$ are played by the constants $\\left(\\texttt{exp}\\left( \\frac{2 \\pi i}{R}(L_l F_m)\\right) \\right)_{l,m}$, and when $l$ or $m$ are $1$  this is indeed $1$, and otherwise they are roots of unity as required. In addition the matrices $\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(B_j L_l)\\right) \\right)_{j,l}$ and $\\left( \\texttt{exp}\\left( \\frac{2 \\pi i}{R}(F_m G_n)\\right) \\right)_{m,n}$ are Hadamard matrices. Thus, the matrix associated with $B\\oplus F, L \\oplus G$ with scaling factor $R$ is a Hadamard matrix, so they are a Hadamard pair.\n\\end{myproof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.4a}]\nAs in Theorem \\ref{standard}, we have that $B=h_0 N^f \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$ and $L=h_1 N^g \\{0 = l_0, l_1 , \\dots ,l_{N-1} \\}$ for some non-negative integers $f$ and $g$ and positive integers $h_0$ and $h_1$ not divisible by $N$, and $\\{ b_i \\}$ and $\\{ l_i \\}$ are complete sets of residues modulo $N$. Here we have decomposed the greatest common divisors of $B$ and $L$ into powers of $N$, and other numbers. Also $R=NS$ where $S$ divides $N^{f+g}h_0h_1$ and $Z_2:=N^{f+g}h_0h_1\/S$ is prime with $N$. We then have $R= N^{f+g+1} h_0 h_1 \/ Z_2$, where $N$ and $Z_2$ are mutually prime.\n\nFirst, we further rearrange things. Notice that since $R$ is an integer, $Z_2$ must divide $h_0 h_1$.  We can write $h_0 = w_0 z_0$ and $h_1 = w_1 z_1$, $Z_2=z_0 z_1$. Then $z_0$ and $z_1$ are mutually prime with $N$. Therefore we may rewrite $B$ in the following way: $B=w_0 z_0 N^f \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$, where, since $z_0$ and $N$ are mutually prime, $z_0 \\{0 = b_0, b_1 , \\dots ,b_{N-1} \\}$ is a complete set of residues modulo $N$. Thus we let $B=w_0  N^f \\{0 = B_0, B_1 , \\dots ,B_{N-1} \\}$, $L=w_1 N^g \\{0 = L_0, L_1 , \\dots ,L_{N-1} \\}$, where $\\{B_k \\}$ and $\\{L_j \\}$ are complete sets of residues modulo $N$, and $R=N^{f+g+1} w_0 w_1$.\n\n\n\n\nLet $B' = T_0 \\oplus T_1 \\oplus T_2 \\oplus T_3$ and $L' = U_0 \\oplus U_1 \\oplus U_2 \\oplus U_3$, where\n\\begin{equation}\nT_0 = \\{ 0,1,2,\\dots,w_0 -1 \\} ; U_0 = \\{ 0,1,2,\\dots,w_1 -1 \\}\n\\end{equation}\n\\begin{equation}\nT_1 = \\{ 0, w_0, 2w_0, \\dots, (N^f -1) w_0 \\} ; U_1 = \\{ 0, w_1, 2w_1, \\dots, (N^g -1) w_1 \\}\n\\end{equation}\n\\begin{equation}\nT_2 =  \\{ 0, w_0 N^{f+1}  , \\dots, (N^g - 1) w_0 N^{f+1}  \\}; U_2 = \\{ 0, w_1 N^{g+1}  , \\dots, (N^f -1) w_1 N^{g+1}  \\}\n\\end{equation}\n\\begin{equation}\nT_3 =  \\{ 0, w_0 N^{f+g+1}  , \\dots, (w_1 - 1) w_0 N^{f+g+1}  \\}; U_3 = \\{ 0, w_1 N^{f+g+1}  , \\dots, (w_0 -1) w_1 N^{f+g+1}  \\}\n\\end{equation}\n\n\nWe shall show that $B' , L'$ is the desired complementary Hadamard pair.\n\n\nFirst, we show that $B \\oplus B' = \\mathbb{Z} (\\mod R)$, and likewise for $L$ and $L'$. Notice that $B \\oplus T_1 = w_0 (\\mathbb{Z} (\\mod N^{f+1}))$. Then, $B \\oplus T_0 \\oplus T_1 = \\mathbb{Z} (\\mod N^{f+1} w_0)$. Then, $B \\oplus T_0  \\oplus T_1 \\oplus T_2 = \\mathbb{Z} (\\mod N^{f+g+1} w_0)$. Lastly, $B \\oplus T_0  \\oplus T_1 \\oplus T_2 \\oplus T_3 = \\mathbb{Z} (\\mod N^{f+g+1} w_0 w_1)$, and we are done. Similar reasoning applies to $L'$.\n\nNow we show that $B' , L'$ are a Hadamard pair with scaling factor $R$. By performing a few cancelations, we notice that $T_0 , U_3$ is a Hadamard pair with scaling factor $R$. Similarly, so is $T_1 , U_2$. In addition, notice that $t_1 u_3$ is a multiple of $R$ for every $t_1 \\in T_1 , u_3 \\in U_3 $. Thus, by Proposition \\ref{prHP}, $T_0 \\oplus T_1 , U_3 \\oplus U_2 $ is a Hadamard pair with scaling factor $R$. Similarly, so is $T_2 \\oplus T_3 , U_1 \\oplus U_0$. Now notice that $tu$ is a multiple of $R$ for every $t \\in T_2 \\oplus T_3 , u \\in U_3 \\oplus U_2 $. Thus, by Proposition \\ref{prHP}, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\nNow we show that $\\text{gcd} (B\\oplus B') = 1$. If $w_0 >1$, $1\\in B'$. If not, if $f=0$, $N \\in B'$ and $B$ contains an element of the form $Nk+1$ so $\\gcd(B\\oplus B')=1$; if $f>0$ then $1\\in B'$, so we are done.\n\nNow we show that the extreme cycles for $B',L'$ are contained in $\\mathbb{Z}$. If $f>0$, $1\\in B'$, so we are done (by Proposition \\ref{pr1.2}). If not, $\\text{gcd}(B')$ divides $w_0 N$, so the extreme cycle points are in $\\mathbb{Z} \/ w_0 N$. Consider two such points, $x$ and $y$, where $x=\\frac{y+l}{R}$ for some $l\\in L'$. Upon multiplying by $R$, we notice that the left hand side is an integer, as is $l$, so $y$ is an integer, and we are done.\n\nThus, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\n\\end{myproof}\n\nDue to the above theorem, we have a complementary Hadamard pair for every Hadamard pair when $N=2$, $3$, and $5$, whenever such a thing is possible. We turn our attention to the case $N=4$.\n\n\n\\begin{myproof}[Proof of Theorem \\ref{th0.5a}]\nThe cases of size 2,3,5 are covered by Theorem \\ref{th0.4a} so we considered the case of size 4. As above, we have that $R=2^{C+M+a+1} d$, $B=2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\}$, and $L=2^M \\{0, n_1, n_1 + 2^a n_2, 2^a n_3\\}$, where $c_i$ and $n_i$ are all odd, $a$ is a positive integer, $C$ and $M$ are non-negative integers, and $d$ divides $c_1 n$, $c_3 n$, $n_2 c$, and $n_3 c$, where $c$ is the greatest common divisor of the $c_k$'s and similarly for $n$.\n\nWe begin by constructing sets $B'$ and $L'$ such that $B\\oplus B' = L\\oplus L' = \\mathbb{Z} (\\mod R)$. Let $B' = T_0 \\oplus T_1 \\oplus T_2 \\oplus T_3$ and $L' = U_0 \\oplus U_1 \\oplus U_2 \\oplus U_3$, where\n\\begin{equation}\nT_0 = 2^{C+1} \\{0,1,2,\\dots2^{a-1} -1 \\} ;   U_0 = 2^{M+1} \\{0,1,2,\\dots2^{a-1} -1 \\} ;\n\\end{equation}\n\\begin{equation}\nT_1 =\\{0,1,2,\\dots2^{C} -1 \\} ;   U_1 =  \\{0,1,2,\\dots2^{M} -1 \\}\n\\end{equation}\n\\begin{equation}\nT_2 = 2^{a+C+1} \\{0,1,2,\\dots2^{M} -1 \\} ;   U_2 = 2^{a+M+1} \\{0,1,2,\\dots2^{C} -1 \\}\n\\end{equation}\n\\begin{equation}\nT_3 =   U_3  = 2^{a+M+C+1} \\{0,1,2,\\dots, d -1 \\} .\n\\end{equation}\nNotice that $\\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\} \\oplus \\{0,2,4,\\dots,2^a -2 \\} = \\mathbb{Z}(\\mod2^{a+1})$. Then $$B\\oplus T_0 = 2^C \\{0, 2^a c_1, c_2, c_2 + 2^a c_3\\} \\oplus  2^C \\{0,2,4,\\dots,2^a -2 \\} = 2^C \\mathbb{Z}_{2^{a+1}}.$$ Thus $B \\oplus T_0 \\oplus T_1 = \\mathbb{Z} (\\mod 2^{a+C+1})$. Therefore, $B \\oplus B' = \\mathbb{Z}(\\mod R)$. Similar logic applies to $L$ and $L'$.\n\nNow we must show $B',L'$ are a Hadamard pair with scaling factor $R$. Consider the polynomial\n\\begin{equation}\nB'(z) \\equiv \\sum_{b' \\in B'} z^{b'}  .\n\\end{equation}\nSince $B'$ is a direct sum of sets, we have\n\\begin{equation}\nB'(z) = \\sum_{t_0 \\in T_0} z^{t_0}  \\sum_{t_1 \\in T_1} z^{t_1}  \\sum_{t_2 \\in T_2} z^{t_2}  \\sum_{t_3 \\in T_3} z^{t_3} .\n\\end{equation}\nNow we let $p_n (z) = \\sum_{k=0}^{n-1} z^{k}$. Then, rewriting the product that is $B'(z)$, we have\n\\begin{equation}\nB'(z) = p_{2^{a-1}} (z^{2^{C+1}}) p_{2^{C}} (z) p_{2^{M}} (z^{2^{a+C+1}}) p_{d} (z^{2^{a+M+C+1}}) .\n\\end{equation}\nNow let $l_1 ' \\neq l_2 ' \\in L'$. We would like to show that if $q = l_1 ' - l_2 '$ then $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$. This in turn would imply that the matrix associated with $B',L'$ and scaling factor $R$ is unitary and thus, $B',L'$ is a Hadamard pair with scaling factor $R$.\n\nAny difference $q$ of distinct elements in $L'$ can be written\n\\begin{equation} \\label{425}\nq= q_1 + 2^{M+1} q_2 + 2^{a+M+1} q_3 + 2^{a+M+C+1} q_4 ,\n\\end{equation}\nwhere $q_1 \\in \\pm \\{0,1,\\dots,2^M -1 \\}$, $q_2 \\in \\pm \\{0,1,\\dots,2^{a-1} -1 \\}$, \n$q_3 \\in \\pm \\{0,1,\\dots,2^C -1 \\}$, and $q_4 \\in \\pm \\{0,1,\\dots,d -1 \\}$, and at least one $q_j$ is non-zero.\n\nNotice that since $p_n (z) (z-1) = z^n - 1$, the zeroes of $p_n$ are exactly the $n$th roots of unity other than $1$. We shall use this to prove by cases that $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$ for any $q \\in L'$.\n\nNow assume $q \\neq 0$ modulo $d$. Notice $$p_d \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right)^{2^{a+C+M+1}} \\right) = p_d \\left( \\text{exp} \\left( \\frac{2 \\pi i}{d}   q \\right) \\right)=0,$$ and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0.$ Thus, we may assume $q= 0$ modulo $d$, and thus we let $q = q_0 d$.\n\nNext assume $q \\neq 0$ modulo $2^M$. Then since $d$ is odd, the same is true of $q_0$. Notice $$p_{2^M} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q_0 d \\right)^{2^{a+C+1}} \\right) = p_{2^M} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^M}   q_0 \\right) \\right)=0,$$ and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$. Thus, we may assume $q = 0$ modulo $2^M d$, so we let $q=q_a 2^M d$. Then, from \\eqref{425}, we can see that $q_1 = 0$, and thus $2^{M+1} d$ divides $q$. Thus we let $q= q_b 2^{M+1} d$.\n\nNext assume $q_b \\neq 0$ modulo $2^{a-1}$. Then  $p_{2^{a-1}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q_b 2^{M+1} d \\right)^{2^{C+1}} \\right) = p_{2^{a-1}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^{a-1}}   q_b \\right) \\right)=0$, and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$. Thus we may assume $q = 0$ modulo $2^{M+a} d$, so examining \\eqref{425}, we see that $q_2=0$. Thus $2^{M+a+1} d$ divides $q$, so we let $q= q_w 2^{M+a+1} d$.\n\nNow assume $q_w \\neq 0$ modulo $2^{C}$. Then  $p_{2^{C}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q_w 2^{M+a+1} d \\right)  \\right) = p_{2^{C}} \\left( \\text{exp} \\left( \\frac{2 \\pi i}{2^{C}}   q_w \\right) \\right)=0$, and thus $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$.\n\nThus $q$ must be a multiple of $R$, otherwise $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$. But a difference of distinct elements in $L'$ contains no such thing, so $B'\\left( \\text{exp} \\left( \\frac{2 \\pi i}{R}   q \\right) \\right)=0$ and thus $B',L'$ are a Hadamard pair with scaling factor $R$. \n\nNext we must show that the greatest common divisor of elements in $B \\oplus B'$ is one. If $C>0$, this is true because $1\\in T_1$. If $C=0$ then $B$ contains  an odd number and since $\\gcd(T_2)$ divides $2^{a+C+1}$ we get that $\\gcd(B\\oplus B')=1$.  \n\nLastly, we must show that the extreme cycles for $B',L'$ are contained in $\\mathbb{Z}$. By construction, the greatest common divisor of $B'$ divides $R$. Therefore all the extreme cycle points must be in $\\mathbb{Z} \/ R$. Consider two such points, $\\frac{x}{R}$ and $\\frac{y}{R}$, consecutive in the cycle. Then we have $\\frac{x}{R} = \\frac{l'+\\frac{y}{R}}{R}$ for some $l' \\in L'$. Multiplying both sides by $R$, we can see that the left hand side is an integer. Therefore, so is the right hand side, so $\\frac{y}{R}$ must be an integer. But $y$ was arbitrary, so we are done.\n\n\\end{myproof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Examples}\n\nIn the following examples, we will frequently refer to the extreme cycles of $L\\oplus L'$. It is to be understood that these cycles are extreme for $(B \\oplus B',L\\oplus L')$ with scaling factor $R$, where, since we are dealing with complementary Hadamard pairs, the greatest common divisor of $B \\oplus B'$ is $1$. We also refer to the digits of a cycle as the cycle itself. \n\n\\begin{example}\\label{ex4.1}\nLet $R=4$, $B=\\{0,2\\}$, $B'=\\{0,1\\}$. Then $\\mu_B$ is the 4-Cantor measure defined in \\cite{JoPe98} and $\\mu_{B'}$ is a contraction by 2 of this measure. Let $L=\\{0,3\\}$ and $L'=\\{0,2\\}$. We check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nIt is easy to check that $(B,L)$ and $(B',L')$ are Hadamard pairs and $B\\oplus B'$ and $L\\oplus L'$ are complete sets of representatives $\\mod 4$. \nBy Proposition \\ref{pr1.2}, the extreme cycles for $(B,L)$ are contained in $\\frac12\\mathbb{Z}\\cap [0,1]$. We can check the points $\\{0,1\/2,1\\}$ one by one and \nwe see that the extreme cycles are $\\{0\\}$ with digits $\\underline 0$ and $\\{1\\}$ with digits $\\uln3$. \n\n\n\nFor $(B',L')$, the extreme cycles are contained in $\\mathbb{Z}\\cap[0,2\/3]$. So we have only one extreme cycle $\\{0\\}$ with digits $\\uln0$. \n\nThus, the condition (ii) in Definition \\ref{def2.1} is satisfied. Condition (iii) is also satisfied. So we have complementary Hadamard pairs. \n\nSince $B\\oplus B'=\\{0,1,2,3\\}$, the attractor $X_{B\\oplus B'}$ is the unit interval $[0,1]$ and $\\mu_{B\\oplus B'}$ is the Lebesgue measure on the unit interval. Therefore the convolution of the measures $\\mu_B$ and $\\mu_{B'}$ is the Lebegue measure on the unit interval. \n\nNext, we find the extreme cycles for $L\\oplus L'=\\{0,2,3,5\\}$. These are contained in $\\mathbb{Z}\\cap[0,5\/3]$. We have $\\frac{1+3}{4}=1$. So the only extreme cycles are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln3$. Since $p(3)=3$ and $p'(3)=0$ and $\\uln3$ is a cycle for $L$ and $\\uln0$ is a cycle for $L'$, Theorem \\ref{th2.3} (v) implies that the spectrum $\\Lambda(L)$ for $\\mu_B$ tiles $\\mathbb{Z}$ by the spectrum $\\Lambda(L')$ for $\\mu_{B'}$.\n\nNote that $\\Lambda(L)$ contains negative numbers: for example $-1$ has the representation $\\uln3$, $-4$ has the representation $0\\uln3$.\n\n\n\nTake now $L=\\{0,1\\}$ and $L'=\\{0,6\\}$. One can check as above that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. The extreme cycle for $(B,L)$ is $\\{0\\}$ with digits $\\underline 0$ and the extreme cycles for $(B',L')$ are $\\{0\\}$ with digits $\\underline 0$ and $\\{2\\}$ with digits $\\underline 6$. \n\nThe spectrum $\\Lambda(L)$ for $\\mu_B$ is the one described in \\eqref{eqspmu4}. We have $L\\oplus L'=\\{0,1,6,7\\}$. The extreme cycles for $(B\\oplus B', L\\oplus L')$ are $\\{0\\}$ with digits $\\underline 0$ and $\\{ 2\\}$ with digits $\\underline 6$. Since $p(\\underline 6)=\\underline 0$ which is an extreme cycle for $(B,L)$ and $p'(\\underline 6)=\\underline 6$ which is an extreme cycle for $(B',L')$, it follows that $\\Lambda(L)$ tiles $\\mathbb{Z}$ with $\\Lambda(L')$.\n\\end{example}\n\n\\begin{example}\\label{ex4.2}\nLet $R=4$, $B=\\{0,2\\}$, $B'=\\{0,1\\}$,  $L=\\{0,1\\}$ , $L'=\\{0,2\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nThe only extreme cycle for $(B,L)$ and $(B',L')$ is $\\{0\\}$. The spectra $\\Lambda(L)$ and $\\Lambda(L')$ are contained in $\\mathbb{N}$. \nSince $L\\oplus L'=\\{0,1,2,3\\}$ there is a non-trivial extreme cycle for $L\\oplus L'$, $1=\\frac{1+3}4$. Therefore we have that $\\uln3$ is an extreme cycle for $L\\oplus L'$. \nBut $p(\\uln3)=\\uln1$ and $p'(\\uln3)=\\uln2$ and these are not extreme cycles for $L$ and $L'$ respectively.   \n\\end{example}\n\n\\begin{example}\\label{ex4.3}\nLet $R=6$, $B=\\{0,1,2\\}$, $B'=\\{0,3\\}$,  $L=\\{0,2,10\\}$ , $L'=\\{0,1\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. \nThe extreme cycles for $(B,L)$ are $\\{0\\}$ with digits $\\uln0$ and $\\{2\\}$ with digits $\\uln{(10)}$. $(B',L')$ has only the trivial cycle.\n\nWe consider $L\\oplus L' = \\{0,1,2,3,10,11\\}$. The extreme cycles are are $\\{0\\}$ with digits $\\uln0$ and $\\{2\\}$ with digits $\\uln{(10)}$. It is clear that $p(\\uln0)$ and $p'(\\uln0)$ are cycles for $L$ and $L'$ respectively. Notice that $p(\\uln{(10)})=\\uln{(10)}$, which is a cycle for $L$, and $p'(\\uln{(10)})=\\uln0$, which is a cycle for $L'$. Therefore, by theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nIf we replace $10$ in $L$ by $4$, we still have complementary Hadamard pairs, but the extreme cycles for $L\\oplus L' = \\{0,1,2,3,4,5\\}$ are different, and now all the extreme cycles for $(B,L)$ and $(B',L')$ are trivial.\nFor $L\\oplus L'$ we still have $\\{0\\}$ with digits $\\uln0$, and now the other cycle is $\\{1\\}$ with digits $\\uln5$. Since $p(\\uln5)=\\uln4$ is not a cycle for $L$ and $p'(\\uln5)=\\uln1$ is not a cycle for $L'$, we have by Theorem \\ref{th2.3} that $\\Lambda(L)\\oplus\\Lambda(L') \\neq \\mathbb{Z}$. \n\\end{example}\n\n\\begin{example}\nLet $R=8$, $B=\\{0,3,4,7\\}$, $B'=\\{0,2\\}$,  $L=\\{0,3,4,7\\}$ , $L'=\\{0,2\\}$. Then it is easy to check that $(B,L)$ and $(B',L')$ are complementary Hadamard pairs. The matrix associated with $(B,L)$ and scaling factor $R$ is interesting: it is\n$$\\begin{pmatrix}\n  1&1&1&1\\\\\n  1&-1&e^{\\pi i \/4}&-e^{\\pi i \/4}\\\\\n  1&-1&-e^{\\pi i \/4}&e^{\\pi i \/4}\\\\\n  1&1&-1&-1\n\\end{pmatrix}.\n$$\nThe extreme cycles for $(B,L)$ are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln{7}$. $(B',L')$ has only the trivial cycle.\n \nWe consider $L\\oplus L' = \\{0,2,3,4,5,6,7,9\\}$. The cycles are are $\\{0\\}$ with digits $\\uln0$ and $\\{1\\}$ with digits $\\uln{7}$. It is clear that $p(\\uln0)$ and $p'(\\uln0)$ are cycles for $L$ and $L'$ respectively. Notice that $p(\\uln{7})=\\uln{7}$, which is a cycle for $L$, and $p'(\\uln{7})=\\uln0$, which is a cycle for $L'$. Therefore, by Theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nIf we replace $2$ in $L'$ by $14$, we still have complementary Hadamard pairs, but the extreme cycles for $L\\oplus L' = \\{0,3,4,7,14,17,18,21\\}$ are different. The extreme cycles for $(B,L)$  are unchanged. The extreme cycles for $(B',L')$ are now $\\{0\\}$ with digits $\\uln{0}$ and $\\{2\\}$ with digits $\\uln{(14)}$.\n\nFor $L\\oplus L'$ we still have $\\{0\\}$ with digits $\\uln0$, and now we also have $\\{1\\}$ with digits $\\uln7$, $\\{2\\}$ with digits $\\uln{(14)}$, and $\\{3\\}$ with digits $\\uln{(21)}$. We have $p(\\uln7)=\\uln7$, which is an extreme cycle for $(B,L)$, and $p'(\\uln7)=\\uln0$, which is an extreme cycle for $(B',L')$. We also have $p(\\uln{14})=\\uln0$, which is an extreme cycle for $(B,L)$, and $p'(\\uln{14})=\\uln{(14)}$, which is an extreme cycle for $(B',L')$. Finally, we have $p(\\uln{21})=\\uln7$, which is an extreme cycle for $(B,L)$, and $p'(\\uln{21})=\\uln{(14)}$, which is an extreme cycle for $(B',L')$. Therefore, by Theorem \\ref{th2.3}, $\\Lambda(L)\\oplus\\Lambda(L')=\\mathbb{Z}$.\n\nSo $\\Lambda(L)$ tiles with two very different tiling sets $\\Lambda(\\{0,2\\})$ and $\\Lambda(\\{0,14\\})$. \n\n\n\\end{example}\n\n\\begin{acknowledgements}\nThis work was partially supported by a grant from the Simons Foundation (\\#228539 to Dorin Dutkay).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{alpha}\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe Standard Model (SM) can explain a multitude of observations. However, several phenomena still require explanations, {\\it e.g.}~the existence and nature of dark matter, the matter--antimatter asymmetry of the universe, and the origin of neutrino masses. \n\nA popular model to explain these beyond the SM physics is minimal supersymmetry (MSSM). Although the MSSM addresses issues of fine-tuning in the Higgs mass and there are dark matter candidates in MSSM, it has been constrained stringently by LHC searches \\cite{CMSsusy, ATLASsusy}.  The MSSM currently also lacks a simple mechanism to generate neutrino masses as well as the baryon asymmetry of the universe. As such, it is necessary to consider supersymmetric models beyond the MSSM. \n\nOne extension of the MSSM that addresses these questions is the $R$-symmetric MSSM \\cite{HALL1991289}. In $R$-symmetric MSSM the superpartners are charged under a global $U(1)_R$ symmetry while their SM counterparts are neutral.  While this global symmetry is unbroken, gauginos cannot be Majorana particles. Additional adjoint fields with opposite $U(1)_R$ charge, with respect to gauginos, are introduced so that gauginos can acquire Dirac masses \\cite{FAYET1975104, FAYET1976135}. $R$-symmetric MSSM addresses SUSY $CP$ and flavor problems by forbidding one-loop diagrams mediated by Majorana gauginos as well as forbidding left-right sfermion mixing \\cite{Kribs:2007ac, Dudas:2013gga}. Apart from gauginos, this model requires electroweak (EW) partners for higgsinos with an opposite $R$-charge so that a $\\mu$-term is allowed. It was shown that the scalar components of these new superfields can help to have a first-order EW phase transition \\cite{Fok:2012fb}. Moreover, new interactions can bring in new sources of \\emph{CP} violation. Hence this model can potentially explain the baryon asymmetry of the universe\\footnote{Another mechanism for generating the baryon asymmetry in such models is oscillations and out-of-equilibrium decays of a pseudo-Dirac bino \\cite{Ipek:2016bpf}.} \\cite{Fok:2012fb}. Furthermore, Dirac gluinos make the fine-tuning problem milder in $R$-symmetric MSSM \\cite{Kribs:2007ac, Fox:2002bu}. \n\nThe global  $U(1)_R$ symmetry is broken because the gravitino acquires a mass. Consequently small $U(1)_R$-breaking Majorana masses for gauginos will be generated through anomaly mediation \\cite{Randall:1998uk, Giudice:1998xp, ArkaniHamed:2004yi}. Since the $U(1)_R$ symmetry is only approximate, gauginos in $R$-symmetric MSSM are pseduo-Dirac fermions, having both Dirac and Majorana masses.  \n\nThe LHC phenomenology of $R$-symmetric MSSM is different than minimal SUSY models. For example in $R$-symmetric MSSM the supersymmetric particles need to be produced in particle--antiparticle pairs since the initial SM state is $U(1)_R$ symmetric. Furthermore some production channels for supersymmetric particles are not available due to the $U(1)_R$ symmetry. Hence collider limits on $R$-symmetric MSSM tend to be less stringent than the ones on MSSM, see \\emph{e.g.}, \\cite{Frugiuele:2012kp, Alvarado:2018rfl,Diessner:2017ske,Kalinowski:2015eca}. \n\nIn this work we study the LHC phenomenology of a version of the $R$-symmetric MSSM in which the $U(1)_R$ symmetry is elevated to $U(1)_{R-L}$, where $L$ is the lepton number. We give the details of the model in Section~\\ref{sec:model}.  It has been shown that in this model the pseudo-Dirac bino can play the role of right-handed neutrinos \\cite{Coloma:2016vod} and light Majorana neutrino masses are generated via an inverse-seesaw mechanism. The smallness of the light neutrino masses is given by a hierarchy between the source of $U(1)_R$-breaking, namely the gravitino mass $m_{3\/2}$, and the messenger scale $\\Lambda_M$.  As benchmark points this requires $m_{3\/2}\\sim$~10~keV and $\\Lambda_M\\sim$~100~TeV. \n\nThe mixing between electroweak gauginos and the SM neutrinos allows the gauginos to decay to gauge bosons and leptons, which can remove the usual $\\slashed{E}_{\\mathrm{T}}$ signature associated with SUSY searches.  In cases where the lepton is a neutrino there is still $\\slashed{E}_{\\mathrm{T}}$ in the event but the kinematics are different from typical weak scale SUSY models.\nWe use current searches for jets+$\\slashed{E}_{\\mathrm{T}}$ at the LHC with $\\sqrt{s}=13~$TeV and  $\\mathcal{L}=36~{\\rm fb}^{-1}$ to find the constraints on squark and bino masses in this model. We focus on the parameter region with $100~{\\rm GeV}<\\ensuremath{M_{\\tilde{B}}}<\\ensuremath{M_{\\tilde{q}}}$. We also forecast our results for $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~{\\rm fb}^{-1}$. The analysis is described in Section~\\ref{sec:LHCpheno}. Our results are shown in  Figure~\\ref{fig:Plotexcatlas} and our conclusions are given in Section~\\ref{sec:conclusions}.\n\n\n\\section{Model}\n\\label{sec:model}\n\nIn this section we review the model that was considered in \\cite{Coloma:2016vod}. This is an extension of $U(1)_R$--symmetric SUSY models \\cite{Kribs:2007ac} where, instead of  the $R$ symmetry, the model has a global $U(1)_{R-L}$ symmetry. The field content and the $U(1)_R$ and $U(1)_{R-L}$ charges of the relevant superfields are given in Table \\ref{table:fields}. Note that in the rest of the text we use $U(1)_R$ instead of $U(1)_{R-L}$ whenever the distinction is not important.\n\n$U(1)_R$--symmetric SUSY is an extension of the MSSM in which the superpartners of the SM particles are charged under a global $U(1)$ symmetry. The SM particles are not charged under this symmetry.  This model was introduced \\cite{Kribs:2007ac} to solve the SUSY \\emph{CP} and flavor problems. Due to the $U(1)_R$ symmetry, Majorana masses for the gauginos are forbidden as well as left-right mixing of sfermions. Hence, \\emph{e.g.}, one-loop diagrams that would generate a large electric dipole moment for fermions are suppressed, solving the SUSY \\emph{CP} problem. Similar arguments follow for the flavor problem.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nSuperfields\t&\t$U(1)_R$\t&\t$U(1)_{R-L}$ \\\\\n\\hline\\hline\n$Q, U^c, D^c$\t&\t 1\t&\t1 \\\\\n$L$\t&\t1\t&\t0 \\\\\n$E^c$\t&\t1\t&\t2\t\\\\\t\n\\hline\n$H_{u,d}$\t&\t0\t&\t0\t\\\\\n$R_{u,d}$\t&\t2\t&\t2\t\\\\\n\\hline\n$W_{\\tilde{B},\\tilde{W},\\tilde{g}}$\t&\t1\t&\t1\t\\\\\n$\\Phi_{S,T,\\mathcal{O}}$\t&\t0\t&\t0\t\\\\\n\\hline\ngravitino\/goldstini & 1 &  1 \\\\ \n\\hline\n\\end{tabular}\n\\caption{The relevant field content of the model. (SM charges are not shown.) $\\Phi_{S,T,\\mathcal{O}}$ are superfields which has the same SM charges as $W_{\\tilde{B},\\tilde{W},\\tilde{g}}$ and their fermionic components, $S, T,\\mathcal{O}$ are the Dirac partners of the bino, wino and the gluino respectively. The fermionic components of the superfields $R_{u,d}$ are the Dirac partners of the Higgsinos $\\tilde{h}_{u,d}$.} \\label{table:fields}\n\\end{table}\n\nPhenomenologically one novel aspect of $U(1)_R$--symmetric SUSY is that gauginos in this model are not Majorana fermions, since they are charged under a global symmetry, but are instead Dirac particles. In order to make gauginos into Dirac fermions, adjoint fields are added for each SM gauge field with opposite $U(1)_R$ charges \\cite{Fox:2002bu}. These new fields, $\\Phi_{S,T,\\mathcal{O}}$ are called singlino, tripletino and octino respectively and their fermionic components, $S,T,\\mathcal{O}$, become the Dirac partners of the bino, weakino and the gluino\\footnote{Dirac gauginos have been studied in the literature extensively. See, \\emph{e.g.}, \\cite{Benakli:2010gi, Benakli:2008pg, Goodsell:2012fm}}.  The Dirac nature of gauginos means that $t$-channel gluino exchange diagrams which contribute to squark pair production are suppressed and the rate for squark production at the LHC is reduced, allowing for lighter squarks.  Such models have been dubbed ``super-safe\" \\cite{Kribs:2012gx}.  Furthermore, the minimal incarnation of Dirac gauginos, supersoft SUSY breaking \\cite{Fox:2002bu}, has only a $D$-term spurion leading to improved renormalization properties.  Sfermions only receive finite contributions to their mass, rather than logarithmically divergent as in $F$-term breaking scenarios.  The ratio between gluino and squark masses is also larger, $m_{\\tilde{g}}\/m_{\\tilde{f}}\\sim 5-10$, than in alternative scenarios. \n\nThe unbroken $R$-symmetry forbids the usual $\\mu$-term.  In order to give mass to higgsinos, superfields $R_{u,d}$, with the same SM charge as the Higgs superfield but opposite $U(1)_R$ charges are added.  While the usual two Higgs doublets $H_{u,d}$ acquire vacuum expectation values (vev), $R_{u,d}$ do not.\n\nHere we will be considering several sources of SUSY breaking, both $F$- and $D$-term.  We envision two sectors each of which separately break SUSY. The first contains both a $D$-term and $F$-term spurion, of comparable size, and is coupled to the fields in the supersymmetric standard model.  The second is not coupled to the standard model, except through gravity and has a higher SUSY breaking scale than the first sector.  We are agnostic as to whether this is in $F$- or $D$-terms, or both, and parametrize the SUSY breaking simply as $F_2$. This second sector will raise the mass of the gravitino ($m_{3\/2}$) and provide an additional goldsitino with tree-level mass $2 m_{3\/2}$ \\cite{Cheung:2010mc,Cheung:2011jq}.\n\n\n\n\n\n\\subsection{SUSY breaking and superpartner masses}\n\nWe focus for now on the effects of the SUSY breaking that is communicated non-gravitationally to the SM.  SUSY is broken in a hidden sector which communicates with the visible sector at the messenger scale $\\Lambda_M$. SUSY breaking is incorporated via the spurions,\n\\begin{align}\nW'_\\alpha = \\theta_\\alpha D~,\\qquad X=\\theta^2 F~.\n\\end{align}\nWe assume that $X$ transforms non-trivially under some symmetry of the SUSY-breaking sector so that gauginos do not have Majorana masses of the form $\\int d^2\\theta (X\/\\Lambda_M)W_\\alpha W^\\alpha$, where $W^\\alpha$ is a SM gauge field strength superfield. $W'_\\alpha$ is the field strength of a hidden $U(1)'$ which gets a $D$-term vev. In this case, Dirac gaugino masses come from the supersoft term \\cite{Fox:2002bu}\n\\begin{align}\\label{eq:supersoftop}\n\\int d^2\\theta\\, \\frac{\\sqrt{2} c_i}{\\Lambda_M} W'_\\alpha W_i^\\alpha\\Phi_i~,\n\\end{align}\nwhere $c_i$ is a dimensionless coefficient, that we take to be $\\mathcal{O}(1)$, and $i=\\tilde{B}, \\tilde{W}, \\tilde{g}$. This operator can be generated by integrating out messenger fields, of mass $\\sim \\Lambda_M$,  charged under both the SM and the $U(1)'$. The Dirac mass of the gaugino is $M_i=c_i D\/\\Lambda_M$.  \nThe operator (\\ref{eq:supersoftop}) also gives a mass to the scalar adjoint, while leaving the pseudoscalar massless\\footnote{Pseudoscalars in the extended superpartners can acquire masses through another soft term of the form $\\int d^2\\theta \\frac{W'_\\alpha W^{'\\alpha}}{\\Lambda_M^2}\\Phi_i^2$~\\cite{Fox:2002bu}.}, and introduces a trilinear coupling between the scalar adjoint, the SM and the $D$-term.  At one loop a scalar charged under gauge group $i$ receives a \\emph{finite} soft mass from the gaugino\n\\begin{equation}\nm^2 = \\frac{C_i \\alpha_i \\left(M_i\\right)^2}{\\pi}\\log 4~,\n\\end{equation}\nwhere $C_i$ is the quadratic Casimir of the scalar and we have assumed the scalar adjoint only receives a mass from (\\ref{eq:supersoftop}). We will be interested in a spectrum with the bino in the $\\mathcal{O}(100~{\\rm GeV} - {\\rm TeV})$ mass range and the squarks in the same range, but heavier than the bino.  If the sfermion masses are entirely from the supersoft operator this means the right-handed sleptons would be below the LEP bound.  Thus, at least for the right handed sleptons, we include additional sources of SUSY breaking through the operator\n\\begin{equation}\n\\int d^4\\theta\\, \\frac{X^\\dagger X}{\\Lambda_M^2}c_{ij} \\Psi^\\dagger_i \\Psi_j~,\n\\end{equation} \nwith $\\Psi_i$ a right handed lepton superfield.  We assume $F\\sim D$ and $c_{ij}\\sim 1$.\nThe squarks can be heavier than the bino from the finite supersoft contributions alone, as long as the gluino is sufficiently heavy, in the multi-TeV mass range.  \n\n\nAs all global symmetries, $U(1)_R$ is broken due to gravity. Anomaly mediation \\cite{Randall:1998uk} generates a Majorana mass for the gauginos proportional to the gravitino mass, $m_{3\/2}$, \n\\begin{align}\nm_i=\\frac{\\beta(g)}{g}m_{3\/2}~,\n\\end{align}\nwhere $\\beta(g)$ is the beta function for the appropriate SM gauge coupling $g$.  The gravitino picks up mass from all sources of SUSY breaking, $m_{3\/2}^2 = \\sum_i (F_i^2 + D_i^2\/2)\/\\sqrt{3} M_{\\rm Pl}^2$.  We assume that the messenger scale $\\Lambda_M$ is below the Planck scale and thus $m_i\\ll M_i$.  We ignore the small anomaly mediated corrections to scalar masses.\nNote that $U(1)_R$-breaking Majorana masses for the Dirac partners, $\\tilde{m}_i \\Phi_i \\Phi_i$, could also be generated. We assume these are much smaller than the Dirac gaugino masses as well. (For LHC studies we will set the Majorana masses to zero.)\nDue to the small anomaly-mediated Majorana gaugino masses, the gauginos are pseudo-Dirac particles. \n\n\n\n\n\\subsection{Neutrino masses}\n\\label{sec:neutrinomasses}\n\nIt has been shown in \\cite{Coloma:2016vod} that the operators,\n\\begin{equation}\n\\frac{f_i}{\\Lambda_M^2}\\int d^2\\theta\\, W'_\\alpha W_{\\tilde{B}}^\\alpha H_u L_i \\ \\ \\ \\text{and}\\ \\ \\ \\frac{d_i}{\\Lambda_M}\\int d^4\\theta\\,  \\phi^\\dagger \\Phi_S H_u L_i\n\\end{equation}\n(where $\\phi=1+\\theta^2 m_{3\/2}$) can generate two non-zero neutrino masses through the Inverse Seesaw mechanism\n\\cite{Mohapatra:1986aw,Mohapatra:1986bd}, with the bino--singlino pair acting as a pseudo-Dirac right-handed neutrino.  These operators can be generated by integrating out two pairs of gauge singlets $N_i,N_i'$, with R-charge 1 and lepton number $\\mp 1$.\n\n\nOnce the Higgs acquires a vev the neutrino-bino mass mixing matrix, in the basis $(\\nu_i, \\tilde{B}, S)$, is \n\\begin{equation}\n\\mathbb{M} =\t\\begin{pmatrix}\n\t\t\t0_{3\\times3}\t&\t\\mathbf{Y}v\t&\t\\mathbf{G}v \\\\\n\t\t\t\\mathbf{Y}^Tv\t&\tm_{\\tilde{B}}\t&\tM_{\\tilde{B}}\t\t\\\\\n\t\t\t\\mathbf{G}^Tv\t&\tM_{\\tilde{B}}\t\t\t&\tm_S\n\t\t\t\\end{pmatrix}~, \\label{eq:Mneutrino}\n\\end{equation}\t\t\t\nwith $Y_i= f_i M_{\\tilde{B}}\/\\Lambda_M$ and $G_i = d_i m_{3\/2}\/\\Lambda_M$.  The mass matrix $\\mathbb{M}$ has an Inverse Seesaw structure with $\\mathbf{G}\\ll \\mathbf{Y}$.  The light neutrino masses do not depend on the Dirac bino mass and at normal ordering they are given by\n\\begin{align}\nm_1 = 0,~~~m_2=\\frac{m_{3\/2}\\, v^2}{\\Lambda_M^2}(1-\\rho),~~~ m_3=\\frac{m_{3\/2}\\, v^2}{\\Lambda_M^2}(1+\\rho)~,\n\\end{align}\nwhere $\\rho = \\hat{\\mathbf{Y}}\\cdot\\hat{\\mathbf{G}}$,  which is determined by the neutrino mass splittings to be $\\simeq0.7$. We ignore the small corrections, $\\mathcal{O}(m_S\/M_D)$, due to Majorana masses.\nParametrically the neutrino masses are\n \\begin{align}\n m_\\nu\\simeq (2-20)\\times10^{-2}~{\\rm eV}\\left(\\frac{m_{3\/2}}{10~{\\rm keV}}\\right)\\left(\\frac{100~{\\rm TeV}}{\\Lambda_M}\\right)^2~.\n \\end{align}\nTo recover the correct neutrino mixing matrix, and by setting all phases in the neutrino sector to zero, $\\mathbf{Y}$ and $\\mathbf{G}$ must have the approximate form\n\\begin{equation}\n\\mathbf{Y}\\simeq \\frac{M_{\\tilde{B}}}{\\Lambda_M}\n\\begin{pmatrix}\n0.35 \\\\\n0.85 \\\\\n0.35\n\\end{pmatrix},\\quad\n\\mathbf{G}\\simeq \\frac{m_{3\/2}}{\\Lambda_M}\n\\begin{pmatrix}\n-0.06 \\\\\n0.44 \\\\\n0.89\n\\end{pmatrix}~. \\label{eq:YG}\n\\end{equation}\n\nLow-energy searches for lepton flavor violation place strong constraints on these couplings. The strongest current constraint comes from ${\\rm Br}(\\mu\\to e\\gamma)$ \\cite{TheMEG:2016wtm} and places a lower bound on the messenger scale $\\Lambda_M>30$~TeV, independent of $M_{\\tilde{B}}$ or $m_{3\/2}$.  Future experiments, \\emph{e.g.} Mu2e \\cite{Bartoszek:2014mya}, will probe $\\Lambda_M\\sim100$~TeV. We use the word ``bi$\\nu$o\" from now on to refer to the pseudo-Dirac bino in order to emphasize that it is involved in neutrino-mass generation.\n\n\n\\subsection{Neutralino mixing}\n\nIn $R$-symmetric models the Higgs sector is extended by two additional $SU(2)$ doublets, $R_{u,d}$, that do not acquire a vev.  Once electroweak symmetry is broken there is mixing between $R_{u,d}$ and the adjoint fermions, $S$ and $T$, in addition to the usual wino-bi$\\nu$o mixing.  However, the neutrinos only mix with the bi$\\nu$o.  Significant neutralino mixing only changes the collider phenomenology and does not affect the generation of neutrino masses, which happens only through bi$\\nu$o--neutrino mixing.  We follow \\cite{Kribs:2008hq} to investigate the neutralino mixing in this model.\n\nThe relevant part of the superpotential for neutralino mixing is \n\\begin{align}\\label{eq:Wneutralino}\n\\mathcal{W}=\\mu_u H_u R_u + \\mu_d H_d R_d + \\Phi_S \\left( \\lambda^u_{\\tilde{B}} H_u R_u + \\lambda^d_{\\tilde{B}} H_d R_d  \\right) + \\Phi_T \\left( \\lambda^u_{\\tilde{W}} H_u R_u + \\lambda^d_{\\tilde{W}} H_d R_d  \\right).\n\\end{align}\nAfter EW symmetry breaking, together with Dirac gaugino masses, kinetic terms and ignoring the small Majorana gaugino masses, (\\ref{eq:Wneutralino}) generates the neutralino mass matrix \n\\begin{align} \\label{eq:Mneutralino}\n\\mathbb{M}_N=\\begin{pmatrix}\n\t\tM_{\\tilde{B}}\t&\t0\t&\t\\frac{g_Y v_u}{\\sqrt{2}}\t&\t-\\frac{g_Y v_d}{\\sqrt{2}} \\\\\n\t\t0\t\t\t&\tM_{\\tilde{W}}\t&\t-\\frac{g_2 v_u}{\\sqrt{2}}\t&\t\\frac{g_2 v_d}{\\sqrt{2}} \\\\\n\t\t\\frac{\\lambda_{\\tilde{B}}^u v_u}{\\sqrt{2}}\t&\t-\\frac{\\lambda_{\\tilde{W}}^u v_u}{\\sqrt{2}}\t&\t\\mu_u\t&\t0\t\\\\\n\t\t-\\frac{\\lambda_{\\tilde{B}}^d v_d}{\\sqrt{2}}\t&\t\\frac{\\lambda_{\\tilde{W}}^d v_d}{\\sqrt{2}}\t&\t0\t&\t\\mu_d\n\\end{pmatrix}\n\\end{align}\nin the basis $( \\tilde{B},\\tilde{W}, \\tilde{R}_u, \\tilde{R}_d )\\times ( S,T,\\tilde{h}_u, \\tilde{h}_d )$, where $\\tilde{R}_{u,d}$ are the fermionic components of the superfield $R_{u,d}$ (see Table \\ref{table:fields}). Here $v_{u,d}\\equiv \\langle H_{u,d}\\rangle$ are the up\/down-type Higgs vevs defined as $v_u^2+v_d^2=v^2\/2 \\simeq(174~{\\rm GeV})^2$ and $M_{\\tilde{B},\\tilde{W}}$ are the bi$\\nu$o and wino Dirac masses defined in (\\ref{eq:supersoftop}).\n\nThe neutralino mass matrix $\\mathbb{M}_N$ has a rather simple form due to the Dirac nature of gauginos. It further simplifies for large $\\tan\\beta \\equiv v_u\/v_d$. In this limit\n\\begin{align}\n\\mathbb{M}_N\\simeq\\begin{pmatrix}\n\t\tM_{\\tilde{B}}\t&\t0\t&\t\\frac{g_Y v}{2}\t&\t0 \\\\\n\t\t0\t\t\t&\tM_{\\tilde{W}}\t&\t-\\frac{g_2 v}{\\sqrt{2}}\t\t&\t0 \\\\\n\t\t\\frac{\\lambda_{\\tilde{B}}^u v}{2}\t&\t-\\frac{\\lambda_{\\tilde{W}}^u v}{2}\t&\t\\mu_u\t&\t0\t\\\\\n\t\t0\t&\t0\t&\t0\t&\t\\mu_d\n\\end{pmatrix}.\n\\end{align}\nIt can immediately be seen that one of the states, with mass $\\mu_d$, decouples. Furthermore, in the limit where $\\lambda_{\\tilde{B}}^u = \\lambda_{\\tilde{W}}^u=0$, there is no mixing between the bi$\\nu$o, weakino and the Higgsinos. For simpicity, we assume a hierarchy $\\mu>M_{\\tilde{B},\\tilde{W}}$ and work in this limit, where the lightest neutralino is a pure bi$\\nu$o. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.7\\textwidth]{spectrum_new.pdf}\n\\caption{Approximate spectrum of particles in the model described in Section \\ref{sec:model}.}\\label{fig:spectrum}\n\\end{figure}\n\n\\subsection{Gravitino\/Goldstino dark matter}\n\\label{sec:goldstini}\n\nAs discussed in Section~\\ref{sec:model}, the model has two independent sectors that break supersymmetry.  The breaking at the lower scale involves a $D$-term spurion but for the purposes of the discussion here it is sufficient to parametrize the two breaking scales as $\\tilde{F}_{(1,2)}$, with $\\tilde{F}^2=F^2+D^2\/2$ and $\\tilde{F}_{2}>\\tilde{F}_{1}$.  A viable neutrino mass spectrum, and the spectrum of superparticles we are interested in, is achieved with $F_1\\sim (10\\ensuremath{\\rm~TeV}\\xspace)^2$ and $F_2\\sim  (10^4\\ensuremath{\\rm~TeV}\\xspace)^2$, as shown in Figure~\\ref{fig:spectrum}.\n\nSince there are two independent sources of SUSY breaking, there are two goldstini \\cite{Cheung:2010mc,Cheung:2011jq}, of which one linear combination is eaten by the gravitino to have mass $m_{3\/2} = \\sqrt{(\\tilde{F}_{1})^2 + (\\tilde{F}_{2})^2}\/\\sqrt{3}M_{\\mathrm{Pl}}\\sim 10\\ensuremath{\\rm~keV}\\xspace$, while the other is twice as heavy, at tree level.  Furthermore, the couplings of the uneaten goldstini are enhanced relative to the gravitino's by a factor of $\\tilde{F}_2\/\\tilde{F}_1$.  \n\nBoth the gravitino, $\\tilde{G}$, and the golstino, $\\zeta$, are lighter than the other $R$-symmetry-odd particles.  The goldstino can decay into a gravitino and SM particles, \\emph{e.g.} $\\zeta\\rightarrow \\tilde{G} \\psi\\bar{\\psi}$.  The lifetime for this process is \n\\begin{equation}\n\\tau_{\\zeta\\rightarrow \\tilde{G}\\psi\\bar{\\psi}} \\sim \\frac{9\\pi^3}{4} \\frac{M_{\\mathrm{Pl}}^4}{m_{3\/2}^5}\\left(\\frac{\\tilde{F}_1}{\\tilde{F}_2}\\right)^2~.\n\\end{equation}\nFurthermore, even though the gravitino in this model is the LSP, it can decay into neutrinos and photons via the neutrino-bi$\\nu$o mixing. The gravitino lifetime is $\\tau = \\Gamma^{-1} \\sim \\frac{M_{\\rm Pl}^2}{\\theta^2 m_{3\/2}^3}\\sim 10^{39}$~s for $m_{3\/2}\\sim10$~keV and the bi$\\nu$o-neutrino mixing angle $\\theta \\sim Y v\/M_{\\tilde{B}}\\sim 10^{-3}$. Thus, for the range of parameters we are interested in, both the gravitino and goldstino are cosmologically stable and may contribute to dark matter.\n\nIt has been shown that a gravitino with mass $O(1-10~{\\rm keV})$ could be a warm dark matter candidate \\cite{Takayama:2000uz, Gorbunov:2008ui, Cheung:2011nn, Monteux:2015qqa}. The parameter region we study in this model suggests that gravitino could be a dark matter candidate if $T_{\\rm reh} \\sim O({\\rm TeV})$. However, with the same parameters, goldstino would be overproduced since its couplings are enhanced by a factor of $\\tilde{F}_2\/\\tilde{F}_1$. The abundance of goldstino depends on the production mechanism and $T_{\\rm reh}$. (Depending on the masses of the gravitino, goldstino and other sparticles, the dominant production channel can be either decays or scatterings.) If $T_{\\rm reh}< M_{\\tilde{B},\\tilde{q}}$, the sparticle abundance, and hence the abundance of goldstinos, will be suppressed. One expects a range of reheat temperatures where there will be just enough goldstino\/gravitino to make up the correct dark matter abundance. Finding this range requires detailed calculations for allowed range of sparticle masses. We leave this for future work.\n\n\n\n\n\\section{LHC phenomenology}\\label{sec:LHCpheno}\n\nIn this section we recast current LHC searches to find the constraints on the model described in Section \\ref{sec:model}. In order to make the LHC analysis more tractable, we assume the following mass hierarchy for the SUSY particles (see Figure \\ref{fig:spectrum}). \n\\begin{itemize}\n\\item Gravitino is the LSP with $m_{3\/2}\\sim O(10~ {\\rm keV})$.\n\\item Next-to-lightest supersymmetric particle (NLSP) is a pure bi$\\nu$o, and the other neutralinos are decoupled. Note that there are two physical bi$\\nu$o states with masses $M_{\\tilde{B}}\\pm \\frac{m_{\\tilde{B}}+m_S}{2}$. For simplicity we take the Majorana masses to be zero in the LHC analysis. Hence the physical bi$\\nu$o mass is $M_{\\tilde{B}}$. \n\\item Squarks are degenerate and heavier than the bi$\\nu$o. We do not apply any flavor tags in the analyses and only consider the first two generations of squarks, which gives a conservative estimate for the rate. \n\\item Slepton masses are of the same order as squarks, and slepton production is irrelevant.\n\\item As expected for an $R$-symmetric model, the gluino and charginos are considerably heavier than the sfermions and the squark production cross section is reduced due to the suppressed $t$-channel gluino contribution. \n\\end{itemize}\n\nThe LHC phenomenology of this model should be compared to both models with right-handed neutrinos and to the MSSM. \n\\begin{enumerate}\n\\item In models with right-handed neutrinos that address the origin and size of the neutrino masses, the SM singlets are only produced in EW processes via their mixing with the SM neutrinos. Due either to small mixing angles between the right-handed neutrinos and the SM neutrinos or to large right-handed neutrino masses, their production rates are greatly suppressed at the LHC. However, the bi$\\nu$o can be produced in decays of colored particles in the model we consider. Hence this is a neutrino-mass model that can currently be probed at the LHC.  \n\n\\item In this model all supersymmetric particles need to be produced in sparticle--antisparticle pairs due to the $U(1)_R$ symmetry. (At 13~TeV LHC, the main $\\tilde{q}\\tilde{q}^\\dagger$-production channel is gluon fusion.) Furthermore, some sparticle-production channels, \\emph{e.g.} t-channel gluino exchange, are not present due again to the $U(1)_R$ symmetry. Hence it is expected that the constraints on this model are weaker than the ones on MSSM \\cite{Kribs:2012gx}. Furthermore in this model the lightest neutralino, namely the bi$\\nu$o, decays promptly and produces a combination of jets, leptons and missing energy.\n\\end{enumerate}\n\n\n\\subsection{Expected signals and search strategies}\n\\label{sec:signals}\n\nDue to the sparticle spectrum we assume, bi$\\nu$os are predominantly produced via squark decays with $Br(\\tilde{q}\\to q \\tilde{B}^\\dagger)=1$. The bi$\\nu$o subsequently decays through one of four possible modes: (i) $\\tilde{B}\\to\\tilde{G}\\gamma $; (ii) $\\tilde{B}\\to W^-\\ell^+$; (iii) $\\tilde{B}\\to Z\\bar{\\nu}$; and \n(iv) $\\tilde{B}\\to h\\bar{\\nu}$. The first decay mode is strongly suppressed by the Planck mass, $\\Gamma(\\tilde{B}\\to\\tilde{G}\\gamma)\\sim \\frac{M_{\\tilde{B}}^5}{M_{\\rm Pl}^2 m_{3\/2}^2}\\sim 10^{-8}$~eV. The rest of the decay modes are only suppressed by the neutrino-bi$\\nu$o mixing angle and their branching ratios are approximately equal to 1\/3. (Note that due to the $U(1)_{R-L}$ symmetry, $\\tilde{B}\\to W^+\\ell^-$ decay is not allowed.)  The total decay width of the bi$\\nu$o is $\\Gamma_{tot}\\sim M_{\\tilde{B}}Y^2\\sim M_{\\tilde{B}}^3\/\\Lambda_M^2\\sim O(10~{\\rm MeV})$ for $M_{\\tilde{B}}=500~$GeV and $\\Lambda_M=100~$TeV. Hence, it decays promptly to final states, which include a combination of jets, leptons and missing energy. We show some of the final states with large branching fractions in Table \\ref{table:BR} and Figures \\ref{fig:signals1}-\\ref{fig:signals2}.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nSignal\t&\tBranching fraction\t&\tLHC searches \\\\\n\\hline\\hline \n$6 j + \\slashed{E}_{\\mathrm{T}}$\t&\t20$\\%$\t&\tATLAS \\cite{ATLAS-CONF-2017-022}, CMS \\cite{Khachatryan:2016kdk, Khachatryan:2016epu, Sirunyan:2017cwe} \\\\\n\\hline\n$6 j + 1\\ell + \\slashed{E}_{\\mathrm{T}}$\t&\t15$\\%$\t&\tATLAS \\cite{Aad:2016qqk, Aaboud:2016lwz}, CMS \\cite{Khachatryan:2016epu, Khachatryan:2016iqn} \\\\\n\\hline\n$4 j + 2\\ell + \\slashed{E}_{\\mathrm{T}}$\t&\t6$\\%$\t&\tATLAS \\cite{Aaboud:2016zpr, Aaboud:2016qeg}, CMS \\cite{Khachatryan:2016iqn, Khachatryan:2017qgo} \\\\\n\\hline\n$4 j + \\slashed{E}_{\\mathrm{T}}$\t&\t5$\\%$\t&\tATLAS \\cite{ATLAS-CONF-2017-022}, CMS \\cite{Khachatryan:2016kdk, Khachatryan:2016epu, Sirunyan:2017cwe} \\\\\n\\hline\n$6 j + 2\\ell$\t&\t3$\\%$\t&\tATLAS \\cite{Aaboud:2016qeg}, CMS \\cite{CMS-PAS-EXO-17-003}\\\\\n\\hline\n\\end{tabular}\n\\caption{Some of the signals that are produced by bi$\\nu$o production and subsequent decays in the model described in Section~\\ref{sec:model} with their branching fractions and relevant LHC searches. Here the leptons $\\ell = e,\\mu$ and $j=u,d,s,c$.} \\label{table:BR}\n\\end{table}\n\n\n\n\n\\begin{figure*}[t]\n  \\centering\n  \\begin{subfigure}[t]{0.5\\textwidth}\n  \\includegraphics[width=\\textwidth]{signal6j.pdf}\\label{fig:6jets}\n  \\caption{6 jets + missing energy}\n  \\end{subfigure}%\n  ~\n    \\begin{subfigure}[t]{0.5\\textwidth}\n  \\includegraphics[width=\\textwidth]{signal4j.pdf}\\label{fig:4jets}\n  \\caption{4 jets + missing energy}\n  \\end{subfigure}\n   \\caption{Final states with jets and missing energy. We recast current SUSY searches at ATLAS and CMS for this signal.} \\label{fig:signals1}\n\\end{figure*}\n\n\\begin{figure*}[t]\n  \\centering\n    \\begin{subfigure}[b]{0.5\\textwidth}\n  \\includegraphics[width=\\textwidth]{signal6j1lep.pdf}\\label{fig:4jets}\n  \\caption{6 jets + 1 lepton + missing energy}\n  \\end{subfigure}%\n    ~\n  \\begin{subfigure}[b]{0.5\\textwidth}\n \\includegraphics[width=\\textwidth]{signal6j2lep.pdf}\\label{fig:6jets2lep}\n \\caption{6 jets + 2 leptons}\n \\end{subfigure}\n  \\caption{Final states with leptons and missing energy. Leptoquark searches are recast for these signals.} \\label{fig:signals2}\n\\end{figure*}\n\n\nWe emphasize the importance of final states with leptons, \\emph{e.g.} $6j+2\\ell$ and $6j + 1\\ell+\\slashed{E}_{\\mathrm{T}}$, as  smoking-gun signals in determining if bi$\\nu$o is the source of neutrino mass generation, see Fig.\\ref{fig:signals2}. The bi$\\nu$o-neutrino mixing angle is $\\theta_i \\simeq \\frac{Y_i v}{M_{\\tilde{B}}}$ where $Y_i$ is given in (\\ref{eq:YG}). The branching ratio of bi$\\nu$o into different lepton species is fully determined by the neutrino mixing parameters. For example, in searches for first- and second-generation leptoquarks, relative rates of $ee:\\mu\\mu = 1:16$ and $e\\nu:\\mu\\nu= 1:2$ are expected\\footnote{Note that these branching fractions are given for the case where the phases in the PMNS matrix are set to zero. The matrix elements, hence the branching ratios, will change for non-zero  phases~\\cite{Gavela:2009cd}.}. \n\n\\subsection{Analysis}\n\nOur model is implemented in \\feynrules{} \\cite{Alloul:2013bka} and the events are generated with \\madgraph{5} \\cite{Alwall:2014hca}, using \\pythia{8} \\cite{Sjostrand:2014zea} for parton shower and hadronization, and \\delphes{} \\cite{deFavereau:2013fsa} for detector simulation at $\\sqrt{s}=13$ TeV and $\\mathcal{L}=36~\\text{fb}^{-1}$. We use the default settings for jets in \\madgraph{5} with $R=0.4, ~p_{Tj}>20$ GeV and $|\\eta_j|<5$. We generate signal events for bi$\\nu$os in the mass range $100~{\\rm GeV}<\\ensuremath{M_{\\tilde{B}}}<\\ensuremath{M_{\\tilde{q}}}$ with a common squark mass for first and second generation squarks, $200~{\\rm GeV}<\\ensuremath{M_{\\tilde{q}}}<1200~$GeV, in 50~GeV mass increments. We set all other sparticle masses to 10~TeV such that they are decoupled. As the bi$\\nu$o mass gets closer to the squark mass, the computational time required to generate events increases. Hence, we do not consider splittings smaller than $25\\ensuremath{\\rm~GeV}\\xspace$, {\\it i.e.}\\ $\\ensuremath{M_{\\tilde{q}}}-\\ensuremath{M_{\\tilde{B}}} \\ge 25~\\ensuremath{\\rm~GeV}\\xspace$. For $\\ensuremath{M_{\\tilde{B}}}\\lesssim 90~$GeV, the gauge bosons are off-shell and the phase space and the energy distribution of the final states are different. We leave a study of light bi$\\nu$os to future work and focus on $\\ensuremath{M_{\\tilde{B}}}>100~$GeV. \n \nWe find that currently the most constraining search is the jets$ + \\slashed{E}_{\\mathrm{T}}$ final state due its large branching ratio and the integrated luminosity used in available analyses.  At the partonic level there are processes leading to 6q$ + \\slashed{E}_{\\mathrm{T}}$ and 4q$ + \\slashed{E}_{\\mathrm{T}}$ final states, see Figure~\\ref{fig:signals2}.  We analyze this search in detail and use it to constrain the parameter space of the bi$\\nu$o model.\n\n\nWe use the $m_{\\rm eff}$-based  analysis given by ATLAS \\cite{ATLAS-CONF-2017-022}. The observable $m_{\\rm eff}$ is defined as the scalar sum of the transverse momenta of the leading jets and missing energy, $\\slashed{E}_{\\mathrm{T}}$. Taken together with $\\slashed{E}_{\\mathrm{T}}$, $m_{\\rm eff}$ strongly suppresses the multijet background. There are 24 signal regions in this analysis. These regions are first divided according to jet multiplicities (2-6 jets).\n Signal regions with the same jet multiplicity are further divided according to the values of $m_{\\rm eff}$ and the $\\slashed{E}_{\\mathrm{T}}\/m_{\\text{eff}}$ or $\\slashed{E}_{\\mathrm{T}}\/\\sqrt{H_T}$ thresholds. \n In each signal region, different thresholds are applied on jet momenta and pseudorapidities to reduce the SM background.\nConstraints on the smallest  azimuthal  separation  between $\\slashed{E}_{\\mathrm{T}}$ and \n the momenta of any of the reconstructed jets further reduces the multi-jet background. Two of the signal regions require two large radius jets and in all  signal regions the required  jet momentum is $p_T>50~$GeV and missing energy $\\slashed{E}_{\\mathrm{T}}>250~$GeV. \n The thresholds on the observables which characterize the signal regions have been chosen to target models with squark or gluino pair production and direct decay of squarks\/gluinos or one-step decay of squark\/gluino via an intermediate chargino or neutralino.\n\nIn order to identify the allowed parameter points we compare the  signal cross section to the measured cross section limits at  95$\\%$ C.L. in all 24 signal regions using the code from \\cite{Asadi:2017qon}. If the signal cross section of a parameter point exceeds the measured  cross section at 95$\\%$ C.L. in at least one bin we take this parameter point to be ruled out. \n  \nWe also analyze the expected exclusion limits at the end of  LHC Run 3 with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~\\text{fb}^{-1}$, by rescaling with the luminosity the expected number of signal and background events, as given in~\\cite{ATLAS-CONF-2017-022}. In order to obtain the allowed parameter region at a high-luminosity LHC we use the median expected exclusion significance \\cite{Kumar:2015tna}\n\\begin{align}\nZ_{exc}=\\Big[2\\left(s-b \\log\\left(\\frac{b+s+x}{2b}\\right)-\\frac{b^2}{\\Delta_b^2}\\log\\left(\\frac{b-s+x}{2b}\\right)\\right)-(b+s-x)(1+\\tfrac{b}{\\Delta_b^2})\n\\Big]^{1\/2}~,\n\\end{align}\nwith\n\\begin{align}\nx=\\big[(s+b)^2-4sb\\tfrac{\\Delta_b^2}{(b+\\Delta_b^2)}\\big]^{1\/2}~,\n\\end{align}\nwhere $s$ is the signal, $b$ is background and $\\Delta_b$ is the uncertainty on the background prediction. For a 95$\\%$ C.L. median exclusion, we require $Z_{exc}>1.645$.  We assume, as a conservative estimate, that the relative background uncertainty after $300~\\text{fb}^{-1}$ remains the same as it is now, as presented in \\cite{ATLAS-CONF-2017-022}.  The estimate that $\\Delta_b\/b$ is constant could be improved upon, especially if the background is estimated from data in sidebands.\n\n\n\\subsection{Results and discussion}\n\\label{sec:results}\n\nWe show 95\\% exclusion limits on squark and bi$\\nu$o masses for current and forecasted searches in Figure~\\ref{fig:Plotexcatlas}.\nWe find that squarks heavier than 950~GeV are not excluded for any bi$\\nu$o mass by current LHC data with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=36~{\\rm fb}^{-1}$. In the mass regions we analyzed, bi$\\nu$o masses 100--150~GeV are not currently excluded for squark masses above 350~GeV, as the resulting jet momenta and missing energy do not pass the search cuts. We also project limits for a high-luminosity LHC with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=300~{\\rm fb}^{-1}$. This forecast shows that even with this luminosity upgrade, as long as the same cuts are used in the analysis, the LHC can probe squark masses up to 1150~GeV. However, bi$\\nu$o masses lighter than 150~GeV for $\\ensuremath{M_{\\tilde{q}}}>800$~GeV will still be allowed. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{Plotexcfore300.pdf\n\\caption{\\label{fig:Plotexcatlas} Current and forecasted 95\\% exclusion limits from searches for  jets$+ \\slashed{E}_{\\mathrm{T}}$ final state in the squark mass ($\\ensuremath{M_{\\tilde{q}}}$) -- bi$\\nu$o mass ($\\ensuremath{M_{\\tilde{B}}}$) plane. The dark red region is excluded by our recast of the ATLAS analysis \\cite{ATLAS-CONF-2017-022} which uses $\\mathcal{L}=36~{\\rm fb}^{-1}$ data at $\\sqrt{s}=$~13~TeV. The red dashed line shows  a  forecast for $\\mathcal{L}=300~\\text{fb}^{-1}$ at $\\sqrt{s}=$~13~TeV. We have not analyzed the parameter ranges in the gray striped regions, which correspond to regions where bi$\\nu$o is heavier than squarks or the bi$\\nu$o is lighter than the SM gauge bosons.}\n\\end{figure}\n\nWe find that constraints in the parameter region we considered come from (4-6)-jet signal regions with small to medium values of $m_{\\rm eff}$ for $\\mathcal{L}=36~{\\rm fb}^{-1}$. For our forecast to $\\mathcal{L}=300~{\\rm fb}^{-1}$, we find that constraints for $\\ensuremath{M_{\\tilde{q}}}<800$~GeV mostly come from a 4-jet region with small $m_{\\rm eff}$ while a 6-jet region with medium  $m_{\\rm eff}$ is most constraining for $\\ensuremath{M_{\\tilde{q}}}>800$~GeV.\n\n\n\nIn discussing our results we emphasize the differences between this model and some other ($R$-symmetric) SUSY models. \n\\begin{enumerate}\n\\item In this model the gluinos are heavy and decoupled. Furthermore due to the $R$-symmetry some important squark production channels are not allowed. Hence, compared to the MSSM~\\cite{SUSYxsec}, the squark--antisquark production cross-section is $O(0.1)$ smaller. \n\\item Due to the sparticle spectrum we assume, squarks decay to a quark and the lightest neutralino 100\\% of the time. The lightest neutralino, which we take to be purely bi$\\nu$o, decays promptly to gauge bosons and leptons due to a broken $U(1)_{R-L}$ symmetry. In comparison to MSSM scenarios where the missing energy is carried by the neutralino, in this model there would be cascade decays and the missing energy is carried by light neutrinos. \n\\item Similarly, due to the large number of jets and how the missing energy is distributed in this model, constraints on squark and bi$\\nu$o masses are expected to be different than some other $R$-symmetric models, \\emph{e.g.} \\cite{Kribs:2012gx}. Although it is not straightforward to make a direct comparison, we point out that in \\cite{Kribs:2012gx} the LSP is massless and the most constraining signal region contains only 2 jets whereas in this model the constraining signal regions contain 4-6 jets. The authors in \\cite{Kribs:2012gx} mention as the LSP mass is increased to 300~GeV, all constraints disappear. However, note that, even with a finite mass, the LSP in that work does not decay.  We emphasize that we do not consider the region where $\\ensuremath{M_{\\tilde{B}}}<100~$GeV. In this region the bi$\\nu$o would decay via off-shell gauge or Higgs bosons. Due to the low mass of the bi$\\nu$o, final states may not pass the missing energy and jet momentum cuts in the current analysis. We leave an analysis of this region to future work.\n\n\\item The closest study to ours is done in \\cite{Frugiuele:2012kp}. In addition to some technical differences between the two models, in~\\cite{Frugiuele:2012kp} the authors fix the lightest neutralino mass to be 1~TeV while we do a scan over both the squark and the bi$\\nu$o masses. In \\cite{Frugiuele:2012kp} the limit on the squark mass is found to be $\\ensuremath{M_{\\tilde{q}}}\\simeq 650~$GeV by using an ATLAS jets+$\\slashed{E}_{\\mathrm{T}}$ analysis \\cite{Aad:2012hm} at $\\sqrt{s}=$~7~TeV with $\\mathcal{L}=4.7~{\\rm fb}^{-1}$ data. In our work we do not consider the region where $\\ensuremath{M_{\\tilde{B}}}> \\ensuremath{M_{\\tilde{q}}}$. In this region the bi$\\nu$o decays off-shell and it is expected that energy will be distributed to jets and missing energy democratically. \n We expect the bound on the squark masses coming from the ATLAS anaylsis we use~\\cite{ATLAS-CONF-2017-022} to be similar to that given by our most constraining signal region,  $5j+\\slashed{E}_{\\mathrm{T}}$, {\\it i.e.}~$\\ensuremath{M_{\\tilde{q}}}> 950$~GeV.\n\\end{enumerate}\n\nWe also analyzed the final state with $6 j + 2\\ell$, which is a possible smoking gun signature for this model as the branching fractions of the bi$\\nu$o to different lepton families is fully determined by the neutrino mixing parameters. We recast the CMS leptoquark analysis \\cite{CMS-PAS-EXO-17-003}, which looks for a final state of two muons and two jets produced in the decay of a leptoquark pair. We find that this analysis currently has a very small exclusion power due to the small signal-to-background ratio ($S\/B\\sim 10^{-2}$). \n\n\\section{Conclusions}\\label{sec:conclusions}\n\nLHC constraints on sparticle masses in the MSSM are becoming more and more stringent.  Avoiding these strong experimental constraints and keeping superpartners light often leads to considering extensions of the MSSM.  These extensions are characterized either by adding additional operators ({\\it e.g.}~R-parity violation) or adding additional fields ({\\it e.g.}~Dirac gauginos).  We studied one such extension, with additional fields, which allows for a global $U(1)_{R-L}$ symmetry on the supersymmetric sector.   This leads to phenomenology associated with both R-parity violation and Dirac gauginos.  It was previously shown in \\cite{Coloma:2016vod}  that the role of right-handed neutrinos can be played by one of these Dirac gauginos, the pseudo-Dirac bi$\\nu$o, and that the observed neutrino mass spectrum can be achieved.  \n\nWe considered a scenario where the lightest neutralino is a pure bi$\\nu$o, and this state is the lightest SM superpartner.  The squarks, which have a QCD production cross section, decay to the bi$\\nu$o.  The mixing of this state with SM neutrinos means that it in turn can decay, despite the presence of a $U(1)_{R-L}$ symmetry.   The bi$\\nu$o decays to a combination of quarks, leptons and missing energy.  We investigated the LHC constraints on this model and found the strongest comes from a recast of the most recent ATLAS analysis with $\\sqrt{s}=13~$TeV and $\\mathcal{L}=36~{\\rm fb}^{-1}$, see Figure~\\ref{fig:Plotexcatlas}. The constraints go up to only $\\ensuremath{M_{\\tilde{q}}}=950~$GeV and squarks as light as 350~GeV are allowed for $\\ensuremath{M_{\\tilde{B}}}=100-150~$GeV. We also forecast constraints for $\\mathcal{L}=300~{\\rm fb}^{-1}$ at $\\sqrt{s}=13~$TeV and show that high-luminosity LHC can probe up to $\\ensuremath{M_{\\tilde{q}}}=1150~$GeV if the same cuts for the jets+$\\slashed{E}_{\\mathrm{T}}$ analysis are used. However, even with the high-luminosity, low bi$\\nu$o masses cannot be excluded. The flavor of the charged lepton in the bi$\\nu$o decay depends upon the neutrino mixing parameters and thus the LHC is potentially sensitive to parameters in the neutrino sector, for instance through flavor ratios in leptoquark searches. \n\n\n\nWhile our analysis indicates that, in these models, the squarks may be as light $950$ GeV for any bi$\\nu$o mass, and as light as 350~GeV for bi$\\nu$o between 100--150 GeV, it is intriguing to wonder if they can be even lighter.  We have not investigated the bounds for bi$\\nu$o mass below $100$ GeV, nor the region with $M_{\\tilde{B}}> M_{\\tilde{q}}$.  It is also an interesting question to understand what are the ideal set of cuts for the jet+$\\slashed{E}_{\\mathrm{T}}$ final state to probe this model. Most importantly, the smoking-gun signals involving lepton final states need careful attention to find the best discovery path for this model. In a separate direction, the viability of gravitino\/goldstino dark matter in this model requires detailed calculations of their production mechanisms given the sparticles masses allowed by LHC data. \n\n\n\n\\section*{Acknowledgements}\n\nWe thank Pilar Coloma for her collaboration in the early stages of this work. We are grateful to Angelo Monteux for sharing his analysis code with us as well as his help with running the code. SI acknowledges support from the University Office of the President via a UC Presidential Postdoctoral fellowship and partial support from NSF Grant No.~PHY-1620638. This work was performed in part at Aspen Center for Physics, which is supported by NSF grant PHY-1607611. JG has received funding\/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674896.\nPF was supported by the DoE under contract number DE-SC0007859 and Fermilab, operated by Fermi Research Alliance, LLC under\ncontract number DE-AC02-07CH11359 with the United States Department of Energy.\n\n\n\\bibliographystyle{JHEP}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSuppose that $\\ell \\geq 5$ is prime. \nMany papers \\cite{Ono-Skinner} \\cite{Bruinier} \\cite{Bruinier-Ono} \\cite{Ahlgren-Boylan} \\cite{Ahlgren-Boylan2} \\cite{Ahlgren-Choi-Rouse} \\cite{Ahlgren-Rouse}\nstudy half-integral weight modular forms with few non-vanishing coefficients modulo $\\ell$ and give applications for divisibility properties of the algebraic parts of the central critical values of modular $L$-functions and the orders of Tate-Shafarevich groups of elliptic curves.\n\nThese results are modulo $\\ell$ analogues of a theorem of Vign\\`{e}ras in characteristic $0$. \nIf $\\lambda$ is a non-negative integer and $N$ is a positive integer with $4\\mid N$, let $M_{\\lambda+\\frac{1}{2}}(\\Gamma_1(N))$ be the space of modular forms of weight $\\lambda+\\frac{1}{2}$ (in the sense of \\cite{Shimura}) on $\\Gamma_1(N)$. \nVign\\`{e}ras \nproved that a form $F(z) \\in M_{\\lambda+\\frac{1}{2}}(\\Gamma_1(N))$ whose coefficients are supported on finitely many square classes of integers is a linear combination of single-variable theta series. The precise result is below (Bruinier \\cite{Bruinier-Vigneras} gave a different proof of this theorem ).\n\n \\begin{theorem}{\\cite{Vigneras}}\\label{thm:Vigneras}\nSuppose that $\\lambda \\geq 0$ is an integer, that $N$ is a positive integer with $4 \\mid N$, and that $F(z) \\in M_{\\lambda+ \\frac{1}{2}}(\\Gamma_1(N))$. If there exist finitely many square-free integers $t_1$,$t_2$,...,$t_m$ for which\n\n\\[\nF(z)=\\sum_{i=1}^{m}\\sum_{n=0}^{\\infty}a(t_{i}n^{2})q^{t_{i}n^{2}} , \\ \\ \\ q= e^{2\\pi i z}\n\\]\n then $\\lambda=0 \\text{ or } 1$ and $F(z)$ is a linear combination of theta series.\n \\end{theorem}\n \n A recent result of Bella\\\"{\\i}che, Green and Soundararajan \\cite{Belliache-Green-Sound} implies for any half-integral weight modular form that the number of coefficients $\\leq X$ which do not vanish modulo $\\ell$\nis $\\gg\\frac{\\sqrt X}{\\log\\log X}$.  It is natural to suspect that the only half-integral weight forms for which the number of non-vanishing coefficients is close to this lower bound are\nthose which  are supported on finitely many square classes modulo $\\ell$.  Forms of half-integral weight on  $\\operatorname{SL}_2(\\mathbb{Z})$ whose coefficients are sparse modulo $\\ell$ play an important role in the recent work of Ahlgren, Beckwith and Raum \\cite{Scarcity} on scarcity of congruences for the partition function.  \n \n Ahlgren, Choi and Rouse \nproved a modulo $\\ell$ analogue of Theorem~\\ref{thm:Vigneras} for forms $f(z)$ in the Kohnen plus-space $S_{\\lambda+\\frac{1}{2}}^{+}(\\Gamma_{0}(4))$. Their main theorem was the following.\n \n \\begin{theorem}{\\cite{Ahlgren-Choi-Rouse}}\\label{Ahlgren-Choi-RouseMainTheorem}\n Suppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime  $v$ of $K$ above $\\ell$. Let $\\mathcal{O}_{v}$ denote the ring of $v$-integral elements of $K$. Suppose that $f \\in S^{+}_{\\lambda+\\frac{1}{2}}(\\Gamma_0(4)) \\cap \\mathcal{O}_{v}[[q]]$ satisfies\n \\[\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^{2})q^{t_{i}n^{2}} \\not \\equiv 0 \\pmod{v},\n \\]\n where each $t_{i}$ is a square-free positive integer. If $\\lambda+\\frac{1}{2} < \\ell(\\ell+1+\\frac{1}{2})$, then $\\lambda$ is even and\n \\[\n f \\equiv a(1)\\sum_{n=1}^{\\infty}n^{\\lambda}q^{n^{2}} \\pmod{v}.\n \\]\n \\end{theorem}\n \n \n In this paper, we study the analogous question for half-integral weight modular forms on $\\operatorname{SL}_{2}(\\mathbb{Z})$. Before we state our main result, we introduce some notation. If $\\lambda \\geq 0$ is an integer, $N$ is a positive integer, and $\\nu$ is a multiplier system on $\\Gamma_0(N)$ in weight $\\lambda+\\frac{1}{2}$, we denote by $S_{\\lambda+\\frac{1}{2}}(N, \\nu)$ the space of cusp forms of weight $\\lambda+\\frac{1}{2}$ and multiplier $\\nu$ on $\\Gamma_0(N)$ (details will be given in the next section). Let $\\nu_{\\eta}$ be the multiplier for the Dedekind eta function defined in \\eqref{etamultiplier}. With this notation, we prove the following theorem.\n\n\n \n \\begin{theorem}\\label{thm:main}\n Suppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime  $v$ of $K$ above $\\ell$. \n Let $\\mathcal{O}_{v}$ denote the ring of $v$-integral elements in $K$. Suppose that $\\lambda$ is a non-negative integer satisfying $\\lambda +\\frac{1}{2} < \\frac{\\ell^{2}}{2}$. Suppose that $r$ is a positive integer with $(r,6)=1$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu^{r}_{\\eta}) \\cap \\mathcal{O}_{v}[[q^{\\frac{r}{24}}]]$ satisfies\n \n \\[\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^2)q^{\\frac{t_{i}n^{2}}{24}}  \\not \\equiv 0 \\pmod v,\n \\]\n where each $t_{i}$ is a square-free positive integer. Then one of the following is true.\n \n \\begin{enumerate}\n \\item \n $f \\equiv a(1)\\displaystyle \\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)n^{\\lambda}q^{\\frac{n^2}{24}} \\pmod v $.\n\nIn this case, $r \\equiv 1 \\pmod{24}$ and $\\lambda$ is even.\n \n  \\item\n  $f \\equiv \\displaystyle a(\\ell)\\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod v $.\n \n In this case, \n  $r \\equiv \\ell \\pmod{24}$ and $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\n \\item\n  $f \\equiv a(1)\\displaystyle \\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)n^{\\lambda}q^{\\frac{n^2}{24}}+a(\\ell)\\sum_{n=1}^{\\infty}\\(\\mfrac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod v $, \n  where $a(1) \\not \\equiv 0 \\pmod{v}$ and $a(\\ell) \\not \\equiv  0 \\pmod{v}$.\nIn this case, $r \\equiv \\ell \\equiv ~1 \\pmod{24}$ and $\\lambda \\equiv \\frac{\\ell -1}{2} \\pmod{\\ell-1}$.  \n \\end{enumerate}\n \n \\end{theorem}\n\n \\begin{remark}\n For an example of case $(1)$ of Theorem~\\ref{thm:main}, let $\\ell \\geq 5$ be prime and $\\lambda$ be a nonnegative integer. \n Lemma~\\ref{Lemma1} below implies that there exists a form $f \\in S_{(\\frac{\\lambda}{2})(\\ell+1)}(1,\\nu_{\\eta})$ such that $f\\equiv\\Theta^{\\frac{\\lambda}{2}}(\\eta)\\pmod{\\ell}$, where $\\Theta$ is the Ramanujan $\\Theta$-operator defined in \\eqref{RamanujanTheta}. We have\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{\\ell}.\n \\]\n For an example of case $(2)$ of Theorem~\\ref{thm:main}, set $f=\\eta^{\\ell}$. Since $\\eta^{\\ell}(z) \\equiv \\eta(\\ell z) \\pmod{\\ell} $, we have\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{\\ell}.\n \\]\n  For an example of case $(3)$, suppose that $\\ell$ is a prime such that $\\ell \\equiv 1 \\pmod{24}$. Lemma~\\ref{Lemma1} implies that there exists a form $g \\in S_{(\\frac{\\ell-1}{4})(\\ell+1)+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that \n  $g \\equiv \\Theta^{\\frac{\\ell-1}{4}}(\\eta) \\pmod{\\ell}$. Set $f=24^{\\frac{\\ell-1}{2}}g+\\eta^{\\ell}E_{\\ell-1}^{\\frac{\\ell-1}{4}}$ . We have\n  \\[\n  f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\frac{\\ell-1}{2}}q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{\\ell}.\n  \\]\n  For this example, note that $\\frac{\\ell-1}{4}(\\ell+1) \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n \\end{remark}\n \\begin{remark}\n The upper bound on $\\lambda$ is sharp. For an example which illustrates this, set $f=\\eta^{\\ell^{2}}$. Then\n \\[\n f \\equiv \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell^{2}n^{2}}{24}} \\pmod{\\ell}.\n \\]\n Note that we have $\\lambda+\\frac{1}{2}=\\frac{\\ell^{2}}{2}$ in this case.\n\\end{remark}\n The paper is organized as follows. In Section $2$, we give some background results on modular forms of integral and half-integral weight. In Section $3$, we prove some preliminary results. In Section $4$, we make a preliminary reduction for the proof of Theorem~\\ref{thm:main}, and in Section $5$, we prove the theorem.\n \\section{Background}\nSuppose that $k \\in \\frac{1}{2}\\mathbb{Z}$, that $N$ is a positive integer, and that $\\chi$ is a Dirichlet character modulo $N$. For a function $f(z)$ on the upper half plane and \n\\[\n \\gamma =\\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\operatorname{GL}_{2}^{+}(\\mathbb{Q}),\n\\]\nwe have the weight $k$ slash operator \n\\[\nf(z)\\big|_k  \\gamma := \\operatorname{det}(\\gamma)^{\\frac{k}{2}}(cz+d)^{-k}f\\(\\frac{az+b}{cz+d}\\).\n\\]\n\nSuppose that $\\ell \\geq 5$ is prime and that $K$ is a number field. Fix an embedding of $K$ into $\\mathbb{C}$ and a prime  $v$ of $K$ above $\\ell$.\nLet $\\mathcal{O}_{v}$ be the ring of $v$-integral elements of $K$.\nIf $\\nu$ is a multiplier system on $\\Gamma_0(N)$, \nwe denote by $M_{k}(N,\\nu)$, $S_{k}(N,\\nu)$\nand $M_{k}^{!}(N, \\nu)$ \nthe spaces of modular forms, cusp forms, \nand weakly holomorphic modular forms \nof weight $k$ and multiplier \n$\\nu$ on $\\Gamma_0(N)$ whose Fourier coefficients are in $\\mathcal{O}_{v}$.\nWhen $k$ is an integer and the multiplier $\\nu$ is trivial, we write $M_{k}(N)$, $S_{k}(N)$\nand $M_{k}^{!}(N)$.\nForms in these spaces satisfy the transformation law\n\n\\[\nf \\big|_k  \\gamma= \\nu(\\gamma)f \\ \\ \\ \\text{ for } \\ \\ \\ \\gamma = \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\Gamma_0(N)\n\\]\nand the appropriate conditions at the cusps of $\\Gamma_0(N)$. \n\n\nThroughout, let $q:=e(z)=e^{2 \\pi i z}$. \nWe define the eta function by\n\\[\n\\eta(z):= q^{\\frac{1}{24}}\\prod_{n=1}^{\\infty}(1-q^{n})\n\\]\nand the theta function by\n\\[\n\\theta(z):= \\sum_{n=-\\infty}^{\\infty}q^{n^2}.\n\\]\nThe eta function has a multiplier $\\nu_{\\eta}$ satisfying\n\n\\[\n\\eta(\\gamma z)=\\nu_{\\eta}(\\gamma)(cz+d)^{\\frac{1}{2}}\\eta(z), \\ \\ \\ \\ \\ \\gamma= \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\operatorname{SL}_{2}(\\mathbb{Z});\n\\]\nthroughout, we choose the principal branch of the square root. For $c>0$, we have the formula \\cite[~$\\mathsection$$4.1$]{Knopp}\n\n\\begin{equation}\\label{etamultiplier}\n\\nu_{\\eta}(\\gamma)=\n \\begin{cases} \n \\(\\frac{d}{c}\\)e\\(\\frac{1}{24}((a+d)c-bd(c^2-1)-3c )\\), & \\text{if } c \\text{ is odd,} \\\\\n\\(\\frac{c}{d}\\)e\\(\\frac{1}{24}((a+d)c-bd(c^2-1)+3d-3-3cd)\\) & \\text{if } c \\text{ is even}.\n\\end{cases}\n\\end{equation}\nFor the multiplier of the theta function we have\n\\[\n\\nu_{\\theta}(\\gamma):= (cz+d)^{-\\frac{1}{2}}\\frac{\\theta(\\gamma z)}{\\theta(z)}=\\(\\frac{c}{d}\\)\\epsilon_{d}^{-1}, \\ \\ \\ \\ \\ \\gamma= \\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right) \\in \\Gamma_0(4),\n\\]\nwhere\n\\[\n\\epsilon_{d}=\n\\begin{cases}\n1, & \\text{if } d \\equiv 1 \\pmod{4}, \\\\\ni, & \\text{if } d \\equiv 3 \\pmod{4}.\n\\end{cases}\n\\]\n\n\nIn the next several paragraphs, we follow the exposition in \\cite{Scarcity}.\nIf $f \\in M_{k}(N,\\chi\\nu_{\\eta}^{r})$, then $\\eta^{-r}f \\in M^{!}_{k-\\frac{r}{2}}(N,\\chi)$.\nThis implies that $f$ has a Fourier expansion of the form\n\n\\begin{equation}\\label{lemma4.2}\nf = \\sum_{n \\equiv r (24)}a(n)q^{\\frac{n}{24}}.\n\\end{equation} \nThese facts together imply the following lemma.\n\\begin{lemma}\\label{integerhalfinteger}\nSuppose that $0 <r < 24$ is an integer with $(r,6)=1$ and that $f \\in S_{k}(N, \\chi\\nu_{\\eta}^{r})$. Then we have \n\\[\n\\eta^{-r}f \\in M_{k-\\frac{r}{2}}(N,\\chi).\n\\]\n\\end{lemma}\n\nWe also have\n\\[\nM_{k}(N,\\chi\\nu_{\\eta}^{r}) \\neq \\{0\\} \\ \\ \\ \\ \\text{ only if } \\ \\ \\ \\ 2k-r \\equiv 1-\\chi(-1) \\pmod{4},\n\\]\nso if $(r,6)=1$, then $k \\not \\in \\mathbb{Z}$.\nIt follows from a computation using \\eqref{etamultiplier} and the formulas\n\n\\[\ne\\(\\frac{1-d}{8}\\)=\\(\\frac{2}{d}\\)\\epsilon_{d} \\ \\ \\text{ and } \\ \\ \\epsilon_{d_1d_2}=\\epsilon_{d_1}\\epsilon_{d_2}(-1)^{\\frac{d_1-1}{2}\\frac{d_2-1}{2}}\n\\]\nfor odd $d$, $d_1$, and $d_2$ that we have\n\\begin{equation}\\label{passing to Shimura's space}\nf(z) \\in S_{k}(N,\\chi \\nu_{\\eta}^{r}) \\Longrightarrow f(24z) \\in S_{k}(576N,\\chi\\(\\tfrac{12}{\\bullet}\\)\\nu_{\\theta}^{r}).\n\\end{equation}\nThis fact, together with the usual Shimura lift on $S_{k}(576, \\(\\frac{12}{\\bullet}\\)\\nu_{\\theta}^{r})$, allows us to define a Shimura lift $\\operatorname{Sh}_{t}$ on $S_{k}(1, \\nu_{\\eta}^{r})$ for each squarefree integer $t$ with $(t,6)=1$. Its action on Fourier expansions is\n\\begin{equation}\\label{ShimuraCorrespondence}\n\\operatorname{Sh}_{t}\\(\\sum a(n)q^{\\frac{n}{24}}\\)= \\sum A_{t}(n)q^{n},\n\\end{equation}\nwhere the coefficients $A_{t}(n)$ are given by\n\\begin{equation}\\label{ShimuraCorrespondenceCoefficients}\nA_{t}(n)= \\sum_{d \\mid n}\\(\\mfrac{-1}{d}\\)^{k-\\frac{1}{2}}\\(\\mfrac{12t}{d}\\)d^{k-\\frac{3}{2}}a\\(\\frac{tn^{2}}{d^{2}}\\).\n\\end{equation}\nThe work of Shimura \\cite{Shimura} and Niwa \\cite{Niwa} shows that, for each squarefree integer $t$ with $(t,6)=1$, we have\n\\[\nf \\in S_{k}(1,\\nu_{\\eta}^{r}) \\implies \\operatorname{Sh}_{t}(f) \\in S_{2k-1}(288).\n\\]\n\nWe make use of Hecke operators on the spaces we will consider. If $k$ is an integer, we denote the Hecke operator on $S_{k}(N)$ by $T(p,k,1)$. If $k \\in \\frac{1}{2}\\mathbb{Z}\\setminus \\mathbb{Z}$ and $4 \\mid N$ and $r$ is a positive integer with $(r,6)=1$, we denote the Hecke operator on $S_{k}(N,\\chi\\(\\frac{12}{\\bullet}\\)\\nu_{\\theta}^{r})$ by $T(p^{2},k, \\chi)$ (these forms are half-integral weight forms in the sense of \\cite{Shimura}). \n\nWe next recall the $U$ and $V$ operators. For a positive integer $m$, we define them on Fourier expansions by\n\\[\n\\(\\sum_{n=1}^{\\infty}a(n)q^{\\frac{n}{24}}\\)\\sl U_{m}:= \\sum_{n=1}^{\\infty}a(mn)q^{\\frac{n}{24}},\n\\]\n\n\\[\n\\(\\sum_{n=1}^{\\infty}a(n)q^{\\frac{n}{24}}\\)\\sl V_{m}:= \\sum_{n=1}^{\\infty}a(n)q^{\\frac{mn}{24}}.\n\\]\nThe following facts appear as Lemma $2.1$ in \\cite{Scarcity}.\n\n\\begin{lemma}\\label{UandVproperties}\nSuppose that $r$ is a positive integer with $(r,6)=1$, that $f \\in M_{k}(N,\\chi\\nu_{\\eta}^{r})$, and that $m$ is a positive integer. Then\n\\[\nf\\sl U_{m}=m^{\\frac{k}{2}-1}\\sum_{t \\ (m)}f\\big|_k  \\left(\\begin{matrix}1 & 24t \\\\0 & m\\end{matrix}\\right) \\ \\ \\ \\text{ and } \\ \\ \\ f\\sl V_{m}=m^{-\\frac{k}{2}}f\\big|_k  \\left(\\begin{matrix}m & 0 \\\\0 & 1\\end{matrix}\\right).\n\\]\n\\end{lemma}\nA computation using Lemma~\\ref{UandVproperties} and \\eqref{etamultiplier} implies that if $(r,6)=1$ and $p \\geq 5$ is prime, then\n\\begin{equation}\\label{Up}\nU_p:M_{k}(N,\\chi\\nu_{\\eta}^{r}) \\rightarrow M_{k}\\(N\\frac{p}{(N,p)},\\chi\\chi_{p}\\nu_{\\eta}^{pr}\\),\n\\end{equation}\n\n\\begin{equation}\\label{Vp}\nV_p:M_{k}(N,\\chi\\nu_{\\eta}^{r}) \\rightarrow M_{k}\\(Np,\\chi\\chi_{p}\\nu_{\\eta}^{pr}\\).\n\\end{equation}\nDenote by $\\chi_{p}=\\(\\frac{\\bullet}{p}\\)$ the quadratic character of modulus $p$ and by $\\chi_{p}^{\\text{triv}}$ the trivial character with modulus $p$. If $(r,6)=1$ and if $\\psi= \\chi_{p} \\text{ or } \\chi_{p}^{\\text{triv}}$, we define the twist of $f \\in M_{k}(N, \\chi\\nu_{\\eta}^{r})$ with Fourier expansion $f= \\displaystyle\\sum_{n=1}^{\\infty}a(n)q^{\\frac{n}{24}}$ by\n\n\\begin{equation}\\label{twist}\nf \\otimes \\psi:= \\sum_{n=1}^{\\infty}\\psi(n)a(n)q^{\\frac{n}{24}}.\n\\end{equation}\n\nIf $p \\geq 5$ is prime, the fact that $p^{2} \\equiv 1 \\pmod{24}$ together with \\eqref{Up} and \\eqref{Vp} implies that we have\n\n\\begin{equation}\\label{chi TRIV}\nf \\in M_{k}(N,\\chi\\nu_{\\eta}^{r}) \\implies f\\otimes \\chi_{p}^{\\text{triv}} \\in M_{k}(Np^{2},\\chi\\nu_{\\eta}^{r}).\n\\end{equation}\nWe also have\n\\[\nf \\otimes \\chi_{p}=\\frac{1}{\\epsilon_{p}\\sqrt{p}}\\sum_{t \\ (p)}\\chi_{p}(t)f\\big|_k  \\left(\\begin{matrix}1 & \\frac{24t}{p} \\\\0 & 1\\end{matrix}\\right).\n\\]\nA similar computation as for \\eqref{Up} and \\eqref{Vp} shows that\n\\begin{equation}\\label{chi P}\nf \\otimes \\chi_{p} \\in M_{k}(Np^{2},\\chi\\nu_{\\eta}^{r}).\n\\end{equation}\n\nWe next review some facts about the algebra of modular forms modulo $\\ell$ and filtrations. Suppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime ideal of $K$ above $\\ell$. Let $\\mathcal{O}_{v}$ be the ring of $v$-integral elements of $K$, and let the residue field be $\\mathbb{F}_{v}$.\nFor Fourier expansions $\\sum a(n)q^{\\frac{n}{24}} \\in \\mathcal{O}_{v}[[q^{\\frac{1}{24}}]]$, \nwe define $\\bar{f}:= \\sum \\bar{a(n)}q^{\\frac{n}{24}} \\in \\mathbb{F}_{v}[[q^{\\frac{1}{24}}]]$, \n and we define \n\n\\[\n\\bar{M_{k}(N, \\nu)}:= \\{\\bar{f}: \\ \\ f \\in M_{k}(N,\\nu)\\},\n\\]\n\n\\[\n\\bar{S_{k}(N, \\nu)}:=\\{\\bar{f}:\\ \\ f \\in S_{k}(N,\\nu)\\}.\n\\]\nIf $k$ is an integer and $f \\in M_{k}(N)$, then we define the filtration of $f$ to be \n\n\\[\n\\omega(f)=\\omega(\\bar{f}):= \\inf\\{k': \\ \\text{there exists } g \\in M_{k'}(N)\\text{ with } \\bar{f}=\\bar{g}\\}.\n\\]\n If $f \\in M_{k}(N, \\nu)$, we define\n\\begin{equation}\\label{RamanujanTheta}\n\\Theta:=\\frac{1}{2\\pi i}\\frac{d}{dz}=q\\frac{d}{dq}.\n\\end{equation}\nWe make use of the following facts from \\cite[~$\\mathsection$$4$]{Gross}.\n\\begin{proposition}\\label{Gross}\nSuppose that $N$ is a positive integer, that $k$ is an integer, that $\\ell \\geq 5$ is prime with $\\ell \\nmid N$, and that $f \\in M_{k}(N)$. Then\n\\begin{enumerate}\n\\item\nThere exists a form $g \\in M_{k+\\ell+1}(N)$ with $\\bar{g}=\\bar{\\Theta(f)}$.\n\\item\n$\\omega(\\Theta(f)) \\leq \\omega(f)+\\ell+1$, with equality if and only if $\\ell \\nmid \\omega(f)$.\n\\item\n$\\omega(f^{i})=i\\omega(f)$ for all $i \\geq 1$.\n\\item\nIf $g \\in M_{k'}(N)$ has $\\bar{f}=\\bar{g}\\neq 0$, then $k \\equiv k' \\pmod{\\ell-1}$.\n\\end{enumerate}\n\\end{proposition}\nFinally, if $k \\geq 2$ is an even integer, we denote the weight $k$ Eisenstein series by $E_k$.\n\n\\section{Preliminary Results}\nWe record here some preliminary results which we require for the proof of Theorem~\\ref{thm:main}.\n\n\\begin{lemma}\\label{Lemma1}\nLet $\\ell \\geq 5$ be a prime and $K$ is a number field. Suppose that $r$ is a positive integer with $(r,6)=1$ and that $f\\in S_{\\lambda+\\frac{1}{2}}(1, \\nu^{r}_{\\eta})$. Then there is a form $F \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1, \\nu^{r}_{\\eta})$\nsuch that $F \\equiv \\Theta (f) \\pmod{\\ell}$.\n\\end{lemma}\n\n\\begin{proof}\nChoose an integer $j$ such that $\\ell j+r \\equiv 0 \\pmod{24}$. Define\n\n\\[\ng:= \\eta^{\\ell j}f \\in S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1).\n\\]\nThere exists a form $G \\in S_{\\lambda+\\frac{\\ell j}{2}+\\ell+1+\\frac{1}{2}}(1)$ such that\n\n\\[\nG \\equiv \\Theta (g) \\equiv \\eta^{\\ell j}\\Theta (f) \\pmod{\\ell}.\n\\]\nLet $d=\\dim(S_{\\lambda+\\frac{\\ell j}{2}+\\ell+1+\\frac{1}{2}}(1))$. The space $S_{\\lambda+\\frac{\\ell j}{2}+\\ell+1+\\frac{1}{2}}(1)$ has a basis $\\{f_1,...,f_d\\}$ with integer coefficients with the property \n\\begin{equation}\\label{eqn:INTEGRALITY}\nf_i(z)=q^{i}+O(q^{i+1}),  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1 \\leq i \\leq d.\n\\end{equation}\nAfter subtracting an appropriate linear combination of these basis elements from $G$, we may assume that $F:= \\frac{G}{\\eta^{\\ell j}} \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1,\\nu^{r}_{\\eta})$. We have\n\n\\[\nF \\equiv \\Theta (f) \\pmod{\\ell}.\n\\]\n\n\\end{proof}\n\n\\begin{lemma}\\label{Lemma2}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field which is Galois over $\\mathbb{Q}$, and that $v$ is a prime of $K$ above $\\ell$. Suppose that $\\lambda$ is a non-negative integer, that $r$ is a positive integer with $(r,6)=1$, and that $g \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ satisfies\n\n\\[\ng \\equiv \\sum_{n=1}^{\\infty}a(n)q^{\\frac{\\ell n}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nThen there exists $\\lambda' \\geq 0$ with $\\lambda'+\\frac{1}{2} \\leq \\frac{1}{\\ell}(\\lambda+\\frac{1}{2})$, and a form $f \\in S_{\\lambda'+\\frac{1}{2}}(1, \\nu_{\\eta}^{r\\ell})$ such that\n\\[\nf \\equiv  g\\sl U_{\\ell} \\equiv  \\sum_{n=1}^{\\infty}a(n)q^{\\frac{n}{24}}  \\pmod v.\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nChoose an integer $j>0$ such that $\\ell j+r \\equiv 0 \\pmod{24}$, and define \n\\[\nh:= \\eta^{\\ell j}g \\in S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1).\n\\]\nSuppose that $x \\in \\mathcal{O}_{v}$ and  that $\\sigma \\in \\operatorname{Gal}(K\/\\mathbb{Q})$ is a Frobenius automorphism for the prime $v$. Then we have $x^{\\sigma} \\in \\mathcal{O}_{v}$ and \n\n\\[\nx^{\\sigma} \\equiv x^{\\ell} \\pmod{v}.\n\\]\nNote that $\\sigma$ preserves the space $S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1)$. Since $U_{\\ell}$ acts as  \n$T(\\ell, \\lambda+\\frac{\\ell j}{2}+\\frac{1}{2},1)$ modulo $v$, we see that $\\bar{h\\sl U_{\\ell}} \\in \\bar{S_{\\lambda+\\frac{\\ell j}{2}+\\frac{1}{2}}(1)}$. We have\n\\[\n\\bar{h^{\\sigma}}=(\\bar{h\\sl U_{\\ell}})^{\\ell}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we know that there exists an integer $\\beta \\geq 0$ such that\n\n\\[\nk:= \\omega(\\bar{h\\sl U_{\\ell}})=\\frac{1}{\\ell}\\omega(\\bar{h^{\\sigma}})=\\frac{1}{\\ell}\\(\\lambda-\\beta(\\ell-1)+\\frac{\\ell j}{2}+\\frac{1}{2}\\).\n\\]\nTherefore, arguing as in the proof of Lemma~\\ref{Lemma1}, we can find a form\n$H \\in S_{k}(1)$ such that $\\bar{H}=\\bar{h\\sl U_{\\ell}}=\\bar{\\eta^{j}(g\\sl U_{\\ell}})$ and\n$f:= \\frac{H}{\\eta^{j}} \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1)$. Then, we see that $f \\in S_{k-\\frac{j}{2}}(1,\\nu_{\\eta}^{r\\ell})$, and we have $\\bar{f}=\\bar{g\\sl U_{\\ell}}$.  \nThe lemma follows since $k-\\frac{j}{2} \\leq \\frac{1}{\\ell}(\\lambda+\\frac{1}{2})$.\n\n\\end{proof}\n\n\\section{Preliminary Reduction}\nBefore proving Theorem~\\ref{thm:main}, we reduce the number of square classes on which our forms may be supported and the number of multipliers which we must consider.\n\n\\begin{proposition}\\label{Proposition1}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime above $\\ell$. Suppose that $\\lambda$ is a non-negative integer, that $r$ is a positive integer with $(r,6)=1$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$. Further, suppose that\n\n\\begin{equation}\\label{finitelymanysquareclasses}\n f \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^2)q^{\\frac{t_{i}n^{2}}{24}}  \\not \\equiv 0 \\pmod v,\n \\end{equation}\n  where each $t_{i}$ is a square-free positive integer. Then\n \\begin{equation}\\label{twosquareclasses}\n f \\equiv \\sum_{n=1}^{\\infty}a(n^2)q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \n \\pmod v.\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFix an $i \\in \\{1,...,m \\}$. We may assume that there exists an integer $n_{i}$ for which $a(t_{i}n_{i}^{2}) \\not \\equiv 0 \\pmod{v}$. Recalling our notation \\eqref{twist} and the facts \\eqref{chi TRIV} and \\eqref{chi P}, we follow the argument in the proof of Lemma $4.1$ of \\cite{Ahlgren-Boylan} to find primes $p_{1},...,p_{n} \\geq 5$, each relatively prime to $n_{i}t_{i}\\ell$ and a form\n\\[\nG_{i} \\in S_{\\lambda+\\frac{1}{2}}(p_{1}^{2}\\cdots p_{n}^{2},\\nu_{\\eta}^{r})\n\\]\nsatisfying\n\\[\nG_{i} \\equiv \\sum_{(n,\\prod p_{j})=1}a(t_{i}n^{2})q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nNote that \n\n\\[\nG_{i}^{24} \\in S_{24\\lambda+12}(p_{1}^{2},...,p_{s}^{2}).\n\\]\nSince \n\\[\nG_{i}^{24} \\equiv \\sum_{n=1}^{\\infty}b(t_{i}n)q^{t_{i}n} \\pmod{v}\n\\]\nfor some coefficients $b(t_{i}n)$, we can apply \nthe following result\nto conclude that $t_{i}=1 \\text{ or } \\ell$.\n\n\\begin{theorem}{\\cite[Thm 3.1]{Ahlgren-Choi-Rouse}}\\label{thm:ACR THM 3.1}\nSuppose that $K$ is a number field and that $v$ is a prime above $\\ell$ with ring of $v$-integral elements $\\mathcal{O}_{v}$. Suppose that $k$ is positive integer and that \n\\[\nf=\\sum_{n=1}^{\\infty}a(n)q^{n} \\in S_{2k}(\\Gamma_0(N)). \n\\]\nIf $t > 1$ satisfies $(t,\\ell N)=1$ and\n\\[\nf \\equiv \\displaystyle \\sum_{n=1}^{\\infty}a(tn)q^{tn} \\pmod{v},\n\\]\nthen $f \\equiv 0 \\pmod v$.\n\\end{theorem}\n\n\\end{proof}\nThe next result reduces the number of multipliers which we must consider.\n\n\\begin{lemma}\\label{lemmatwomultipliers}\n\nSuppose that $r$ is a positive integer with $(r,6)=1$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ satisfies \\eqref{twosquareclasses}. Then we have\n\\begin{equation}\\label{twomultipliers}\nr \\equiv 1 \\pmod{24} \\ \\ \\ \\ \\text{ or } \\ \\ \\ \\ r \\equiv \\ell \\pmod{24}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $f$ satisfies \\eqref{twosquareclasses}, it follows that either $a(n^{2}) \\neq 0$ or $a(\\ell n^{2}) \\neq 0$ for some positive integer $n$. It follows from \\eqref{lemma4.2} and the fact that $r^{2} \\equiv 1 \\pmod{24}$ whenever $(r,6)=1$ that we have \\eqref{twomultipliers}. \n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{thm:main}}\nThe proof of Theorem~\\ref{thm:main} will proceed in several steps. We first consider the case when $r \\equiv 1 \\pmod{24}$ and $\\lambda$ is even.\n\n\\begin{theorem}\\label{Theorem1}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime above $\\ell$. Suppose that $\\lambda$ is a non-negative integer and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta})$ has the form \\eqref{twosquareclasses}. If $\\lambda$ is even and $\\lambda < 2\\ell^{2}+\\ell-1$, then\n\n\\[\n\\sum_{\\ell \\nmid n}a(n^{2})q^{\\frac{n^{2}}{24}} \\equiv a(1)\\sum_{\\ell \\nmid n}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nDefine $\\bar{\\lambda}:= \\lambda \\pmod{\\ell-1}$. By Lemma~\\ref{Lemma1}, we have forms $g(z) \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1,\\nu_{\\eta})$ and \n$h(z) \\in S_{(\\ell+1)\\frac{\\bar{\\lambda}+2}{2}+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that\n\n\\[\ng=\\sum_{n=1}^{\\infty}c(n)q^{\\frac{n}{24}} \\equiv \\Theta(f) \\equiv \\sum_{n=1}^{\\infty}\\frac{n^2}{24}a(n^2)q^{\\frac{n^2}{24}} \\pmod{v}\n\\]\nand\n\\[\nh=\\sum_{n=1}^{\\infty}b(n)q^{\\frac{n}{24}} \\equiv 24^{\\frac{\\bar{\\lambda}}{2}}a(1)\\Theta^{\\frac{\\bar{\\lambda}+2}{2}}(\\eta) \\equiv a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)\\frac{n^{\\bar{\\lambda}+2}}{24}q^{\\frac{n^2}{24}} \\pmod{v}.\n\\]\n\nIt suffices to show that $g \\equiv h \\pmod{v}$.  To this end, we make use of this theorem, which follows from an argument of Bruinier and Ono \\cite[Thm 3.1]{Bruinier-Ono} (see \\cite[Thm 2.1]{Ahlgren-Choi-Rouse}).\n\n\\begin{theorem}\\label{Bruinier-Ono}\nSuppose that $N$ is a positive integer with $4 \\mid N$. Suppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$. \nSuppose that $\\lambda$ is a non-negative integer and that $r$ is a positive integer with $(r,6)=1$. Suppose that\n\n\\[\nf(z)=\\sum_{n=1}^{\\infty}a(n)q^{n} \\in S_{\\lambda+\\frac{1}{2}}(N,\\chi\\(\\tfrac{12}{\\bullet}\\) \\nu_{\\theta}^{r}),\n\\]\nthat $\\ell \\nmid N$, and that $p \\nmid N\\ell$ is prime. If there exists $\\epsilon_{p} \\in \\{ \\pm 1\\}$ such that\n\n\\[\nf(z) \\equiv \\sum_{\\( \\frac{n}{p}\\) \\in \\{0, \\epsilon_{p}\\}}a(n)q^{n} \\pmod v,\n\\]\nthen we have\n\\[\n(p-1)f(z)\\sl T\\(p^2,\\lambda +\\mfrac{1}{2},\\chi\\) \\equiv \\epsilon_{p}\\chi(p)\\(\\tfrac{(-1)^{\\lambda}}{p}\\)(p^{\\lambda}+p^{\\lambda-1})(p-1)f(z) \\pmod v.\n\\]\n\\end{theorem}\n\n\n\n\nBy \\eqref{passing to Shimura's space}, we can apply Theorem~\\ref{Bruinier-Ono} to $g(24z)$ to conclude that, for odd primes $p \\geq 5$ with $p \\not \\equiv 0,1 \\pmod{\\ell}$, we have\n\n\\begin{equation}\\label{applicationofBruinierOno}\ng(24z) \\sl T\\(p^2,\\lambda+\\ell+1+\\mfrac{1}{2},1\\) \\equiv \\(\\frac{12}{p}\\)(p^{\\bar{\\lambda}+2}+p^{\\bar{\\lambda}+1})g(24z) \\pmod{v}.\n\\end{equation}\nSuppose that $n$ is a positive integer satisfying $(n,6)=1$ which is divisible only by primes $p \\not \\equiv 0,1 \\pmod{\\ell}$.\n If $p$ is such a prime, write $n=p^{a}n_{0}$ if $p^{a} \\mid \\mid n$.\nThe definition of the Hecke operator on $S_{\\lambda+\\ell+1+\\frac{1}{2}}(576,(\\frac{12}{\\bullet})\\nu_{\\theta}^{r})$ implies that we have\n\\[\nc(n^2p^2)+p^{\\bar{\\lambda}+1}\\(\\frac{12n^2}{p}\\)c(n^2)+p^{2\\bar{\\lambda}+3}c\\(\\frac{n^2}{p^2}\\) \\equiv \\(\\frac{12}{p}\\)(p^{\\bar{\\lambda}+2}+p^{\\bar{\\lambda}+1})c(n^2) \\pmod{v},\n\\]\nand an induction argument on $a$ then implies that\n\\[\nc(p^{2a}n_{0}^{2}) \\equiv \\(\\frac{12}{p}\\)^{a}p^{a(\\bar{\\lambda}+2)}c(n_{0}^{2}) \\pmod{v}.\n\\]\n\nThus, we have\n\\[\nc(n^{2}) \\equiv \\(\\frac{12}{n}\\)n^{\\bar{\\lambda}+2}c(1) \\equiv \\(\\frac{12}{n}\\)\\frac{n^{\\bar{\\lambda}+2}}{24}a(1) \\equiv b(n^2) \\pmod{v}.\n\\]\nThis shows that the coefficients $c(n^{2})$ and $b(n^{2})$ agree whenever $n$ is a positive integer such that $(n,6)=1$ which is divisible only by primes $p \\geq 5$ with $p \\not \\equiv 0,1 \\pmod{\\ell}$. \n\n\nNow define\n\\[\nk:= \\max\\{\\lambda+\\ell+1,(\\ell+1)\\frac{\\bar{\\lambda}+2}{2}\\}. \n\\]\nThese numbers agree modulo $\\ell-1$ by virtue of $\\lambda$ being even, so by multiplying $g$ or $h$ by an appropriate power of $E_{\\ell-1} \\equiv 1 \\pmod{\\ell}$, we see that there exist forms $g_{1}$ and $h_{1}$ in $S_{k+\\frac{1}{2}}(1,\\nu_{\\eta})$ such that $g_{1} \\equiv g \\pmod{v}$ and $h_{1} \\equiv h \\pmod{v}$. \nThus, to prove the theorem, it suffices to show that $g_{1} \\equiv h_{1} \\pmod{v}$.\nNote that $c(n^2) \\equiv b(n^2) \\equiv 0 \\pmod{v}$ for positive integers $n$ such that $(n,6) \\neq 1$ by \\eqref{lemma4.2}, and that $c(n^2)$ and $b(n^2)$ vanish modulo $\\ell$ whenever $n$ is divisible by $\\ell$. This implies that $c(n) \\equiv b(n) \\pmod{v}$ whenever $n < (2\\ell+1)^{2}$.\n Thus,\n$\\eta^{-1}(g_{1}-h_{1}) \\in M_{k}(1)$ is of the form\n\n\\[\n\\eta^{-1}(g_{1}-h_{1}) \\equiv cq^{\\frac{\\ell^{2}+\\ell}{6}}+ \\cdots \\pmod{v}\n\\]\nfor some $c \\in \\mathcal{O}_{v}$. By arguing as in the proof of Lemma~\\ref{Lemma1}, we may assume that $\\eta^{-1}(g_{1}-h_{1}) \\in S_{k}(1)$. To prove that $g_{1} \\equiv h_{1} \\pmod{v}$, it suffices to show by \\cite[Thm 1]{Sturm} that \n\n\\[\n\\frac{k}{12} < \\frac{\\ell^{2}+\\ell}{6}.\n\\]\nSince $\\bar{\\lambda} < \\ell$, we have\n\n\\[\n\\frac{(\\ell+1)(\\bar{\\lambda}+2)}{24}<\\frac{\\ell^{2}+\\ell}{6}.\n\\]\nSince $\\lambda < 2\\ell^{2}+\\ell-1$, we have\n\\[\n\\frac{\\lambda+\\ell+1}{12}<\\frac{\\ell^{2}+\\ell}{6}.\n\\]\nThe result follows.\n\\end{proof}\n\nWe now consider what happens when $\\lambda$ is odd.\n\n\\begin{proposition}\\label{oddcases}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$. Suppose that $\\lambda$ is a non-negative integer, that $r$ is a positive integer with $(r,6)=1$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ has the form\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nIf $\\lambda$ is odd, then $\\Theta(f) \\equiv 0 \\pmod{v}$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose by way of contradiction that $\\Theta(f) \\not \\equiv 0 \\pmod{v}$. By Lemma~\\ref{Lemma1}, there exists $g \\in S_{\\lambda+\\ell+1+\\frac{1}{2}}(1,\\nu_{\\eta}^{r})$ such that\n\n\\[\ng \\equiv \\sum_{n=1}^{\\infty} \\frac{n^{2}}{24}a(n^{2})q^{\\frac{n^{2}}{24}} \\not \\equiv 0 \\pmod{v},\n\\]\nso there exists $n_{0}$ such that $a(n_{0}^{2}) \\neq 0$.\nBy \\eqref{lemma4.2}, we have $r \\equiv 1 \\pmod{24}$. By Lemma~\\ref{integerhalfinteger}, we then have $\\eta^{-1}f \\in M_{\\lambda}(1)=\\{0\\},$ which is a contradiction. \nThus, $\\Theta(f) \\equiv 0 \\pmod{v}$.\n\\end{proof}\n\nWe require one more result before proving Theorem~\\ref{thm:main}.\n\n\\begin{proposition}\\label{Spicy}\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field which is Galois over $\\mathbb{Q}$, and that $v$ is a prime of $K$ above $\\ell$. \nSuppose that $r$ is a positive integer with $(r,6)=1$ and that $g \\in S_{\\lambda'+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ satisfies \n\n\\[\n g \\equiv \\sum_{n=1}^{\\infty}a(n^2)q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod v. \n \\]\nIf $\\lambda' < \\frac{\\ell-1}{2}$, then $\\lambda'=0$, $r=1$, and $g=c\\eta$ for \nsome $c \\in \\mathcal{O}_{v}$. \n\\end{proposition}\n\n\\begin{proof}\nFirst assume that $\\lambda'=0$. Assume without loss of generality that $0<r <24$. By Lemma~\\ref{integerhalfinteger}, we have $\\eta^{-r}g \\in M_{\\frac{1}{2}-\\frac{r}{2}}(1)$, from which $r=1$ follows. Thus, $\\eta^{-1}g \\in M_0(1) = \\mathcal{O}_{v}$, and the result follows.\n\n\nNow assume that $1 \\leq \\lambda' < \\frac{\\ell-1}{2}$.\nSuppose first that $\\lambda'$ is odd. Proposition~\\ref{oddcases} implies that $\\Theta (g) \\equiv 0 \\pmod{v}$, so we have \n\\[\ng \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2}n^{2})q^{\\frac{\\ell^{2}n^{2}}{24}} \\pmod{v}.\n\\]\nBy Lemma~\\ref{Lemma2}, there exists $f \\in S_{\\lambda^{*}+\\frac{1}{2}}(1,\\nu_{\\eta}^{r\\ell})$ with \n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2}n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v},\n\\]\nand $\\lambda^{*}+\\frac{1}{2} \\leq \\frac{1}{\\ell}(\\lambda^{'}+\\frac{1}{2})$. Since $\\lambda^{*} \\geq 0$, we have $\\frac{1}{2} \\leq \\frac{1}{\\ell}(\\lambda^{'}+\\frac{1}{2})$, which would imply that $\\lambda' \\geq \\frac{\\ell-1}{2}$, so $\\lambda'$ cannot be odd.\n\n\nNow assume that $2 \\leq \\lambda' < \\frac{\\ell-1}{2}$  and $\\lambda'$ is even. Applying Theorem~\\ref{Theorem1}, we see that \n\n\\[\n\\Theta^{\\ell-1}(g) \\equiv a(1) \\sum_{\\ell \\nmid n}\\(\\frac{12}{n}\\)n^{\\lambda'}q^{\\frac{n^{2}}{24}} \\pmod{v}, \n\\]\nwhich implies that\n\\begin{equation}\\label{somethingkewl}\na(n^{2}) \\equiv a(1) \\(\\frac{12}{n}\\)n^{\\lambda'} \\pmod{v} \\ \\ \\ \\text{ if } \\ \\ \\ \\ell \\nmid n.\n\\end{equation}\n\nWe show that $a(1) \\equiv 0 \\pmod{\\ell}$. Assume to the contrary that $a(1) \\not \\equiv 0 \\pmod{\\ell}$. Consider the Shimura lift\n\n\\begin{equation}\nG:= \\operatorname{Sh}_{1}(g)=\\sum_{n=1}^{\\infty}b(n)q^{n} \\in S_{2\\lambda'}(288)\n\\end{equation}\nfrom \\eqref{ShimuraCorrespondence}. \nIf $\\ell \\nmid n$, then \\eqref{ShimuraCorrespondenceCoefficients} and \\eqref{somethingkewl} give\n\\begin{equation}\\label{*}\nb(n) \\equiv a(1)\\(\\frac{12}{n}\\)\\sum_{d \\mid n}d^{\\lambda'-1}\\(\\frac{n}{d}\\)^{\\lambda'}=a(1)\\(\\frac{12}{n}\\)n^{\\lambda'-1}\\sigma_{1}(n).\n\\end{equation}\nThis is congruent to the $n^{\\text{th}}$ coefficient of \n\\[\n-\\frac{a(1)}{24}\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes \\(\\frac{12}{\\bullet}\\)\\) \\equiv -\\frac{a(1)}{24}\\Theta^{\\lambda'-1}\\(E_{2}\\otimes\\(\\frac{12}{\\bullet}\\)\\) \\pmod{\\ell}.\n\\]\n\nNote that $E_{2}\\otimes(\\frac{12}{\\bullet}) = (E_{2}-2E_{2}\\sl V_{2})\\otimes\\(\\frac{12}{\\bullet}\\)$. \nThis implies that the filtration of $E_{2}\\otimes (\\frac{12}{\\bullet})$ on level $288$ is $2$, since $E_{2}-2E_{2}\\sl V_{2} \\in M_{2}(2)$ \\cite[~$\\mathsection$$1.2$]{DiamondShurman}.\nBy $(2)$ of Proposition~\\ref{Gross}, we have\n\\[\n\\omega\\(\\bar{\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes\\(\\frac{12}{\\bullet}\\)\\)}\\)=\\ell(\\lambda'-1)+\\lambda'+1.\n\\]\nWe also have\n\\[\nG \\equiv GE_{\\ell-1}^{\\lambda'-1} \\in S_{\\ell(\\lambda'-1)+\\lambda'+1}(288).\n\\]\nBy $(1)$ of Proposition~\\ref{Gross}, there exists $H \\in S_{\\ell(\\lambda'-1)+\\lambda'+1}(288)$ with\n\\[\n\\bar{H}= \\bar{GE_{\\ell-1}^{\\lambda'-1}+\\frac{a(1)}{24}\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes\\(\\frac{12}{\\bullet}\\)\\)} \\in \\bar{S_{\\ell(\\lambda'-1)+\\lambda'+1}(288)}.\n\\]\nLet $\\sigma \\in \\operatorname{Gal}(K\/\\mathbb{Q})$ be a Frobenius element for $v$. Note that $\\bar{H}$ has the form $\\sum \\bar{a(n)}q^{\\ell n}$, so we have\n\\[\n\\bar{H}^{\\sigma}=(\\bar{H\\sl U_{\\ell}})^{\\ell}. \n\\]\nBy $(3)$ of Proposition~\\ref{Gross}, we have\n\\[\n\\omega(\\bar{H})=\\omega(\\bar{H}^{\\sigma})=\\ell\\omega(\\bar{H\\sl U_{\\ell}}).\n\\]\nIf $a(1) \\not \\equiv 0 \\pmod{\\ell}$, then $\\omega(\\bar{\\frac{a(1)}{24}\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes\\(\\frac{12}{\\bullet}\\)\\)}) > \\omega(\\bar{GE_{\\ell-1}^{\\lambda'-1}})$ since $\\omega(GE_{\\ell-1}^{\\lambda'-1}) \\leq 2\\lambda'$ and $\\lambda'$ is even. This would imply that\n$\\omega(\\bar{H})=\\omega(\\bar{\\Theta^{\\lambda'-1}\\(E_{\\ell+1}\\otimes\\(\\frac{12}{\\bullet}\\)\\)})= \\ell(\\lambda'-1)+\\lambda'+1$. This contradicts the fact that $\\omega(\\bar{H})$ is a multiple of $\\ell$.\nThus, $a(1) \\equiv 0 \\pmod{\\ell}$. By \\eqref{*}, we have $\\Theta(g) \\equiv 0 \\pmod{v}$. The result now follows as in the odd case.\n\n\\end{proof}\n\nNow we prove Theorem~\\ref{thm:main}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nSuppose that $\\ell \\geq 5$ is prime, that $K$ is a number field, and that $v$ is a prime of $K$ above $\\ell$.\nWe may assume that $K$ is Galois over $\\mathbb{Q}$. \nSuppose that $r$ is a positive integer satisfying $(r,6)=1$, that $\\lambda$ is a non-negative integer satisfying $\\lambda+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$, and that $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta}^{r})$ has the property that\n\n\\[\nf \\equiv \\sum_{i=1}^{m}\\sum_{n=1}^{\\infty}a(t_{i}n^{2})q^{\\frac{t_{i}n^{2}}{24}} \\not \\equiv 0 \\pmod{v}.\n\\]\nBy Proposition~\\ref{Proposition1} and Lemma~\\ref{lemmatwomultipliers}, we may assume that\n\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\not \\equiv 0 \\pmod{v}\n\\]\nand that either $r \\equiv 1 \\pmod{24}$ or $r \\equiv \\ell \\pmod{24}$. So, we need only consider the cases when $f \\in S_{\\lambda+\\frac{1}{2}}(1, \\nu_{\\eta})$ and when $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$ with $\\ell \\not \\equiv 1 \\pmod{24}$.\n\nSuppose that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta})$. Assume that $\\lambda$ is even. If $\\lambda=0$, then $f=c\\eta$ for some $c \\in \\mathcal{O}_{v}$. This implies that\n\\[\nf = a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{n^{2}}{24}},\n\\] \nwhich has the form of case $(1)$ of Theorem~\\ref{thm:main}, so assume that $\\lambda > 0$.\nTheorem~\\ref{Theorem1} then implies that\n\\begin{equation}\\label{**}\n\\Theta^{\\ell-1}(f)=\\sum_{\\ell \\nmid n}a(n^{2})q^{\\frac{n^{2}}{24}} \\equiv a(1) \\sum_{\\ell \\nmid n}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\end{equation}\nDefine $\\bar{\\lambda}:=\\lambda \\pmod{\\ell-1}.$ By Lemma~\\ref{Lemma1}, we have\n\\begin{equation}\\label{R.L.C.}\n\\bar{\\Theta^{\\ell-1}(f)}=\\bar{24^{\\frac{\\bar{\\lambda}}{2}}a(1)\\Theta^{\\frac{\\bar{\\lambda}}{2}}(\\eta)} \\in \\bar{S_{\\frac{\\bar{\\lambda}}{2}(\\ell+1)+\\frac{1}{2}}(1,\\nu_{\\eta})}.\n\\end{equation}\nThe fact that\n\\[\n\\(\\frac{\\bar{\\lambda}}{2}\\)(\\ell+1)+\\frac{1}{2} < \\frac{\\ell^{2}}{2}\n\\]\nimplies that $\\bar{f-\\Theta^{\\ell-1}(f)} \\in \\bar{S_{\\lambda'+\\frac{1}{2}}(1,\\nu_{\\eta})}$, where $\\lambda'+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$. Since\n\\[\nf-\\Theta^{\\ell-1}(f) \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2}n^{2})q^{\\frac{\\ell^{2} n^{2}}{24}} \\pmod{v},\n\\]\nwe apply Lemma~\\ref{Lemma2} to conclude that there exists $g \\in S_{\\lambda^{*}+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$ \nwith $\\lambda^{*}+\\frac{1}{2} < \\frac{\\ell}{2}$ satisfying\n\\[\ng \\equiv \\(f-\\Theta^{\\ell-1}(f)\\)\\sl U_{\\ell} \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{n^{2}}{24}}+\\sum_{n=1}^{\\infty}a(\\ell^{2} n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nIf $g \\equiv 0 \\pmod{v}$, then\n\\[\nf \\equiv \\Theta^{\\ell-1}(f)\\pmod{v},\n\\]\nand, by \\eqref{**}, this proves that\n\\[\nf \\equiv a(1) \\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}} \\pmod{v}.\n\\]\nThis has the form of case $(1)$ of Theorem~\\ref{thm:main}.\n\nIf $g \\not \\equiv 0 \\pmod{v}$, then Proposition~\\ref{Spicy} implies that $\\lambda'=0$ and $g=c\\eta$ for some $c \\in \\mathcal{O}_{v}$, which means that\n\n\\[\nf-\\Theta^{\\ell-1}(f) \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nThus, we have\n\n\\begin{equation}\\label{deletthis}\nf \\equiv a(1)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)n^{\\lambda}q^{\\frac{n^{2}}{24}}+a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\end{equation}\n\nSince $g \\not \\equiv 0 \\pmod{v}$, we have $a(\\ell) \\not \\equiv 0 \\pmod{v}$. Proposition~\\ref{Spicy} applied to $f$ implies that $\\ell \\equiv 1 \\pmod{24}$. \nIf $a(1) \\equiv 0 \\pmod{v}$, then \\eqref{deletthis} has the form of case $(2)$ of Theorem~\\ref{thm:main}.\nThis is equivalent to the congruence\n\n\\[\nf \\equiv a(\\ell)\\eta^{\\ell} \\pmod{v}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we have $\\lambda \\equiv \\omega(\\bar{\\eta^{-1}f})  \\equiv \\omega(\\bar{\\eta^{\\ell-1}}) \\equiv \\frac{\\ell-1}{2}\\pmod{\\ell-1}$.\n\nIf $a(1) \\not \\equiv 0 \\pmod{v}$, then \\eqref{deletthis} has the form of case $(3)$ of Theorem~\\ref{thm:main}. This is equivalent to the congruence\n\n\n\\[\nf \\equiv 24^{\\frac{\\lambda}{2}}a(1)\\Theta^{\\frac{\\lambda}{2}}(\\eta)+a(\\ell)\\eta^{\\ell} \\pmod{v}.\n\\]\nBy $(4)$ of Proposition~\\ref{Gross}, we have $\\omega(\\bar{\\eta^{-1}f}) \\equiv \\omega(\\bar{\\eta^{-1}\\Theta^{\\frac{\\lambda}{2}}(\\eta)}) \\equiv \\lambda \\pmod{\\ell-1}$. This implies that\n$\\omega(\\bar{\\eta^{\\ell-1}}) \\equiv \\lambda \\pmod{\\ell-1}$. Since $\\omega(\\bar{\\eta^{\\ell-1}})=\\frac{\\ell-1}{2}$, we have $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\n \n Now assume that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta})$ and that $\\lambda$ is odd. By Proposition~\\ref{oddcases}, we have\n \\[\n f \\equiv \\sum a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n \\] \nBy Lemma~\\ref{Lemma2} (since $\\lambda+\\frac{1}{2} < \\frac{\\ell^{2}}{2}$), there exists $g \\in S_{\\lambda'+\\frac{1}{2}}(1, \\nu_{\\eta}^{\\ell})$ such that $g \\equiv f\\sl U_{\\ell} \\pmod{v}$ and $\\lambda'+\\frac{1}{2} < \\frac{\\ell}{2}$. \nProposition~\\ref{Spicy} implies that $\\lambda'=0$ and $g =c\\eta$ for some $c \\in \\mathcal{O}_{v}$. Thus,\n\\[\nf \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nSince $f \\not \\equiv 0 \\pmod{v}$ has a Fourier expansion of the form $\\eqref{lemma4.2}$, we have \n$r \\equiv \\ell \\equiv 1 \\pmod{24}$\nin this case. As above, this implies that $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$, which implies that $\\lambda$ is even. This is a contradiction, so $\\lambda$ cannot be odd in this case.\n\nFinally, suppose that $\\ell \\not \\equiv 1 \\pmod{24}$ and that $f \\in S_{\\lambda+\\frac{1}{2}}(1,\\nu_{\\eta}^{\\ell})$. \nBy \\eqref{lemma4.2}, we have\n\\[\nf \\equiv \\sum_{n=1}^{\\infty}a(\\ell n^{2})q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nBy Lemma~\\ref{Lemma2}, there is $g \\in S_{\\lambda'+\\frac{1}{2}}(1,\\nu_{\\eta})$ with $\\lambda'+\\frac{1}{2}< \\frac{\\ell}{2}$ such that $g \\equiv f\\sl U_{\\ell} \\pmod{v}$.\nProposition~\\ref{Spicy} implies that $\\lambda'=0$ and that $g=c\\eta$ for some $c \\in \\mathcal{O}_{v}$. Thus,\n\\[\nf \\equiv a(\\ell)\\sum_{n=1}^{\\infty}\\(\\frac{12}{n}\\)q^{\\frac{\\ell n^{2}}{24}} \\pmod{v}.\n\\]\nThis has the form of case $(2)$ of Theorem~\\ref{thm:main}. As above, we have $\\lambda \\equiv \\frac{\\ell-1}{2} \\pmod{\\ell-1}$.\n\\end{proof}\n\n \\section{Acknowledgements}\nThe author would like to thank Scott Ahlgren for suggesting this project and for advice and guidance for this work. The author would also like to thank the referee for carefully reading this manuscript and making helpful comments which improved its exposition. Finally, the author would like to thank the Graduate College Fellowship program at the University of Illinois at Urbana-Champaign and the Alfred P. Sloan Foundation for their generous research support. \n \n\n \n \n \\bibliographystyle{amsalpha}\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nThe transition from carbon-based electricity production to renewable sources provides pressing arguments for investing in more transmission capacity between European countries \\cite{kristiansen2018}. Renewable power sources, such as wind farms and photovoltaic panels are often more efficiently placed at certain geographical locations with, e.g., higher levels of wind or sunshine than others. A well-developed international transmission network can help achieve a system where production can take place where it is opportune, and consumption where it is needed \\cite{UN2006multi}. Furthermore, renewable sources typically have higher levels of uncertainty in the associated production levels \\cite{konstantelos2014valuation}. International transmission lines can help balance the power production among geographical regions and thus help mitigate production uncertainty through geographical diversification \\cite{hasche2010general}.\n\nAn important obstacle to achieving the desired interconnected transmission network is the fact that the welfare benefits and costs associated with transmission expansions are often unevenly distributed among countries \\cite{mezHosi2016model}. In fact, situations can arise in which an investment in transmission capacity that is beneficial from a Europe-wide system perspective is detrimental to the economic welfare in an individual country. If this negatively affected country is one of the countries hosting the proposed transmission expansion, then it may block the investment and as a consequence, hurt the system as a whole \\cite{huppmann2015national}. This issue motivates the need for compensation mechanisms that distribute welfare gains in order to convince all countries to follow through on the transmission expansion plan and thus help achieve the system optimum \\cite{olmos2018transmission}.\n\nWe consider welfare compensation mechanisms in international transmission expansion planning (TEP) under uncertainty. In the literature, several studies have been performed that determine compensation amounts that should be paid to compensate countries \\textit{in expectation} \\cite{hogan2018primer,jansen2015alternative,konstantelos2017integrated,kristiansen2018}. However, by restricting attention to expected values, important effects resulting from uncertainty are neglected. The \\textit{actual} benefits\/costs depend on realization of uncertain elements, such as renewable production levels and electricity prices at different nodes. Hence, if uncertainty is ignored when constructing compensation mechanisms, countries run the risk that in some scenarios, they compensate more than they benefit or are not compensated enough to cover their welfare loss. \n\nIn this paper, we explore the potential of different compensation mechanisms to mitigate the risk associated with investing in a new transmission line. We use a fixed, lump sum payment as a benchmark and investigate whether other compensation mechanisms can perform better in terms of the risk faced by the countries involved as a consequence of the investment in the new transmission line. One alternative mechanism is proposed in the literature \\cite{kristiansen2018}, in the form of  power purchase agreements (PPAs). PPAs are contracts that essentially give a certain country a virtual, fixed price at which it can trade in the power market \\cite{kristiansen2018}. Deviations of the spot price from this fixed price are then used to determine the compensation amount to be paid or received. We show that in a stochastic setting only one country can receive a PPA. This is in contrast with the deterministic case, in which multiple PPAs can be constructed, one for each country, such that they balance each other exactly.\n\nBesides these two mechanisms from the literature, we also propose two novel compensation mechanisms. We aim to construct compensation mechanisms that achieve risk mitigation by using scenario information to determine the compensation amount. For this purpose, we propose to base the compensations on economic measures related to the economic value of the proposed transmission line investment in the realized scenario. Specifically, we use the amount of flow through the new transmission line and its economic value as measures to base our novel compensation mechanisms on.\n\nWe compare the different compensation mechanisms numerically, using a case study of the Northern-European electricity market, focusing on a new transmission line between Norway and Germany. We test the ability of each compensation mechanism to mitigate the risk associated with the transmission expansion investment for each of the affected countries. Here, we consider risk both in terms of the variability of the welfare effect of the investment and in terms of the expected welfare loss as a result of the investment. We run experiments in two settings: a setting with bilateral compensations between Norway and Germany only and a setting with multilateral compensations between all countries that are significantly affected by the proposed investment. \n\nIn both cases, we show that a theoretically ideal mechanism, that equally shares welfare benefits in every scenario, significantly outperforms lump sum payments in terms of its ability to mitigate risk of the countries involved. This demonstrates the potential for risk mitigation through alternative compensation mechanisms. Out of the other compensation mechanisms, our novel value-based mechanism appears to perform best. In particular, it consistently outperforms the lump sum payments. Hence, we show that risk mitigation can actually be achieved by using scenario-dependent compensation mechanisms. Finally, for PPAs, the question of which country receives the PPA turns out to be crucial: a Germany-based PPA performs bad for both Norway and Germany, while a Norway-based PPA shifts risk from Germany to Norway, compared to the lump sum.\n\nThe remainder of this paper is structured as follows. In Section~\\ref{sec:literature_review} we review the literature on welfare distribution in TEP. Next, in Section~\\ref{sec:illustrative_examples} we provide simple, illustrative examples that motivate the need for welfare compensation. In Section~\\ref{sec:TEP} we present and solve a transmission expansion model in the Northern-European market. The resulting optimal transmission expansion plan serves as a test environment for our investigation into different welfare mechanisms in Section~\\ref{sec:compensation_mechanisms}. In this section, which constitutes the core of the paper, we discuss existing compensation mechanisms, propose several novel mechanisms, and test their performance in the case study from Section~\\ref{sec:TEP}. Section~\\ref{sec:conclusion} concludes the paper. Finally, Section~\\ref{sec:mathematical_model} and Section~\\ref{sec:data} in the appendix contain a description of the mathematical model formulation and the data used, respectively, in the TEP model of Section~\\ref{sec:TEP}. \n\n\\section{Literature review}\n\\label{sec:literature_review}\n\nTEP is an active topic of research within the field of operations research \\cite{kristiansen2018}. Much of the literature is aimed at developing methods to find good candidates for TEP investments \\cite{latorre2003,hemmati2013comprehensive}. Typically, finding an optimal combination of investments is a challenging task, both from a modeling and a computational point of view. One major difficulty is that many relevant parameters, such as demand levels and renewable energy production, may be uncertain, necessitating stochastic models \\cite{Zhao2009flexible}. Furthermore, investments are typically of a discrete nature, which introduces the need for integer decision variables \\cite{alguacil2003,delatorre2008}. Finally, in order to properly model the effect of the transmission expansion on the power market, explicit modeling of the market participants through equilibrium models may be required \\cite{gabriel2012}.  All these factors make TEP a challenging area of research; see \\cite{mahdavi2018transmission} for a recent review of the literature on TEP.\n\nIn this paper we are not mainly interested in finding the best transmission expansion plan, however, but in making this plan practically \\textit{achievable} by constructing welfare-sharing mechanisms that allocate costs and benefits in such a way that all the relevant actors are willing to follow through on the proposed expansion plan. In the literature, most effort in this direction has been spent on creating mechanisms to share the \\textit{investment costs} \\cite{erli2005transmission,konstantelos2017integrated,nylund2014regional,roustaei2014transmission}. Currently, the most-used cost-sharing strategy is the \\textit{equal share principle} \\cite{huang2016mind}. According to this principle, the investment costs are split equally between the two countries hosting a new cable. Before the year 2016, all except for two transmission expansion projects in the EU followed this principle \\cite{huang2016mind}. Recently, though, there has been a trend towards the so-called \\textit{beneficiaries pay principle} \\cite{ACER,FERC2012order1000}. According to this principle, each country should pay a share of the investment cost proportional to its benefit from the investment. A specific allocation method that follows this principle is the \\textit{net positive benefit differential} \\cite{hogan2018primer, konstantelos2017}. The hope is that such a benefit-based allocation method leads to better incentives and ultimately more investments that are beneficial to the system as a whole.\n\nOne shortcoming of these cost allocation methods, however, is that they ignore potential \\textit{welfare losses} as a result of transmission expansion. As discussed in the introduction, even if the investment costs are zero, some countries might be worse off as a result of transmission expansion, which is especially problematic if one of these countries is hosting the proposed transmission expansion. Recently, a few authors have recognized this problem and proposed methods to share welfare gains and losses \\cite{konstantelos2017integrated,kristiansen2018,churkin2019can,churkin2021review}. These authors take a cooperative game theory perspective and propose compensation amounts between countries, based on different conceptions of fairness, e.g., the net postitive benefit differential \\cite{konstantelos2017integrated}, the Shapley value \\cite{kristiansen2018}, or the nucleolus \\cite{churkin2019can}. \n\nTo the best of our knowledge, all papers in this literature consider a single compensation amount based on the \\textit{expected} welfare benefits to the various countries. That is, they ignore the possibility of compensations varying per scenario and the resulting potential for risk mitigation, as discussed in the introduction. Rather, most papers seem to implicitly assume a lump sum payment.\n\nOne alternative mechanism has been proposed in the literature: power purchase agreements (PPAs) \\cite{kristiansen2018}. Such a PPA is a contract based on a fixed price $\\pi^{\\text{PPA}}$ for purchasing power for each country. The country then pays an amount proportional to the net flow into the country times the difference between the spot price and the predetermined price $\\pi^{\\text{PPA}}$ to a fund. If the price $\\pi^{\\text{PPA}}$ would be less beneficial to the country than the actual spot price at which it trades, then the country pays into the fund. In the reverse case it receives money from the fund. The PPA prices $\\pi^{\\text{PPA}}$ are determined up front such that in expectation, each country receives their ``fair'' share of the total welfare. This PPA-based mechanism does depend on the \\textit{actual} behavior of the system (in terms of flows and prices). Hence, this mechanism could potentially be able to mitigate risk resulting from the transmission expansion faced by the relevant countries. Although Kristiansen et al. \\cite{kristiansen2018} propose to use PPAs as a welfare compensation mechanism, they don't recognize their potential for risk mitigation. In fact, the authors assume a deterministic setting. In Section~\\ref{sec:compensation_mechanisms} we show that the construction of PPAs is fundamentally different in a stochastic setting.\n\nIn the remainder of this paper, we contribute to this literature by testing the performance of various compensation mechanisms in terms of their ability to mitigate the risk of investing in a new transmission cable for the countries involved. We consider both mechanisms from the literature (lump sum payments and PPAs) and novel mechanisms (based on the flow through the new transmission line and its economic value).\n\n\n\\section{Illustrative examples} \\label{sec:illustrative_examples}\n\nIn this section we present two simple, illustrative examples to motivate our study of welfare compensation mechanisms. In Section~\\ref{subsec:two-node} we consider a two-node system and show that in the absence of other nodes, there is no need for welfare distribution. In Section~\\ref{subsec:three-node} we extend the model to a three-node system and provide an example in which transmission expansion is desirable from a system point of view, but negatively affects one of the hosting countries. This motivates the need for compensation mechanisms studied in this paper.\n\n\n\n\n\\subsection{Two-node system: basics} \\label{subsec:two-node}\n\nFirst we consider a network consisting of two country nodes, as illustrated in Figure~\\ref{subfig:network_configurations_2_nodes}. Each node $i=1,2$ contains power suppliers and consumers, represented by a linear supply and demand curve $S_i(\\pi_i)$ and $D_i(\\pi_i)$, respectively. We assume perfect competition with all market participants being price takers. Moreover, we assume that that initially, there is no transmission capacity between the two nodes. Hence, initially every node is an independent market and initial equilibrium prices $\\pi^*_1$ and $\\pi^*_2$ occur where the supply and demand curves meet. See Figure~\\ref{fig:surplus_two_nodes} for an illustration. The green and red areas represent the consumer surplus (CS) and producer surplus (PS), respectively, in each node. Note that $\\pi^*_1 > \\pi^*_2$, indicating that power is more scarce in node 1 as compared to node 2. \n\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{0.25\\columnwidth}\n        \\centering\n        \\input{Figures\/Network2Nodes.tex}\n        \\caption{}\n        \\label{subfig:network_configurations_2_nodes}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.5\\columnwidth}\n        \\centering\n        \\input{Figures\/Network3NodesRadial.tex}\n        \\caption{}\n        \\label{subfig:network_configurations_3_nodes}\n    \\end{subfigure}\n    \\caption{Network configurations used in the examples in Section~\\ref{sec:illustrative_examples}.}\n    \\label{fig:network_configurations}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\input{Figures\/Surplus2Nodes.tex}\n    \\caption{Surpluses in the two-node example in Section~\\ref{subsec:two-node}.}\n    \\label{fig:surplus_two_nodes}\n\\end{figure}\n\n\nBefore introducing the possibility of transmission expansion, it is useful to define the import\/export curve corresponding to node 1 and 2. We define the import curve for node 1 as $I_1(\\pi_1) := D_1(\\pi_1) - S_1(\\pi_1)$ and the export curve for node 2 as $E_2(\\pi_2) := S_2(\\pi_1) - D_2(\\pi_2)$. Here, $I_1(\\pi_1)$ represents the amount that node 1 is willing to import at price $\\pi_1$, while $E_2(\\pi_2)$ is the amount that node 2 is willing to export at price $\\pi_2$. See Figure~\\ref{fig:intput_output} for an illustration of the import\/export graph corresponding to the example in Figure~\\ref{fig:surplus_two_nodes}. On the horizontal axis we have the variable $f$, representing flow from node 2 to 1 (i.e., import to 1\/export from 2). Observe that, by definition, $I_1(\\pi^*_1) = 0$ and $E_2(\\pi^*_2) = 0$. Moreover, the figure shows that if transmission capacity between node 1 and 2 were unlimited, the combined market would clear at a common price $\\pi_1 = \\pi_2 = \\bar{\\pi}$ with an associated flow of $\\bar{f}$. \n\\begin{figure}[h]\n    \\centering\n    \\input{Figures\/Capacity2Nodes.tex}\n    \\caption{Import\/export graph for the two-node example in Section~\\ref{subsec:two-node}.}\n    \\label{fig:intput_output}\n\\end{figure}\n\n\nNow consider a social planner that can invest in transmission capacity $x$ between node 1 and 2. We assume that transmission capacity $x$ can be any non-negative real number and that the marginal cost of investment $C$ is constant. For a given investment $x$, the transmission system operator (TSO) will earn congestion rent: the TSO buys power at the low-price node and sells at the high-price node. It earns an amount $CR = (\\pi_1 - \\pi_2) f$, where $f$ again represents the flow from node 2 to 1. If $x \\geq \\bar{f}$, then the transmission capacity constraint $f \\leq x$ is be non-binding and we obtain the common price $\\bar{\\pi}$ and flow $\\bar{f}$ described above. Now suppose that $x < \\bar{f}$. Then, the transmission capacity constraint is binding and a price difference will arise between the two nodes. An illustration is given in Figures~\\ref{fig:surplus_two_nodes} and \\ref{fig:intput_output} for a capacity of $x^\\prime$. The associated prices, demand and supply levels, and flow are denoted by a prime. \n\nThe welfare gains associated with this capacity are visualized in Figure~\\ref{fig:surplus_two_nodes}. The welfare gain to node 1 is represented by the triangle defined by the three black dots in the left graph, and similarly for node 2. The congestion rent is also part of the welfare gain, and is represented by the rectangle in the left graph. Note that it is not necessarily allocated to a specific region, although it is shown in the diagram representing node 1 in the picture. Equivalently, the welfare effects are visualized in Figure~\\ref{fig:intput_output}. Here, the upper triangle represents the welfare gain to node 1 and has the same area as the left triangle in Figure~\\ref{fig:surplus_two_nodes}. The lower triangle represents the welfare gain to node 2 and has the same area as the right triangle in Figure~\\ref{fig:surplus_two_nodes}. Finally, the rectangle represents the congestion rent and has the same size as the rectangle in Figure~\\ref{fig:surplus_two_nodes}. \n\nFigure~\\ref{fig:intput_output} reveals how much a welfare-maximizing social planner should invest. The welfare gains are represented by the colored trapezoid. Clearly, the marginal welfare gains are decreasing in $x$. The marginal cost is constant at $C$. If $\\pi_1^* - \\pi_2^* \\leq C$, then an investment of $x=0$ is optimal. In the more interesting case where $\\pi_1^* - \\pi_2^* > C$, it is optimal to invest an amount $x$ such that $\\pi_1 - \\pi_2 = C$, and congestion rent equals investment cost. Hence, as expected from economic theory, in the social optimum marginal revenue equals marginal cost. \n\nLooking at the welfare effects of transmission expansion, we see that in each node there are winners and losers. In node 1 consumers benefit from the lower price, while producers are hurt. In node 2 the effects are reversed. However, it is clear from Figure~\\ref{fig:intput_output} that the total welfare effects of transmission expansion, represented by the areas of the two triangles, are non-negative for both nodes. In the next subsection we will see that the latter result need not hold for systems with more than two nodes.\n\n\n\\subsection{Three-node system: welfare issues} \\label{subsec:three-node}\n\nNext, we consider a network consisting of three nodes, illustrated in Figure~\\ref{subfig:network_configurations_3_nodes}. The purpose is to provide an example in which transmission expansion would be beneficial for the three-node system as a whole, but one of the nodes would be worse off than without the added transmission capacity. In particular, we show that this can be the case for one of the nodes that is hosting the new transmission capacity, which means that they will be able to block the transmission expansion and thus, hurt the system as a whole.\n\nThe three-node example we consider is designed to be the simplest possible example that manifests the welfare loss problem outlined in the previous paragraph. Let node 1 be a supply node, meaning that it has only supply, represented by a linear supply curve, and no demand. In contrast, let node 2 and 3 be demand nodes, meaning that they have only demand, represented by a linear demand curve, and no supply. The supply and demand curves are given by $S_1(\\pi_1) = \\pi_1$, $D_2(\\pi_2) = 6 - \\pi_2$, $D_3(\\pi_3) = 6 - \\pi_3$. As in the two-node setting, we assume perfect competition and price-taking behavior for all market participants.\n\nWe consider an initial situation in which there is a transmission line between node 1 and 2 of unlimited capacity and no transmission lines between other pairs of nodes. In this situation, node 1 and 2 form a single market with a common price $\\pi$ and an equilibrium is found where the supply curve of node 1 meets the demand curve of node 2: $\\pi_1 = \\pi_2 = \\pi^* = 3$, $s_1^* = d_2^* = q^* = 3$. This situation is illustrated in the top graph in Figure~\\ref{fig:surplus_three_nodes}. Note that the import\/export graphs are given by $I_2 = D_2$ and $E_1 = S_1$. The green area represents the consumer surplus in node 2, while the red are represents the producer surplus in node 1. The welfare distribution in this old situation is described in the top half of Table~\\ref{tab:welfare_distribution}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\input{Figures\/Surplus3Nodes.tex}\n    \\caption{Import\/output graph for the three-node example in Section~\\ref{subsec:three-node}}\n    \\label{fig:surplus_three_nodes}\n\\end{figure}\n\n\nNext, suppose there is the possibility of opening a transmission line between node 2 and 3. Suppose the cost of building this line is zero and it has unbounded capacity. If the line is built, then node 2 and 3 can be seen as a single demand node and together, node 1, 2 and 3 form a single, joint market, without any constraints on transmission between nodes. Hence, we can find the market equilibrium by aggregating the two demand curves to obtain the joint demand curve $D(\\pi) = D_2(\\pi) + D_3(\\pi) = 12 - 2\\pi$. The new equilibrium is found at $\\pi_1 = \\pi_2 = \\pi_3 = \\pi^* = 4$, and $s_1 = 2 d_2 = 2 d_3 = q^* = 4$. The new situation is illustrated in the bottom graph in Figure~\\ref{fig:surplus_three_nodes}. Note that the import\/export graphs are given by $I_{2,3} = D$ and $E_1 = S_1$. The joint consumer surplus in node 2 and 3 is represented by the green area, while the producer surplus in node 1 is given by the red area. \n\nThe welfare distribution in the new situation is described in Table~\\ref{tab:welfare_distribution}. Note that the total welfare of the entire system has increased. This is as expected, since the new situation has fewer (transmission) constraints than the old situation. However, observe that the total welfare of node 2 has decreased. The new connection with the demand node 3 has made power scarcer and thus, increased the price. This higher price hurts the consumers in node 2, while there are no producers in node 2 to benefit from the higher price. Hence, node 2 is worse off with the new connection than without. Importantly, node 2 is one of the two end points of the cable and hence, it can be expected to be able to block the connection. Moreover, in practice, a direct connection between node 1 and 3 might be infeasible (or very expensive) for geographical reasons.\n\nIn this situation, node 1 and 3 may decide to spend part of their welfare gains to compensate node 2 for its welfare loss. Indeed, they have sufficient welfare gains (3.5 for node 1 and 2 for node 3) to compensate the welfare loss of 2.5 in node 2. Importantly, such a compensation may help achieve the social optimum, which is desirable from a system point of view. This example motivates our investigation into welfare compensation schemes in the remainder of this paper.\n\n\\begin{table}[t]\n    \\centering\n    \\begin{tabular}{@{}llrrrr@{}}\n    \\toprule\n    \\textbf{}  & \\textbf{} & \\textbf{1} & \\textbf{2} & \\textbf{3} & \\textbf{System} \\\\ \\midrule\n    \\multirow{3}{*}{Old situation} & CS & 0 & 4.5 & 0 & 4.5 \\\\\n    & PS & 4.5 & 0 & 0 & 4.5 \\\\ \\cmidrule(l){2-6} \n    & TW & 4.5 & 4.5 & 0 & 9 \\\\ \\midrule\n    \\multirow{3}{*}{New situation} & CS & 8 & 0 & 0 & 8  \\\\\n    & PS & 0 & 2 & 2 & 4 \\\\ \\cmidrule(l){2-6} \n    & TW & 8 & 2 & 2 & 12 \\\\ \\bottomrule\n    \\end{tabular}%\n    \\caption{Welfare distribution -- in terms of consumer surplus (CS), producer surplus (PS), and total surplus (PS) -- in the three-node example of Section~\\ref{subsec:three-node}.}\n    \\label{tab:welfare_distribution}\n\\end{table}\n\n\n\n\n\\section{Transmission expansion planning}\n\\label{sec:TEP}\n\nIn this section we use a TEP model to find a system-optimal transmission expansion plan in a case study of the Northern European power market. The results from this model are used in Section~\\ref{sec:compensation_mechanisms} to investigate the performance of various welfare compensation mechanisms. The goal here is not to find the best possible transmission expansion plan in real life; more sophisticated models exist in the literature that are likely more capable for that purpose (see, e.g., \\cite{mahdavi2018transmission}). Instead, the goal is to find a reasonable candidate transmission expansion plan that can serve as a basis for an analysis of different welfare compensation mechanisms.\n\nThe mathematical model can be described as a mathematical program with equilibrium constraints (MPEC), in which a social planner determines the optimal transmission expansion plan from a system welfare point of view, while taking the optimal subsequent behavior of all market participants into account through optimality conditions in the form of equilibrium constraints. A full description of the mathematical model is given in Section~\\ref{sec:mathematical_model} in the appendix.\n\nOur mathematical model is used to analyze a case study of the Northern European power market, focusing on possible investment in a new transmission line between the price zones NO2 and DE, representing part of Norway and the whole of Germany, respectively. Figure~\\ref{fig:LineDiagram} provides a schematic overview of the power system used in the case study. We allow for investments in transmission cables within Norway, too (i.e., between all zones colored red). However, we disallow investments in any other cables, in order to be able to isolate the effects of investments in the new NO2-DE line.\n\n\n\\begin{figure}[t]\n\\centering\n\\input{Figures\/NodeMap.tex}\n\\caption{Line diagram of the power system in the case study.}\n\\label{fig:LineDiagram}\n\\end{figure}\n\nThe data used to parametrize the model is based on historical data on investments in and operation of the Northern European power market. A detailed description of the data used in the case study can be found in Section~\\ref{sec:data} in the appendix.\n\n\n\\subsection{Results}\n\\label{subsec:results}\nWe first consider a benchmark setting in which we disallow investment in the NO2-DE line, but allow for investments within Norway. In this benchmark a social planner invests in 221 MW of extra capacity in the NO1-NO2 line. This comes at an annualized investment cost of 8.0 million euros.\n\nNext, we allow investment in the NO2-DE line, as well as the other lines within Norway. We find that a social planner would invest in a capacity expansion of 4147 MW in the NO2-DE cable. This comes at an annualized investment cost of 294 million euros per year. As expected, this is equal to the expected yearly congestion rent earned on the new line. Besides investment in the NO2-DE line, a social planner would also invest in 464 MW of extra capacity the NO1-NO2 cable, at an annual investment cost of 16.9 million euros. Note that this investment is larger than in the benchmark setting. The rationale behind this additional investment is to export the electricity produced in NO1 (or imported into NO1) through NO2 to DE and the rest of Europe.\n\nCompared to the benchmark, the social planner solution leads to a system-wide social welfare increase of 84.6 million euros per year. This constitutes a return on investment of 27.9\\%. Hence, from a system point of view, it is highly desirable to invest in a new transmission cable. However, some countries -- Germany in particular -- do not profit from the extra capacity and might block the investment plan.\n\nThe welfare effects of the transmission investment are illustrated in Figure~\\ref{fig:welfare_country}. Three countries benefit significantly from the investment: Norway, Austria, and France. In Norway, producers benefit from their additional capacity to export electricity to mainland Europe, while in Austria and France, consumers benefit from the resulting lower prices. On the other hand, two countries are significantly negatively affected: Germany and Denmark. In both countries, producers suffer from the lower prices as a consequence of cheap Norwegian electricity entering the market. This price drop results in both lower profit margins and loss of sales: some of the demand in Germany will be satisfied through Norwegian power entering the country through the new NO2-DE line. One benefit of this, however, is that the power produced in Norway is more than 95\\% renewable \\cite{norwayministryofpetroliumenergy2016}, while in Germany it is only about 45\\% \\cite{germanumweltsbundesamt2021}. Hence, the new line can indirectly contribute to German sustainability goals. \n\nNevertheless, the negative welfare impact of the capacity expansion in Germany may well be an obstacle for realization of the transmission expansion. Since Germany is one of the hosting countries, it can block the investment if it deems the new line to be detrimental to its welfare. However, since the system welfare effects are positive, it is in principle  possible to construct compensation mechanisms that result in a net welfare gain for Germany. Notably, since the net welfare gain in Norway (88.2 million annually)  exceeds the net welfare loss in Germany (85.1 million annually), a bilateral compensation scheme between these two countries is a viable option.\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.75]{Figures\/plot_welfare_country_adjusted.png}\n    \\caption{Total welfare effects of the NO2-DE cable per country}\n    \\label{fig:welfare_country}\n\\end{figure}\n\n\n\n\\section{Compensation mechanisms}\n\\label{sec:compensation_mechanisms}\n\nIn this section we investigate different compensation mechanisms that may be used to compensate countries for welfare losses resulting from transmission expansion investments. First, in Section~\\ref{subsec:incentive_effects} we shortly discuss potential incentive effects of compensation mechanisms. Next, in Section~\\ref{subsec:mechanisms} we list and discuss a number of existing compensation mechanisms and we propose two novel mechanisms. We discuss the rationale behind them and their expected performance in a stochastic setting. Finally, in Section~\\ref{subsec:performance} we numerically test the performance of the different mechanisms using the case study from Section~\\ref{sec:TEP}.\n\n\\subsection{Incentive effects}\n\\label{subsec:incentive_effects}\n\nOne important potential effect of compensation mechanisms is that they might skew incentives of actors in the power market. That is, compensation mechanisms may create incentives for certain actors in the power market to change their behavior in order to receive a larger compensation amount. Importantly, such incentive distortions deviate from ``pure'', market-based incentives and hence, may steer the market equilibrium away from a perfect market equilibrium. This may well have negative effects on the total welfare in the system as a whole. In this paper, we steer away from this issue as much as possible, but a short discussion is in place.\n\nIn practice, the question whether compensation mechanisms affect incentives depends on \\textit{who receives the compensation}. However, in the literature this issue seems to have been ignored. Welfare compensations are modeled as transfers of money between \\textit{countries}, but what agent within a country should receive the money is typically not specified. Given the fact that the proposed compensations are meant for compensating the \\textit{total welfare} in a country, it seems most reasonable to assume that they are paid between \\textit{governments}, which represent the countries as a whole. Since governments do not trade in power markets directly, it seems to be safe to assume that compensations between governments do not skew incentives of any market players. \n\nHowever, in practice, governments may decide to use the compensations to, e.g., change taxes\/subsidies on power, in order to compensate the groups within the country that are affected by the transmission line investment (e.g., consumers or producers). In this case, the compensation may skew incentives of market players. Moreover, if the tax\/subsidy change \\textit{depends on the actual amount of compensation received} by the government, then not only the expected amount of compensation, but also the particular compensation \\textit{mechanism} may skew incentives of market players. \n\nIn this paper, we assume that governments do not use compensations in a way that affects the incentives of players in the energy market. This assumption is in line with the paradigm of avoiding protection and aiming for perfect competition, on which the European power market is founded \\cite{olmos2018transmission}. It allows us to steer away from the issue of incentive distortions as much as possible and investigate the \\textit{risk} effects of compensation mechanisms in isolation. We believe that the topic of incentive distortions resulting from compensation mechanisms is a complicated and interesting topic in its own right and deserves attention in future research.\n\n\n\\subsection{Mechanisms}\n\\label{subsec:mechanisms}\n\n\n\\subsubsection{Lump sum payment: issues}\n\nThe most straightforward compensation mechanism is a lump sum payment. This consists of a fixed payment from one country to another. It has the benefits of being very simple and completely predictable. However, in the presence of uncertainty, a lump sum has some drawbacks. In a stochastic setting, the \\textit{actual} welfare effect resulting from investment in the new line is uncertain. Hence, the lump sum payment should be based on the \\textit{expected} welfare effect. However, there might well be a discrepancy between this expected welfare effect and its actual, realized value. As a result, there may be scenarios in which the lump sum compensation to Germany is not enough or in which Norway must compensate Germany, even though in reality it does not profit from the new line. The potential of such scenarios might make countries hesitant to accept the lump sum mechanism.\n\nMore generally speaking, uncertainty about the actual welfare effects introduces \\textit{risk} for the countries involved. A lump sum mechanism ignores this risk completely by focusing on only the expected value. Other mechanisms might be able to deal with risk in a smarter way. Ideally, a mechanism compensates countries more in scenarios in which they are hurt more by the new line and vice versa. This would reduce the risk of the countries involved, potentially making them more willing to accept the compensation mechanism.\n\nIn order to be able to construct a risk-mitigating mechanism, a few conditions need to hold. Firstly, the new line's welfare effects in the compensating countries and compensated countries should be negatively correlated, such that they can share their risk between scenarios. For the relevant countries in our case study, Norway and Germany, this correlation turns out to be $-0.34$; see Table~\\ref{tab:corr_coalition}. This moderately negative correlation suggests that there is indeed a potential for risk sharing, although probably only to a modest degree. The relationship between the welfare effects in Norway and Germany is presented in more detail in the scatter plot in Figure~\\ref{fig:scatter_NO_DE_welfare_delta}. Note that just over half of the dots, representing scenarios, are to the top-right of the red 45 degree line. In those cases, there is enough total benefit in both countries combined that could theoretically be shared such that neither country suffers from the new line. For the other half of the scenarios this is not possible, however. This again suggests that there is indeed room for risk sharing, although to a limited degree.\n\nSecondly, for a compensation mechanism to be able to exploit such a negative correlation, it should \\textit{depend on the scenario}. That is, the compensation amount should be different under different circumstances. Evidently, a lump sum payment lacks this property, so alternatives are warranted.\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between the welfare effects of the new transmission line in different countries.}\n\\label{tab:corr_coalition}\n\\begin{tabular}{llllll}\n\\toprule\n            & \\textbf{NO} & \\textbf{AT} & \\textbf{FR} & \\textbf{DE} & \\textbf{DK} \\\\ \\midrule\n\\textbf{NO} & 1.00        & 0.29        & 0.23        & -0.34       & -0.63       \\\\\n\\textbf{AT} & 0.29        & 1.00        & 0.98        & -0.27       & -0.63       \\\\\n\\textbf{FR} & 0.23        & 0.98        & 1.00        & -0.31       & -0.57       \\\\\n\\textbf{DE} & -0.34       & -0.27       & -0.31       & 1.00        & 0.36        \\\\\n\\textbf{DK} & -0.63       & -0.63       & -0.57       & 0.36        & 1.00        \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\n\n\\subsubsection{Power purchase agreement}\nOne scenario-dependent compensation mechanism proposed in the literature \\cite{kristiansen2018} is a so-called power purchase agreement (PPA). This entails giving a certain country, say country A, a virtual, fixed price $\\pi^{\\text{PPA}_A}$ for importing\/exporting power through the new transmission line. Then, after trading at the spot price, the country is compensated such that it is as though the country had traded at the PPA price $\\pi^{\\text{PPA}_A}$. Mathematiaclly, we define the compensation to country $A$ in scenario $\\omega$ by \n\\begin{align*}\n    C_{A}^{\\text{PPA}_A,\\omega} = \\sum_{t \\in \\mathcal{T}} f_{AB}^{\\omega,t} (\\pi^{\\text{PPA}_A} - \\pi_A^{\\omega,t}).\n\\end{align*}\nAssuming country $A$ tends to export (i.e., typically $f_{AB}^{\\omega,t}$ is positive), a low PPA price yields a negative compensation to country A, while a high PPA price yields a positive compensation. The reverse holds if country $A$ tends to import. More general definitions of PPAs exist, which give a country a virtual, fixed price for its trade through \\textit{all} transmission cables it is connected to \\cite{kristiansen2018}. However, we find it unreasonable to expect that a country would be willing to essentially take over all of another country's price risk in a compensation scheme for a single transmission line investment. Hence, we choose to restrict the definition to focus on trade through the new transmission line only. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.4]{Figures\/scatter_NO_DE_welfare_Delta.png}\n    \\caption{Scatter plot of the effect of the new line on the welfare in Norway and Germany. The dashed red line represents all points for which the aggregated welfare effect for Norway and Germany is zero.}\n    \\label{fig:scatter_NO_DE_welfare_delta}\n\\end{figure}\n\n\nOne important property of a PPA is that it is connected to a specific country. This means that one country gets a PPA, i.e., trades at a virtual fixed price, while the other country pays the difference. In a deterministic setting it is possible to construct two PPAs, one for each country, such that the compensation to be paid to the first country according to its PPA exactly matches the compensation to be paid by the second country according to its PPA. This property is called \\textit{budget balancedness} \\cite{narahari2014game}. A multi-country PPA mechanism is proposed in \\cite{kristiansen2018}. However, in a stochastic setting, a multi-country PPA mechanism cannot be constructed, as this would require different PPA prices for every scenario. We illustrate this in the following example.\n\n\\begin{example}\nConsider a network consisting of two countries, A and B, with no initial transmission capacity. There is a proposed investment in a cable of capacity 10 between A and B. We consider two simplified, equally likely, one-period scenarios. In scenario 1 (indicated by the superscript 1), the flow through the new cable is $f_{AB}^1 = 10$, the prices in A and B are $\\pi_A^1 = 1$, $\\pi_B^1 = 2$, and the welfare effects of the new cable are $\\Delta \\text{TW}_A^1 = -10$ and $\\Delta \\text{TW}_B^1 = 20$. In scenario 2, we have $f_{AB}^2 = 10$, $\\pi_A^2 = 1$, $\\pi_B^2 = 3$, $\\Delta \\text{TW}_A^2 = -10$, and $\\Delta \\text{TW}_B^1 = 40$. Before realization of the uncertainty, we need to construct two PPAs that share the welfare gains\/losses equally in expectation. Note that $\\mathbb{E}[\\Delta \\text{TW}_A] = -10$ and $\\mathbb{E}[\\Delta \\text{TW}_B] = +30$. Hence, on average, A should receive a compensation $C_A$ of 20 by B. For A, this yields the following equation for its PPA price:\n\\begin{align*}\n    \\mathbb{E}[C_A^{\\text{PPA}_A}] = \\mathbb{E}[f_{AB} (\\pi^{\\text{PPA}_A} - \\pi_A)] = 20,\n\\end{align*}\nwhich, after some calculation, yields $\\pi^{\\text{PPA}_A} = 3$. Similarly, for country B we have the equation\n\\begin{align*}\n    \\mathbb{E}[C_B^{\\text{PPA}_B}] = \\mathbb{E}[f_{BA} (\\pi^{\\text{PPA}_B} - \\pi_A)] = -20,\n\\end{align*}\nwhich yields $\\pi^{\\text{PPA}_B} = 4.5$. We observe that using these PPA prices, the compensation received by A equals the compensation paid by B, i.e., $\\mathbb{E}[C_A^{\\text{PPA}_A}] = -\\mathbb{E}[C_B^{\\text{PPA}_B}]$. So buget balancedness holds in expectation. \n\nNow consider a single scenario, e.g., scenario 1. For this scenario we have $C_A^{\\text{PPA}_A,1} = f^1_{AB} (\\pi^{\\text{PPA}_A} - \\pi^1_A) = 10 (3 - 1) = 20$. For B, we have $C_B^{\\text{PPA}_B,1} = f^1_{BA} (\\pi^{\\text{PPA}_B} - \\pi^1_B) = - 10 (4.5 - 2) = - 25$. Note that $C_A^{\\text{PPA}_A,1} \\neq - C_B^{\\text{PPA}_B,1}$, so the compensations do not add to zero. Similarly, for scenario 2 we observe $C_A^{\\text{PPA}_A,2} = 20$ and $C_B^{\\text{PPA}_B,2} = -15$, which also do not add to zero. We conclude that in a stochastic setting, budget balancedness generally does not hold for PPAs. \\hfill \\qedsymbol\n\\end{example}\n\nAs a consequence of this lack of budget balancedness in a stochastic setting, we must pick one country that receives the PPA, i.e., that gets to trade at a virtual, fixed price. This is also the reason for the subscript $A$ in $C_i^{\\text{PPA}_A}$, $i=A,B$, and $\\pi^{\\text{PPA}_A}$. Now, it is not hard to show that a Norway-based PPA yields larger compensations if $\\pi^{\\text{NO2}}$ is higher. Similarly, a Germany-based PPA yields larger compensations if $\\pi^{\\text{DE}}$ is higher. In Table~\\ref{tab:price_correlations} we observe that in both these situations, Norway tends to profit more from the new line, while Germany suffers more. Hence, in situations where higher compensations are desired, both PPAs indeed yield higher compensations. This gives us some confidence that the PPAs might be able to succeed in mitigating risk for the countries involved. Based on the fact that the correlations with the NO2 price are stronger than those with the DE price, we also expect the Norway-based PPA to perform better than the Germany-based PPA.\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between welfare effect of the new transmission line in Norway and Germany and different price measures.}\n\\label{tab:price_correlations}\n\\begin{tabular}{llll}\n\\toprule\n& \\textbf{$\\pi^{\\text{NO2}}$} & \\textbf{$\\pi^{\\text{DE}}$} \\\\ \\midrule\n\\textbf{$\\Delta \\text{TW}_{\\text{NO}}$} & 0.23 & 0.52 \\\\\n\\textbf{$\\Delta \\text{TW}_{\\text{DE}}$} & -0.70 & -0.13 \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\\subsubsection{Novel mechanisms}\n\\label{subsubsec:novel_mechanisms}\n\nBesides the lump sum and PPAs, we investigate the potential of other, novel compensation mechanisms. We aim for risk-mitigating mechanisms. Ideally, the mechanism is such that the compensation amount mimics the relative welfare effects in the countries.\n\nOne possibility would be to \\textit{compute} the actual welfare effects using an economic model (such as the one presented in this paper) and base the compensation on this value. The advantage of such a model is that it is likely as close to compensating actual welfare effects as we can plausibly get, and hence, has the greatest potential for risk sharing. Specifically, we define the \\textit{ideal} mechanism as the mechanism that directly shares the welfare benefits from the new transmission line among the participating countries in every scenario, according to some distribution rule represented by the coefficients $\\lambda_i \\geq 0$, $i \\in I$, with $\\sum_{i \\in I} \\lambda_i = 1$. That is, for every scenario $\\omega \\in \\Omega$, the compensation to country $i$ in the set $I$ of participating countries is given by\n\\begin{align*}\n    C^{\\text{ideal},\\omega}_i = \\lambda_i \\bigg(\\sum_{j \\in I} \\Delta \\text{TW}^\\omega_j\\bigg)  - \\Delta \\text{TW}^\\omega_i.\n\\end{align*}\nThe coefficients $\\lambda_i$, $i \\in I$, should be chosen in such a way that in expectation, the total welfare gains are distributed according to the predetermined distribution rule (e.g., the Shapley value or an equal-share principle).\n\nThere are a number of downsides to this theoretically ideal mechanism, though. The main issue is that the compensation amounts depend on the welfare benefits to the countries \\textit{as computed by the model}. Inevitably, however, the model will be imperfect and thus, the compensation levels are not ``correct''. Hence, conflicts may arise over the model to be used, which may undermine trust between the parties involved. Moreover, even if the model were perfect, it might be seen as a black box by non-experts; a simpler mechanism might be preferred. \n\nTo construct other novel, risk-sharing compensation mechanisms, we take the following approach. We search for economic measures that (1) depend on the scenario, (2) relate to the new transmission line, and (3) are correlated with the welfare effects in the hosting countries. If we find such measures, then we can base a compensation mechanism on them.\n\nBased on the first two conditions listed above, we propose two candidate measures: the amount of flow through the new line (referred to as \\textit{flow}), and the economic value of the flow through the new line (referred to as \\textit{flow value}). We define flow and flow value such that flow in the direction $\\text{NO2}\\to\\text{DE}$ is counted as positive and flow in the opposite direction as negative. Moreover, note that the flow value is ambiguous: in periods in which the line is congested (i.e., flow equals transmission capacity), there is a price differential between the two connected nodes. In such cases, we compute the flow value by using the \\textit{average} of the NO2 price and the DE price.\n\n\n\\begin{table}[]\n\\centering\n\\caption{Correlations between two flow-based measures and the welfare effect of the new transmission line in different countries.}\n\\label{tab:corr_country_measure}\n\\begin{tabular}{llllll}\n\\toprule\n\\textbf{}                 & \\textbf{NO} & \\textbf{AT} & \\textbf{FR} & \\textbf{DE} & \\textbf{DK} \\\\ \\midrule\n\\textbf{flow}             & 0.40        & -0.20       & -0.25       & 0.45        & 0.12        \\\\\n\\textbf{flow value} & 0.73        & 0.29        & 0.26        & -0.18       & -0.29      \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\n\nTo investigate the potential of these three candidate measures, we compute for each measure its correlations with the welfare effects of the new line in the hosting countries; see Table~\\ref{tab:corr_country_measure}. Ideally, these correlations are strong and of opposite sign. We observe that flow value is positively correlated with the welfare effect of the new line in all three benefiting countries (Norway, Austria, and France), while negatively correlated with the welfare effect in the suffering countries (Germany and Denmark). Zooming in on the two hosting countries, we observe that it is particularly strongly correlated with the welfare effect in Norway. Moreover, in Figure~\\ref{fig:scatter_flow_value_vs_NO_DE_welfare_delta} we observe that high values for the flow value correspond to scenarios in which Norway benefits most from the new line, while Germany tends to suffer. Based on these observations, flow value appears to be a good candidate measure to base a compensation mechanism on. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.4]{Figures\/scatter_flow_value_vs_NO_DE_welfare_delta.png}\n    \\caption{Scatter plot of the total value (using the hourly average of the NO2 and DE price) of the flow through the NO2-DE line and the welfare effect of the new line for Norway and Germany. Every dot\/cross represents a scenario.}\n    \\label{fig:scatter_flow_value_vs_NO_DE_welfare_delta}\n\\end{figure}\n\nFor flow, on the other hand, the signs of the correlations do not line up with the sign of the expected welfare effect in each country. Moreover, the strongest correlation has a magnitude of 0.45, which is less than then 0.73 we observe for flow value. Hence, flow seems to be a weaker candidate measure to base a compensation mechanism on. Nevertheless, for the sake of completeness we will keep it in our list of candidates and test its performance rigorously.\n\nBased on these two measures, we propose the following novel compensation mechanisms: the \\textit{flow-based} compensation mechanism, under which country $i$ receives a compensation amount of\n\\begin{align*}\n    C_i^{\\text{flow},\\omega} = \\alpha_i \\sum_{t \\in \\mathcal{T}} f_{\\text{NO2-DE}}^{\\omega,t},\n\\end{align*}\nfor some $\\alpha_i \\in \\mathbb{R}$, and the \\textit{value-based} compensation mechanism, under which country $i$ receives\n\\begin{align*}\n    C_i^{\\text{value},\\omega} = \\beta_i \\sum_{t \\in \\mathcal{T}} f_{\\text{NO2-DE}}^{\\omega,t} \\cdot \\bar{\\pi}^{\\omega,t},\n\\end{align*}\nfor some $\\beta_i \\in \\mathbb{R}$, where $\\pi_t := \\frac{1}{2} ( \\pi^{NO2}_t + \\pi^{DE}_t )$. The values of $\\alpha_i$ and $\\beta_i$, respectively, are chosen in such a way that on average, every country receives a compensation based on some predetermined rule (e.g., the Shapley value). In both cases, to achieve budget balancedness, the values of $\\alpha$ and $\\beta$, respectively, over all countries $I$ participating in the compensation scheme should add to zero, i.e., $\\sum_{i \\in I} \\alpha_i = 0$ and $\\sum_{i \\in I} \\beta_i = 0$.\n\n\n\\subsection{Performance}\n\\label{subsec:performance}\nWe now numerically investigate the performance of the various compensation mechanisms on the case study from Section~\\ref{sec:TEP}. We implemented the following compensation mechanisms: no compensation, lump-sum compensation, a Norway-based PPA, a Germany-based PPA, a flow-based mechanism, a value-based mechanism, and a theoretically ideal model-based mechanism. In order to have a fair comparison, we parametrized each compensation mechanism in such a way that it yields the same expected compensation amount. The amount is such that the expected net welfare effect of the new line after compensation is equal in both countries. Incidentally, this coincides with the Shapley value proposed in \\cite{kristiansen2018}. Since the expected compensation amount is equal among the mechanisms, a risk neutral country should be indifferent about them.\n\nHowever, the performance of the mechanisms may vary in terms of the resulting risk that the countries are exposed to. We assume all countries are risk-averse and hence, we prefer mechanisms that reduce the amount of risk faced by each country as a consequence of the transmission investment. Note that the term ``risk'' is ambiguous; it is sometimes interpreted as analogous to ``variability'', and sometimes as ``likelihood and\/or magnitude of losses'' \\cite{rockafellar2007coherent}. In our analysis we don't choose one interpretation over the other, but include measures corresponding to both interpretations of risk.\n\n\\subsubsection{Bilateral compensation}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.75]{Figures\/boxplot_comp_adjusted.png}\n    \\caption{Boxplot of the compensation amount paid by Norway to Germany under various compensation mechanisms. The green dashed lines indicate averages, while the orange solid lines indicate medians.}\n    \\label{fig:boxplot_comp}\n\\end{figure}\n\nFirst, we consider a setting with bilateral compensations between the two hosting countries Norway and Germany. We start by analyzing the compensation amount itself under various mechanisms. In Figure~\\ref{fig:boxplot_comp} we present a boxplot of the compensation paid by Norway to Germany. Note that indeed, the lump sum leads to a fixed compensation amount, while the other mechanisms vary per scenario. The PPAs show a significantly lager variability than the flow- and value-based mechanisms. Hence, the latter two are more predictable, which may be seen as an advantage.\n\nMore important, however, is how the different mechanisms translate into net welfare effects of the new line. In particular, we are interested in the corresponding risk faced by each country. We first focus on variability-based measures of risk. For this purpose, we refer to the boxplots in Figure~\\ref{fig:boxplot_DE_NO} and the first two columns of Table~\\ref{tab:compensation_statistics}. Note that the theoretical ideal mechanism is able to significantly reduce these measures of variability, though not completely. This is due to the fact that there is variability in the combined welfare effects of the new line, aggregated over both countries. Hence, we should not expect to be able to eliminate all variability using compensation mechanisms, although the ideal mechanism shows that significant reduction is theoretically possible.\n\nNext, we observe that a Germany-based PPA increases the risk faced by both countries, even that of Germany. Hence, the variability in the the compensation amount observed in Figure~\\ref{fig:boxplot_comp} does not cancel out the variability in the welfare effects before compensation. This may be caused by the fact that the correlations between the welfare effects of the new line in Norway and Germany and the price in Germany are not large enough. \n\nA Norway-based PPA, however, reduces the risk faced by Germany significantly, to a level close to the theoretical ideal mechanism, while increasing that faced by Norway. Here, at least for Germany, the correlation between the welfare effect in Germany and the NO2 price is apparently large enough for the variability in the compensation amount to cancel out variability in the welfare effect of the new cable in Germany. For Norway, however, the corresponding effect is not strong enough, which might be explained by the weaker correlation between the NO2 price and the welfare effect of the new line on Norway, observed in Table~\\ref{tab:price_correlations}. Hence, the net effect of the Norway-based PPA is that it shifts risk from Germany to Norway. This may or may not be desirable, depending on the risk preferences of the two countries. \n\nFinally, we observe that the value-based mechanism outperforms all other measures for both countries, except for the Norway-based PPA, which is the best option for Germany. Notably, the value-based mechanism succeeds in reducing the amount of risk faced by both Norway and Germany, compared to a lump sum. Hence, it succeeds in its purpose of risk mitigation. However, the improvement over the lump sum is only modest. Comparing the value-based mechanism to the theoretical ideal, we see that there is still a significant potential for improvement.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Various measures capturing the level of risk faced by Germany and Norway as a result of the new line, after compensation by different mechanisms. The measures are: standard deviation of the compensation amount, standard deviation of the net total welfare effect, probability of welfare loss, expected welfare loss, CVaR of welfare loss (with parameter 0.80). All numbers (except for the percentages) are in millions of euros annually.}\n\\label{tab:compensation_statistics}\n\\begin{tabular}{l|l|ll|ll|ll|ll}\n\\toprule\nmechanism & \\textbf{std(C)} & \\multicolumn{2}{c|}{\\textbf{std($\\Delta \\text{NTW}$)}} & \\multicolumn{2}{c|}{\\textbf{P(L)}} & \\multicolumn{2}{c|}{\\textbf{E{[}L{]}}} & \\multicolumn{2}{c}{$\\textbf{CVaR}_{\\textbf{80}}\\textbf{(L)}$} \\\\ \n    &       & DE    & NO    & DE   & NO   & DE    & NO   & DE    & NO    \\\\ \\midrule\nno comp.   & 0.0   & 124.8 & 188.8 & 80\\% & 43\\% & 109.1 & 40.5 & 212.1 & 140.4 \\\\\nlump sum   & 0.0   & 124.8 & 188.8 & 63\\% & 47\\% & 49.2  & 78.6 & 125.4 & 227.1 \\\\\nPPA DE     & 130.5 & 156.4 & 212.1 & 57\\% & 53\\% & 58.9  & 82.5 & 145.2 & 275.5 \\\\\nPPA NO     & 211.2 & 94.9  & 209.7 & 37\\% & 53\\% & 34.8  & 87.2 & 150.8 & 272.0 \\\\\nflow       & 22.4  & 136.4 & 181.1 & 57\\% & 43\\% & 52.4  & 76.0 & 144.8 & 230.8 \\\\\nflow value & 19.3  & 122.8 & 175.2 & 57\\% & 47\\% & 48.1  & 73.9 & 131.2 & 213.8 \\\\\nideal\t   & 130   & 93.6  & 93.6  & 47\\% & 47\\% & 38.0  & 38.0 & 124.8 & 124.8 \\\\\n\\bottomrule\n\\end{tabular}%\n\\end{table*}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[b]{\\columnwidth}\n        \\centering\n        \\includegraphics[scale=0.75]{Figures\/boxplot_DE_adjusted.png}\n        \\caption{Germany}\n        \\label{subfig:boxplot_DE}\n    \\end{subfigure}\\\\\n    \\begin{subfigure}[b]{\\columnwidth}\n        \\centering\n        \\includegraphics[scale=0.75]{Figures\/boxplot_NO_adjusted.png}\n        \\caption{Norway}\n        \\label{subfig:boxplot_NO}\n    \\end{subfigure}\n    \\caption{Boxplots of the net welfare effects of the new line for Germany and Norway under various compensation mechanisms. The green dashed lines indicate averages, while the orange solid lines indicate medians.}\n    \\label{fig:boxplot_DE_NO}\n\\end{figure}\n\n\n\n\nNext, in the remaining columns of Table~\\ref{tab:compensation_statistics} we consider loss-oriented measures. We compute the probability of a net welfare loss, the expected net welfare loss (only scenarios with losses contribute to this value; gains are regarded as zeros), and the $80\\%$ conditional value at risk (CVaR) \\cite{rockafellar2002conditional} of the net welfare loss, representing the expected value of the $20\\%$ worst cases. We observe that the theoretical ideal achieves in mitigating risk significantly, although not completely. The fact that it cannot completely eliminate the risk of loss was already argued in Section~\\ref{subsec:mechanisms} and illustrated by Figure~\\ref{fig:scatter_NO_DE_welfare_delta}. However, Table~\\ref{tab:compensation_statistics} shows that significant improvements are possible, at least in theory. \n\nTurning to the other mechansims, we observe that the Germany-based PPA performs bad on all fronts. For the Norway-based PPA we see a similar performance as before (good for Germany, bad for Norway), except according to the CVaR measure: in terms of CVaR it performs relatively bad for Germany as well. The interpretation is that although the expected losses to Germany are small, the very worst scenarios are worse than under other mechanisms. Next, the value-based mechanism outperforms all other mechanisms (except the theoretical ideal) according to most measures. It outperforms the flow-based mechanism on all measures except the probability of a loss in Norway. Moreover, it outperforms the lump sum on all but one measure: CVaR for Germany. Hence, again, it seems to succeed in its purpose of mitigating risk for most countries. However, comparing it with the theoretical ideal mechanism, there is again significant room for improvement.\n\n\n\n\n\nWe can draw the following conclusions. Firstly, the theoretical ideal mechanism shows that there is indeed a significant potential for mitigating risk by using scenario-dependent compensations. Secondly, a Germany-based PPA performs by far the worst of all measures considered. It increases the risk faced by the countries compared to a lump sum. However, a Norway-based PPA does yield good results for Germany. It basically transfers part of the risk from Germany to Norway. Such behavior might or might not be desirable, depending on the relative risk preferences of both countries. Next, the value-based mechanism consistently outperforms the flow-based mechanism. Hence, including the additional price information allows the mechanism to succeed better in risk sharing. \n\nFinally, the value-based mechanism outperforms most other mechanisms according to most measures. In particular, it outperforms the lump sum on all measures except CVaR for Germany. Hence, it seems to succeed in achieving what it was constructed to do: reducing the risk faced by both hosting countries. However, the value-based mechanism only improves upon a lump sum by a modest degree. Comparing this with the theoretical ideal mechanism, which performs much better, we conclude that there is still significant room for improvement. In particular, a mechanism based on some measure exhibiting stronger correlations with the welfare effects in Norway and Germany than flow value does in Table~\\ref{tab:corr_country_measure}, might achieve higher levels of risk mitigation.\n\n\n\n\\subsubsection{Multilateral compensation}\n\nIn practice, countries hosting a planned transmission expansion might want to involve other countries that are affected by the planned investment. One reason for this may be to avoid conflict that might hurt future cooperation. In our case study in particular, there are three other countries that are significantly affected by the proposed transmission expansion: Austria, France, and Denmark. Even though the welfare gain for Norway is sufficient to compensate Germany on its own, the majority of the welfare gains actually falls on other countries, most notably Austria and France. Moreover, besides Germany, Denmark is another high-production country that is hurt by the cheap Norwegian power entering the European market. Hence, involving all these five countries in a compensation scheme might be a desirable course of action.\n\nGiven this motivation, we now investigate the performance of different compensation mechanisms in a five-country coalition consisting of the countries mentioned above. We compare lump sum payments to the flow- and value-based mechanisms proposed in Section~\\ref{subsubsec:novel_mechanisms}. Note that the definition of PPAs is unclear in this multilateral setting. For instance, suppose we give Norway a PPA, then it is unclear how we should use this to determine the compensation paid from Austria to Denmark. For this reason, we disregard PPAs in this analysis. Finally, in practice, the share of the benefits allocated to each country is a topic for negotiations. In this case study, we assume an equal-share principle, which entails that every mechanism is parametrized such that the expected net total welfare effect of the new transmission line after compensation is equal in all five countries (25.6 million euros annually).\n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\caption{Various measures capturing the level of risk faced by each coalition country as a result of the new line, after compensation by different mechanisms. All numbers (except for the percentages) are in millions of euros annually.}\n\\label{tab:compensation_statistics_coalition}\n\\begin{tabular}{cllllllll}\n\\toprule\n\\multicolumn{1}{l}{\\textbf{}} &\n  \\textbf{mechanism} &\n  \\textbf{std(C)} &\n  \\textbf{std($\\Delta\\text{NTW}$)} &\n  \\textbf{P(L)} &\n  \\textbf{E{[}L{]}} &\n  \\textbf{$\\text{CVaR}_{\\text{0.8}}\\text{(L)}$} \\\\ \\midrule\n\\multirow{4}{*}{NO} & no comp.   & 0.0  & 188.8 & 43\\% & 40.5  & 140.4 \\\\\n                    & lump sum   & 0.0  & 188.8 & 43\\% & 67.7  & 203.1 \\\\\n                    & flow       & 16.2 & 183.0 & 43\\% & 66.2  & 205.8 \\\\\n                    & flow value & 14.0 & 178.9 & 43\\% & 64.5  & 193.5 \\\\ \n                    & ideal      & 159.8& 43.2  & 27\\% & 7.0   & 33.0  \\\\ \\midrule\n\\multirow{4}{*}{AT} & no comp.   & 0.0  & 79.7  & 0\\%  & 0.0   & 0.0   \\\\\n                    & lump sum   & 0.0  & 79.7  & 40\\% & 15.5  & 61.4  \\\\\n                    & flow       & 23.0 & 87.2  & 47\\% & 20.1  & 72.4  \\\\\n                    & flow value & 19.9 & 76.3  & 43\\% & 15.3  & 50.3  \\\\\n                    & ideal      & 63.5 & 43.2  & 27\\% & 7.0   & 33.0  \\\\ \\midrule\n\\multirow{4}{*}{FR} & no comp.   & 0.0  & 54.2  & 0\\%  & 0.0   & 0.0   \\\\\n                    & lump sum   & 0.0  & 54.2  & 30\\% & 8.0   & 38.2  \\\\\n                    & flow       & 11.5 & 58.1  & 37\\% & 9.7   & 42.2  \\\\\n                    & flow value & 9.9  & 52.5  & 37\\% & 7.3   & 31.5  \\\\ \n                    & ideal      & 47.8 & 43.2  & 27\\% & 7.0   & 33.0  \\\\ \\midrule\n\\multirow{4}{*}{DE} & no comp.   & 0.0  & 124.8 & 80\\% & 109.1 & 212.1 \\\\\n                    & lump sum   & 0.0  & 124.8 & 60\\% & 34.2  & 101.4 \\\\\n                    & flow       & 28.6 & 140.0 & 53\\% & 40.3  & 128.2 \\\\\n                    & flow value & 24.7 & 122.8 & 53\\% & 34.9  & 110.9 \\\\ \n                    & ideal      & 125.4& 43.2  & 27\\% & 7.0   & 33.0  \\\\ \\midrule\n\\multirow{4}{*}{DK} & no comp.   & 0.0  & 36.1  & 97\\% & 60.2  & 112.7 \\\\\n                    & lump sum   & 0.0  & 36.1  & 20\\% & 5.4   & 27.0  \\\\\n                    & flow       & 22.1 & 44.6  & 30\\% & 9.0   & 39.4  \\\\\n                    & flow value & 19.1 & 35.7  & 27\\% & 5.4   & 25.5  \\\\\n                    & ideal      & 70.1 & 43.2  & 27\\% & 7.0   & 33.0  \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table*}\n\nIn Table~\\ref{tab:compensation_statistics_coalition} we present performance measures for the various compensation mechanisms. First, we observe that under all mechanisms (except the theoretical ideal), Norway and Germany face by far the largest levels of risk. This can be understood by the fact that since they host the new cable, the cable affects these countries most directly. Next, we observe that the value-based mechanism outperforms the flow-based mechanism on almost all measures for all countries. This is in line with our results in the bilateral case and can again be explained by the more desirable correlations observed in Table~\\ref{tab:corr_country_measure} \n\nThe question remains which of the remaining two mechanisms, lump sum and the value-based mechanism, is preferable. In terms of a variability-based definition of risk, the value-based mechanism is clearly the winner. It has a smaller standard deviation of net total welfare effects than the lump sum for all countries. Hence, it indeed succeeds in risk mitigation. Next, in terms of a loss-based definition of risk, the results are again in favor of the flow-value based mechanism, though less strongly so. In terms of expected loss and CVaR of loss, all countries except for Germany are better off with the value-based mechanism. The reason for underperformance for Germany may be the weak correlation between flow value and the welfare effect of the new line in Germany; see Table~\\ref{tab:corr_country_measure}.\n\nOverall, we conclude that the value-based mechanism consistently outperforms the flow-based mechanism and also outperforms the lump sum in terms of most measures for most countries. All in all, the value-based mechanism seems the most promising in terms of mitigating risk of the countries involved. However, comparing it to the theoretical ideal mechanism, there is again significant room for improvement.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe investigated the potential of existing and novel welfare compensation mechanisms in TEP under uncertainty. The simplest existing mechanism, lump sum payments, does not take uncertainty into account at all. Other mechanisms, under which the compensation amount depends on the scenario, can potentially exploit negative correlations between the welfare effects of a new transmission line in benefiting and suffering countries in order to mitigate the risk of all parties involved. \n\nWe conducted numerical experiments in a case study of the Northern-European power sector. We find a system-optimal investment in a new transmission line between Norway and Germany that benefits Norway but hurts Germany in terms of expected total welfare. We considered both bilateral compensations (between Norway and Germany alone) and multilateral compensations (involving other affected countries, too). In both settings, we observed that a theoretically ideal, model-based mechanism can significantly reduce the levels of risk faced by the countries involved. This highlights the relevance of our research and the potential of mitigating risk by using scenario-dependent compensation mechanisms.\n\nWe analyzed one scenario-dependent mechanism from the literature: PPAs. We first show that in a stochastic setting, budget balancedness does not hold for PPAs. Hence, one should select a single country that receives the PPA. Moreover, in the numerical experiments we observed that a Germany-based PPA \\textit{increases} risk for both countries, while a Norway-based PPA shifts risk from Germany to Norway. We conclude that when considering a PPA, one should be careful in selecting the country at which the PPA is based.\n\nWe also tested two novel mechanisms, based on the flow through the new line and its economic value. In both the bilateral and multilateral setting, our novel value-based mechanisms performs best in terms of mitigating risk for the countries involved. It appears to do so by successfully exploiting negative correlations in the welfare effects of the new transmission line between benefiting and suffering countries. In particular, the value-based mechanism outperforms the lump sum payment. However, the improvement is only moderate. Comparing it with the theoretically ideal mechanism, we see that there is still significant potential for improvement. We expect that the level of outperformance may be higher if the negative correlations mentioned above are stronger or if the correlations between the economic value of flow through the new transmission line and the welfare effects of this line in the neighboring countries are stronger. \n \nWhile we deem the value-based mechanism most promising, its performance may well depend on the specific practical setting at hand. Therefore, in practical situations, we suggest to run an analysis like the one presented in this paper before choosing a particular compensation mechanism. Note that if the proposed transmission expansion plan is found by running a TEP model, then the proposed analysis can be performed by a simple extension of this model. One might use this paper as a blueprint for such an analysis. In any case, we suggest to include our value-based mechanism as one of the candidate compensation mechanisms.\n\nFuture research might focus on further investigating the performance of various compensation mechanisms in settings beyond the case study investigated in our paper. In particular, it would be interesting to test our hypothesis that the value-based mechanism performs better in situations with a strong correlation between the economic value of the flow through the new transmission line and its welfare effects in the neighboring countries. In addition, novel mechanisms may be developed based on measures exhibiting such strong correlations, or mixtures of several mechanisms may be tested. Moreover, it would be interesting to investigate the performance of different compensation mechanisms in a setting with investments in multiple transmission lines simultaneously, rather than the single-line setting used in this paper.\n\nAnother avenue for future research might be to investigate the incentive-distorting effects of welfare compensations and various mechanisms in particular, as sketched in Section~\\ref{subsec:incentive_effects}. It would be interesting to see how much welfare compensations may cause the system to deviate from the equilibrium and what the welfare effects of these deviations are. Such an analysis might be able to identify the types of government interventions (taxes\/subsidies) and compensation mechanisms that minimize this issue of incentive distortions.\n\n\n\\paragraph{Acknowledgements}\nWe want to thank THEMA Consulting for the data they shared with us.\n\n\n\\begin{appendices}\n\n\\section{Mathematical model}\n\\label{sec:mathematical_model}\n\nOur TEP model consists of two levels. In the lower level, the producers, consumers and the TSO act in the electricity market. We assume that each of these actors maximizes their own surplus and we assume perfect competition with all actors being price takers. The equilibrium problem arising from these interacting optimization problems can be formulated as a mixed-complementarity program (MCP) consisting of the Karush-Kuhn-Tucker (KKT) conditions of each actor's optimization problem, combined with a market clearing condition. In the upper level, a social planner is endowed with the task of deciding the transmission expansion investment levels. We assume that the social planner acts as a Stackelberg leader that tries to maximize total welfare of the entire system, while taking the optimal decisions of the followers into account. Together, the bi-level model constitutes an MPEC.\n\n\\subsection{Notation}\n\\noindent\\textbf{Sets:}\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$\\mathcal{N}$ & Set of nodes (indexed by $n$) \\\\\n$\\mathcal{L}$ & Set of lines (indexed by $l$) \\\\\n$\\mathcal{G}_n$ & Set of generators in node $n$ (indexed by $g$) \\\\\n$\\mathcal{R}_n$ & Set of renewables in node $n$ (indexed by $r$) \\\\\n$\\mathcal{S}$ & Set of seasons (indexed by $s$) \\\\\n$\\mathcal{T}$ & Set of time periods (indexed by $t$) \\\\\n$\\mathcal{T}_S$ & Set of time periods in season $s$ (indexed by $t$) \\\\\n$\\Omega$ & Set of scenarios (indexed by $\\omega$) \\\\\n\\end{tabularx}\n\\\\\n\n\\noindent\\textbf{Parameters:}\n\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$P_\\omega$ & Probability for scenario $\\omega$ \\\\\n$I^{R}_{\\omega rt}$ & Production profile for renewable $r$ in scenario $\\omega$ in time period $t$ \\\\\n$G^{R}_{r}$ & Installed capacity for renewable $r$ $[\\SI{}{\\mega\\watt}]$\\\\\n$C^{I,R}_{r}$ & Investment cost for renewable $r$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$C^{I,G}_{g}$ & Investment cost for generator $g$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$C^{G}_{gt}$ & Marginal cost for generator $g$ in time period $t$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n$G^{Max}_{g}$ & Installed generation capacity for generator $g$ [$\\SI{}{\\mega\\watt}$]\\\\\n$Q^{Max}_{\\omega gs}$ & Production limit for generator $g$ in scenario $\\omega$ in season $s$ [$\\SI{}{\\mega\\watt h}$]\\\\\n$A_{nl}$ & Node-line incidence matrix entry for node $n$ and line $l$\\\\\n$C^{I,L}_{l}$ & Investment cost for line $l$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$]\\\\\n\\end{tabularx}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$F^{Max}_{l}$ & Maximum line capacity for line $l$ [$\\SI{}{\\mega\\watt}$]\\\\\n$D^{A}_{\\omega nt}$ & Slope of inverse demand function for node $n$ in scenario $\\omega$ in time period $t$\\\\\n$D^{B}_{\\omega nt}$ & Constant of inverse demand function for node $n$ in scenario $\\omega$ in time period $t$\\\\\n\\end{tabularx}\n\\\\\n\n\\noindent\\textbf{Variables:}\n\n\\begin{tabularx}{0.95\\linewidth}{@{}>{\\bfseries}l@{\\hspace{.5em}}X@{}}\n$y^R_{r}$ & Capacity expansion for renewable $r$ [$\\SI{}{\\mega\\watt}$] \\\\\n$y_{g}$ & Capacity expansion for generator $g$ [$\\SI{}{\\mega\\watt}$] \\\\\n$q_{\\omega gt}$ & Production for generator $g$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$f_{\\omega lt}$ & Flow in line $l$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$x_{l}$ & Capacity expansion for line $l$ [$\\SI{}{\\mega\\watt}$] \\\\\n$d_{\\omega nt}$ & Demand in node $n$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\mega\\watt}$] \\\\\n$\\pi_{\\omega nt}$ & Price in node $n$ in scenario $\\omega$ in time period $t$ [$\\SI{}{\\text{\\euro}\\per\\mega\\watt}$] \n\\end{tabularx}\n\n\n\n\\subsection{Lower-level problem}\n\\label{subsec:lower-level_problem}\nIn this subsection we describe the lower-level equilibrium problem. This equilibrium problem can be represented as an MCP consisting of the KKT conditions of the optimization problems of all the actors in the market, combined with a market clearing constraint. We will not present the KKT conditions explicitly, but simply state each actor's optimization problem.\n\n\n\\subsubsection{Conventional energy producer problem}\n\\label{subsubsec:producer_problem}\nEvery conventional generating unit (power plant) is modeled as an independent generating company maximizing its operational profits minus investment costs. Each generating company can freely choose its generation levels to maximize their profits. Moreover, it has the possibility to invest in additional generating capacity if needed. Eq.~\\eqref{opt:GenCoProblem} describes the optimization problem for generator $g \\in \\mathcal{G}_n$ located in node $n \\in \\mathcal{N}$, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, s \\in \\mathcal{S}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{q_{\\omega gt}, y_{g}}\n{\\sum_{\\omega \\in \\Omega}\\sum_{t \\in \\mathcal{T}}P_\\omega\\left(\\pi_{\\omega nt}-C^G_{gt}\\right)q_{\\omega gt}-C^{I,G}_{g}y_g\\label{opt:GenCoObjective}}\n{\\label{opt:GenCoProblem}}\n{}\n\\addConstraint{q_{\\omega gt}}{\\leq G^{Max}_{g}+y_{g}\\label{opt:GenCoProdCap}}\n\\addConstraint{\\sum_{t\\in\\mathcal{T}_S}q_{\\omega gt}}{\\leq Q^{Max}_{\\omega gs}\\label{opt:GenCoEnergyLimit}}\n\\addConstraint{q_{\\omega gt}, y_{g}}{\\geq 0\\label{opt:GenCoLargerThanZero}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:GenCoObjective} consists of maximizing the expected revenue minus the investment costs in new generating capacity. Operating costs and investment costs are both assumed to be linear. Eq.~\\eqref{opt:GenCoProdCap} states that the production must be no more than the existing generating capacity plus the invested generating capacity. Eq.~\\eqref{opt:GenCoEnergyLimit} states that the total production in a season must be no more than the available quantity, which may vary per season. Finally, Eq.~\\eqref{opt:GenCoLargerThanZero} states that the generation and investment quantities must be non-negative. \n\n\n\n\\subsubsection{Renewable energy producer problem}\n\\label{subsubsec:RES_problem}\nThe renewable energy companies want maximize their profit earned from generating and selling renewable energy. In contrast with the conventional power producers, they are not able to freely choose their production levels; these are determined by the wind and solar profile. They can choose to invest in additional generating capacity, though. The optimization problem for renewable energy producer $r \\in \\mathcal{R}_n$ located in node $n \\in \\mathcal{N}$ is given by Eq.~\\eqref{opt:RESProblem}.\n\\begin{maxi!}[1]\n{y^R_{r}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{t\\in\\mathcal{T}}P_\\omega\\pi_{\\omega nt}\\left(G^R_{r}+y^R_{r} \\right) I^R_{\\omega rt}-C^{I,R}_r y^R_r\\label{opt:RESObjective}}\n{\\label{opt:RESProblem}}\n{}\n\\addConstraint{y_{r}^R}{\\geq 0\\label{opt:RESConNonneg}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:RESObjective} consists of maximizing the expected revenue minus the investments in new renewable capacity. Investment costs are assumed to be linear. Eq.~\\eqref{opt:RESConNonneg} states that capacity investment is non-negative.\n\n\\subsubsection{Consumer problem}\n\\label{subsubsec:consumer_problem}\nThe consumers aim to satisfy their demand for power at the lowest possible cost. In other words, they want to maximize the consumer surplus in each node $n$. Assuming a linear demand function, Eq.~\\eqref{opt:ConsumerProblem} represents the maximization problem for the consumers located in node $n$.\n\n\\begin{maxi!}[1]\n{d_{\\omega nt}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{t\\in\\mathcal{T}}P_\\omega\\left(\\frac{1}{2}D^A_{\\omega nt}d_{\\omega nt}+D^B_{\\omega nt}-\\pi_{\\omega nt}\\right)d_{\\omega nt}\\label{opt:ConsumerObjective}}\n{\\label{opt:ConsumerProblem}}\n{}\n\\end{maxi!}\nEq.~\\eqref{opt:ConsumerObjective} is the objective function for the consumers, representing the consumer surplus.\n\n\n\\subsubsection{TSO Problem}\n\\label{subsubsec:TSO_problem}\nWe assume a single TSO that operates all lines. The TSO maximizes the expected congestion rent for all lines, and sets the line flows accordingly. Recall that we assume that the TSO is a price taker. Hence, it will not use its market power and it will basically act as a dummy player (this is confirmed by our finding that in the equilibrium solution the TSO makes zero profit in expectation). The optimization problem for the TSO is given by Eqs.~\\eqref{opt:TSOProblem}, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, l \\in \\mathcal{L}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{f_{\\omega lt}, x_{l}}\n{-\\sum_{\\omega \\in \\Omega}\\sum_{i \\in \\mathcal{N}}\\sum_{l\\in \\mathcal{L}}\\sum_{t \\in \\mathcal{T}}P_{\\omega}A_{nl}f_{\\omega lt}\\pi_{\\omega nt}\\label{opt:TSOObjective}}\n{\\label{opt:TSOProblem}}\n{}\n\\addConstraint{f_{\\omega lt}}{\\leq F^{Max}_{l}+x_{l}\\label{opt:TSOflowPos}}\n\\addConstraint{f_{\\omega lt}}{\\geq -F^{Max}_{l}-x_{l}\\label{opt:TSOflowNeg}}\n\\end{maxi!}\nThe objective function in Eq.~\\eqref{opt:TSOObjective} consists of the expected congestion rent earned from all lines. Eqs.~\\eqref{opt:TSOflowPos} and \\eqref{opt:TSOflowNeg} state that the flow in a line must not exceed the current capacity plus the invested capacity. Note that the optimization model can be equivalently separated into optmization problems for each line. Hence, the assumption of a single TSO is without loss of generality.\n\n\n\\subsubsection{Market clearing}\nThe market clearing constraint is used to connect the market actors' decisions together. It guarantees that the market clears, i.e., that supply meets demand. It is given by\n\n\\begin{equation}\n d_{\\omega nt}+\\sum_{l\\in \\mathcal{L}}A_{nl}f_{\\omega lt}=\\sum_{g\\in \\mathcal{G}_n}q_{\\omega gt}+\\sum_{r\\in \\mathcal{R}_n}\\left(G^R_{r}+y^R_{r}\\right)I^R_{\\omega rt}  \\label{opt:MarketClearing}  \n\\end{equation}\nIn particular, Eq.~\\eqref{opt:MarketClearing} states that for a given node, the sum of demand and net outgoing flows must be equal to the total amount of power generated from both conventional and renewable sources. The market price is given by the dual variable $\\pi_{\\omega nt}$ corresponding to this constraint.\n\n\\subsection{Upper-level problem} \\label{subsec:upper-level_problem}\nThe upper-level problem consists of the social planner's transmission expansion problem. The social planner chooses the investment levels $x_l$, $l \\in \\mathcal{L}$, that maximize the expected net total welfare in the system. Here, we define net total welfare as gross total welfare (consisting of producer surplus, consumer surplus, and congestion rent) minus investment cost (which are assumed to be linear). The social planner acts as a Stackelberg leader that takes the other actors' optimal responses into account. \n\nLet $TW(x,q,y,y^R,d,f,\\pi)$ denote the expected gross total welfare corresponding to the decision vectors $x,q,y,y^R,d,f$ and price vector $\\pi$. That is, $TW$ is equal to the sum of the objective functions of the optimization problems of all producers, consumers, and the TSO. Let $KKT_1, \\ldots, KKT_4$ denote the sets of KKT conditions corresponding to the problems in Eqs.~\\eqref{opt:GenCoProblem}-\\eqref{opt:TSOProblem}, respectively. Then, the social planner's problem can be described as an MPEC of the form\n\n\\begin{maxi!}[1]\n{x_{l} \\geq 0}\n{TW(x,q,y,y^R,d,f,\\pi) - \\sum_{l\\in\\mathcal{L}}C^{I,L}_l x_l}\n{\\label{opt:GovProblem}}\n{\\label{opt:GovObjective}}\n\\addConstraint{KKT_1, \\ldots, KKT_4}{\\label{opt:GovKKT}}\n\\addConstraint{\\text{Market clearing condition Eq. } \\eqref{opt:MarketClearing}}{\\label{opt:GovMarketClearing}}\n\\end{maxi!}\nEq.~\\eqref{opt:GovObjective} represents the social planner's objective, consisting of maximizing the net total welfare. Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} are the equilibrium constraints arising from the lower-level MCP.\n\n\\subsubsection{Quadratic programming reformulation}\nThe social planner's optimization problem, given by the MPEC Eq.~\\eqref{opt:GovProblem}, can be reformulated as a single quadratic program. To show this, first define $TW^*(x)$ as the value of $TW(x,q,y,y^R,d,f,\\pi)$ at an optimal solution to the lower-level MCP defined by Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} (this value is unique, as we will prove shortly). Then, the social planner's problem consists of maximizing net total welfare:\n\n\\begin{maxi!}[1]\n{x_{l} \\geq 0}\n{TW^*(x) - \\sum_{l\\in\\mathcal{L}}C^{I,L}_l x_l}\n{\\label{opt:GovProblemReformulated}}\n{}\n\\end{maxi!}\n\n\n\nSimilar to the classical result by \\cite{samuelson1952}, it can be shown that the lower-level MCP is equivalent to a central planner quadratic optimization problem in which total welfare is maximized. The proof of this equivalence, which we omit for brevity, is through the observation that the KKT conditions to the quadratic program, which are necessary and sufficient, are equivalent to the mixed-complementarity Eqs.~\\eqref{opt:GovKKT}-\\eqref{opt:GovMarketClearing} defining the lower-level MCP. Observe that $TW^*(x)$ denotes the optimal value of the quadratic program, which proves that this value is indeed unique. \n\nSubstituting the lower-level quadratic programming formulation for $TW^*(x)$ into Eq.~\\eqref{opt:GovProblemReformulated}, and combining the upper and lower level decisions into a single optimization problem yields a quadratic programming reformulation for the social planner's MPEC. It is given by Eq.~\\eqref{opt:CentralPlanner}, where the constraints are defined $\\forall \\: \\omega \\in \\Omega, g \\in \\mathcal{G}_n, r \\in \\mathcal{R}_n, n \\in \\mathcal{N}, l \\in \\mathcal{L},  s \\in \\mathcal{S}, t \\in \\mathcal{T}$.\n\n\\begin{maxi!}[1]\n{\\substack{q_{\\omega gt}, y_{g} \\\\ y^R_{r}  d_{\\omega nt} \\\\ f_{\\omega lt}, x_{l}}}\n{\\sum_{\\omega\\in\\Omega}\\sum_{i\\in\\mathcal{N}}\\sum_{t\\in\\mathcal{T}}P_\\omega\\left(\\frac{1}{2}D^A_{\\omega nt}d_{\\omega nt}+D^B_{\\omega nt}\\right)d_{\\omega nt}}{\\label{opt:CentralPlanner}}{} \\nonumber\n\\breakObjective{-\\sum_{\\omega\\in\\Omega}\\sum_{g\\in\\mathcal{G}_n}\\sum_{t\\in\\mathcal{T}}P_\\omega C^G_{gt}q_{\\omega gt} -\\sum_{g\\in\\mathcal{G}_n}C^{I,G}_{g}y_g} \\nonumber\n\\breakObjective{-\\sum_{r\\in\\mathcal{R}_n}C^{I,R}_ry^R_r-\\sum_{l\\in\\mathcal{L}}C^{I,L}_lx_l \\label{opt:CentralPlannerObj3}} \n\\addConstraint{q_{\\omega gt}}{\\leq G^{Max}_{g}+y_{g} \\label{opt:CentralPlannerConFirst}}\n\\addConstraint{\\sum_{t\\in\\mathcal{T}_S}q_{\\omega gt}}{\\leq Q^{Max}_{\\omega gs}}\n\\addConstraint{d_{\\omega nt}+\\sum_{l\\in \\mathcal{L}}A_{nl}f_{\\omega lt}}{=\\sum_{g\\in \\mathcal{G}_n}q_{\\omega gt}+\\sum_{r\\in \\mathcal{R}_n}\\left(G^R_{r}+y^R_{r}\\right)I^R_{\\omega rt}}\n\\addConstraint{f_{\\omega lt}}{\\leq F^{Max}_{l}+x_{l}}\n\\addConstraint{f_{\\omega lt}}{\\geq -F^{Max}_{l}-x_{l}}\n\\addConstraint{q_{\\omega gt}, d_{\\omega nt}}{\\geq 0}\n\\addConstraint{y_{g}, y^R_{r}, x_{l}}{\\geq 0 \\label{opt:CentralPlannerConLast}}\n\\end{maxi!}\nHere, the objective function in Eq.~\\eqref{opt:CentralPlannerObj3} consists of the sum of the objective functions of all market participants' optimization problems less investment cost in new transmission lines. The constraints in Eqs.~\\eqref{opt:CentralPlannerConFirst}--\\eqref{opt:CentralPlannerConLast} are a concatenation of the constraints from all market actors' individual optimization problems and the market clearing constraint.\n\n\n\n\n\\section{Data}\n\\label{sec:data}\nThe model presented in Section~\\ref{sec:mathematical_model} is solved using a power system consisting of Norway, Germany and their neighbouring regions. In this section we describe the data used to parametrize the model.\n\nMarginal investment costs of generators and transmission lines are estimated using data on historical investments \n\\cite{Blumsac2022Basic,schroder2013current,NorNed2008longest,Statnett2013Cooperation,Powerengineeringinternational2012Germany,Northconnect2022FAQ} and annualized using economic lifetimes based on \\cite{schroder2013current} using an assumed interest rate of 4\\%. The estimates are given in  Table~\\ref{tab:InvestmentCosts in sec:Results}. Since our main concern is the interaction between Norway and Germany, all capacity expansions in transmission and generation in and between other countries than Norway and Germany are fixed at zero.\n\n\\begin{table}[t]\n\\centering\n\\caption{Marginal annualized investment costs [\\SI{}{\\text{\\EUR{}}\\per\\mega\\watt}] for generating technologies and transmission lines.}\n\\label{tab:InvestmentCosts in sec:Results}\n\\begin{tabular}{@{}lrllr@{}}\n\\toprule\n\\multicolumn{2}{c}{\\textbf{Generators}} & \\multicolumn{1}{c}{\\textbf{}} & \\multicolumn{2}{c}{\\textbf{Lines}} \\\\ \\midrule\nCoal                & 136,559           &                               & NO1-NO2          & 36,399          \\\\\nLignite             & 140,826           &                               & NO1-NO3          & 56,874          \\\\\nCCGT                & 70,413            &                               & NO1-NO5          & 52,324          \\\\\nOther gas           & 38,407            &                               & NO2-NO5          & 53,461          \\\\\nSolar               & 66,224            &                               & NO5-NO3          & 79,623          \\\\\nWind                & 78,735            &                               & NO3-NO4          & 130,809         \\\\\n                    &                   &                               & NO2-DE           & 70,864          \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nInitial generation and transmission capacities are based on data from the EMPIRE model in \\cite{backe2022empire}, the ENTSO-E Transparicy platform\n\\cite{EntsoE-Capacities} and from privately communicated data from THEMA Consulting Group. In line with policy goals of phasing out or reducing production from nuclear plants, some nations have their nuclear plant capacities reduced or completely removed. Germany and Belgium have their nuclear capcities completely removed, while France have a reduced capacity in order to mimic normal operating conditions as well as future capacity goals. Using plant efficiencies, together with privately communicated fuel- and $\\mathrm{CO_2}$-price time series from THEMA Consulting Group, we have approximated seasonal marginal operational costs for the thermal power plants. These costs are given in Table \\ref{tab:MarginalCostGen}.\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Marginal costs [\\SI{}{\\text{\\EUR{}}\\per\\mega\\watt}] for generating technologies.}\n\\label{tab:MarginalCostGen}\n\\begin{tabular}{@{}lrrrr@{}}\n\\toprule\n \\multicolumn{1}{c}{\\textbf{}}& \\multicolumn{1}{c}{\\textbf{Winter}}&\\multicolumn{1}{c}{\\textbf{Spring}}&\\multicolumn{1}{c}{\\textbf{Summer}}&\\multicolumn{1}{c}{\\textbf{Autumn}}\n  \\\\ \\midrule\nCoal    & 35,2 & 29,8 & 30,1 & 35,4 \\\\\nLignite & 38,9 & 32,9 & 33,3 & 39,4 \\\\\nCCGT    & 41   & 33,8 & 35,2 & 39,3 \\\\\nGas     & 63,1 & 52,1 & 54,2 & 60,4 \\\\\nNuclear & 15   & 15   & 15   & 15   \\\\\nHydro   & 0    & 0    & 0    & 0   \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nScenario-specific data are sampled from time series from NordPool \\cite{Nordpool2022Historical} and Open Power Systems Data \\cite{openpowersystemsdata2022timeseries}. The linear demand curves are constructed from demand- and price data. Assuming that inverse demand is written as $\\pi = ad + b$, the parameters are calculated as\n\\begin{align*}\n    a = \\frac{1}{\\varepsilon} \\frac{|P|}{D}, \\qquad b = \\left(1 - \\frac{1}{\\varepsilon}\\right) |P|,\n\\end{align*}\nwith $P$ and $D$ being the historical price and demand for a particular hour, respectively, and a price elasticity of demand of $\\varepsilon = -0.05$, which is in line with estimates in, e.g., \\cite{matar2018households}. We use the absolute value $|P|$ to account for historical hours with negative prices. This method ensures that demand is downward sloping with an elasticity close to \\SI{-0.05}{} in cases when model outcomes are close to historical outcomes. Production profiles for solar and wind plants scaled as a factor between 0 and 1 are found from \\cite{RenewablesNinja}.\n\nThe problem instance consists of a total of 30 scenarios. Each scenario is sampled randomly with historical data from the years 2013-2017. We sample hourly data from consecutive one-week periods, one for each of the four seasons. In this way, each scenario gets four 168-hour seasons resulting in a total of 672 hours in each scenario.\n\n\\end{appendices}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nA typical source for the construction of a Kasparov $K$-homology\ncycle is an elliptic differential complex.  If the elliptic complex\nis equivariant with respect to the action of a group $\\mathsf{G}$, and\nif moreover the group action satisfies an additional conformality\nproperty (see below), then one can obtain an element of equivariant $K$-homology. But if\n$\\mathsf{G}$ is a semisimple Lie group of rank greater than one, non-trivial examples of such complexes cannot\nexist (\\cite{Puschnigg}). This paper describes a means of constructing an equivariant\n$K$-homology class from the\nBernstein-Gelfand-Gelfand complex for\n$\\SL(3,\\mathbb{C})$---a differential complex which is neither elliptic nor\nconformal, but which satisfies some weaker (`directional') form of\nthese conditions.\n\nThe motivation for this construction comes from the Baum-Connes\nconjecture. Although an understanding of the conjecture is not\nessential to this paper, it is useful for perspective.  The conjecture asserts that for a second countable locally compact group $\\mathsf{G}$, the \n {\\em assembly map}\n$$\n  \\mu_{\\Gamma} : K^\\Gamma(\\underline{E}\\Gamma) \\to K(C^*_r\\Gamma).\n$$\nis an isomorphism, thus giving a `topological computation' of the $K$-theory of the reduced group $C^*$-algebra.  For a fuller description of the conjecture and its many consequences, we refer the reader to the expository article \\cite{HigsonICM} and the foundational paper \\cite{BCH}.\n\nThe conjecture has been proven for a wide class of groups, amongst which we mention in particular the discrete subgroups of simple Lie groups of real rank one. A notable unknown, however, is the group $\\SL(3,\\mathbb{Z})$.  More broadly, the conjecture is unknown for general discrete subgroups of semisimple Lie groups of rank greater than one.\n\nFor subgroups of rank one semisimple groups $\\mathsf{G}$, the proofs in each case centre on a canonical idempotent $\\gamma$ in the representation ring $R(\\mathsf{G}):=KK^\\mathsf{G}(\\mathbb{C},\\mathbb{C})$. (See \\cite{Kas-Lorentz} for $\\mathsf{G}=\\SO_0(n,1)$, \\cite{JK} for $\\mathsf{G}=\\SU(n,1)$, \\cite{Julg}\\footnote{The first proof of Baum-Connes for discrete subgroups of $\\Sp(n,1)$ was due to V.~Lafforgue, but used a somewhat different approach.} for $\\Sp(n,1)$).  For our purposes, the most convenient way to describe this idempotent $\\gamma$ is via the following fact.\n\n\\begin{theorem}[Kasparov]\n\\label{thm:split_surjection}\nLet $\\mathsf{G}$ be a semisimple Lie group and $\\mathsf{K}$ a maximal compact subgroup.  The restriction map $R(\\mathsf{G}) \\to R(\\mathsf{K})$ is a split surjection of rings.\n\\end{theorem}\n\nThe unit in $R(\\mathsf{K})$ is the class of the trivial $\\mathsf{K}$-representation, and its image under the splitting is an idempotent in $R(\\mathsf{G})$.  This is $\\gamma$.  \n\nIf $\\gamma=1\\in R(\\mathsf{G})$ then the restriction map is an isomorphism.  In this case, the `Dirac-dual Dirac method' of Kasparov implies that the Baum-Connes conjecture holds for all discrete subgroups of $\\mathsf{G}$.  This is the approach taken in the papers cited above, although in the case of $\\Sp(n,1)$ a weaker notion of `triviality' for $\\gamma$ must be used.\n\nThe idempotent $\\gamma$ was originally defined via equivariant $K$-homology for the proper $\\mathsf{G}$-space $\\mathsf{G\/K}$ (\\cite{Kas88}).  In the rank-one proofs mentioned above, however, $\\gamma$ is more conveniently constructed using the compact space $\\mathsf{G\/B}$, where $\\mathsf{B}$ is a minimal parabolic subgroup.  This can be explained by the fact that the induced representations from $\\mathsf{B}$ give a natural topological parameterization of (the relevant subset of) representations of $\\mathsf{G}$, namely the generalized principal series, including the complementary series.  It is also pertinent that $\\mathsf{B}$ is amenable, so itself satisfies Baum-Connes.\n\nIt is instructive to consider the construction of $\\gamma$ in the simple example $\\mathsf{G}=\\SL(2,\\mathbb{C})$.  One begins with the Dolbeault complex for the homogeneous space $\\mathsf{G}\/\\mathsf{B} \\cong \\CC \\mathrm{P}^1$:\n$$\n  \\Omega^{0,0}\\CC \\mathrm{P}^1 \\xrightarrow{\\overline{\\partial}} \\Omega^{0,1}CP^1.\n$$   \nThis is a $\\mathsf{G}$-equivariant elliptic complex.  Importantly, though, $\\CC \\mathrm{P}^1$ does not admit a $\\mathsf{G}$-invariant Riemannian metric.  The action is conformal (with respect to the natural $\\mathsf{K}$-equivariant metric), and the translation representation of $\\mathsf{G}$ on $L^2\\Omega^{0,\\bullet}\\CC \\mathrm{P}^1$ can be made unitary by the introduction of a scalar Radon-Nikodym factor.  But the operator $D := \\overline{\\partial} + \\overline{\\partial}^*$ will not be $\\mathsf{G}$-equivariant, not even in the weak sense of defining an unbounded equivariant Fredholm module.  Somewhat magically though, replacing $D$ by its operator phase results in a bounded equivariant Fredholm module.  For this to work it is crucial that the $\\mathsf{G}$-action is conformal\\footnote{In general, the conformality requirement is even stronger:  the ratio of the Radon-Nikodym factors in degrees $p$ and $p+1$ must be independent of $p$.  We will not explain this further.} on the Hermitian bundles $\\Omega^{0,p}\\CC \\mathrm{P}^1$.\n\nIn order to maintain this crucial conformality property for the other rank one cases, one must use increasingly complicated subellitpic differential complexes --- the Rumin complex for $\\SU(n,1)$; a quaternionic analogue thereof for $\\Sp(n,1)$ --- and corresponding nonstandard pseudodifferential calculi.  We remark that $K$-homological constructions using even nonstandard pseudodifferential calculi typically result in finitely-summable Fredholm modules.  Puschnigg \\cite{Puschnigg} has shown that simple Lie groups of higher rank do not admit any nontrivial finitely summable Fredholm modules.\n\nThis motivates our construction using the Bernstein-Gelfand-Gelfand (`BGG') complex.\n\n\\begin{theorem}[Bernstein-Gelfand-Gelfand]\nLet $\\mathsf{G}$ be a complex semisimple group and $\\mathsf{B}$ a minimal parabolic subgroup.  For any finite dimensional holomorphic representation $V$ of $\\mathsf{G}$ there is a differential complex, consisting of direct sums of homogeneous line bundles over $\\mathsf{G\/B}$ and $\\mathsf{G}$-equivariant differential operators between them, which resolves $V$.\n\\end{theorem}\n\nThe bundles in each degree here are not conformal, but their component line bundles are individually conformal.   (Trivially, any group action on a Hermitian line bundle is conformal.)  The question is whether this structure is enough to produce an element of equivariant $K$-homology.  In this paper, we answer this question affirmatively in the case of $\\mathsf{G}=\\SL(3,\\mathbb{C})$.  We thereby obtain an explicit construction of the splitting map $ R(\\mathsf{K}) \\to R(\\mathsf{G})$, and in particular a construction of $\\gamma$, which factors through $KK^\\mathsf{G}( C(\\mathsf{G\/B}), \\mathbb{C})$.\n\nThe construction is based upon harmonic analysis of $\\SU(3)$ rather than some nonstandard pseudodifferential calculus.  An indication of the difficulties of a purely pseudodifferential approach is given in Chapter 5 of \\cite{Yuncken-thesis}.  In fact, our construction could be made without any reference to pseudodifferential operators at all, though pseudodifferential theory has become so central to index theory that to do so might seem somewhat eccentric.\n\nMuch of the required harmonic analysis has been developed in \\cite{Yuncken:PsiDOs} in the broader context of $\\SU(n)$ ($n\\geq2$).  We expect that the results of this paper should be extendable the groups $\\SL(n,\\mathbb{C})$, and indeed to complex semisimple groups in general.  The main technical difficulty in the case of $\\SL(n,\\mathbb{C})$ is an appropriate version of the the operator partition of unity of Lemma \\ref{lem:operator_po1} of this paper.  For general semisimple groups, the required directional harmonic analysis is yet to be developed.\n\n\n\nAs for the Baum-Connes Conjecture itself, it is known that $\\gamma\\neq1$ for any group $\\mathsf{G}$ which has Kazhdan's property $T$. Therefore,\na direct translation of Kasparov's method cannot prove the\nBaum-Connes conjecture for simple Lie groups of rank greater than\none---some subtle variation of Kasparov's argument would be\nrequired. Nevertheless, it is expected that the present construction\nwill be useful for further study of the Baum-Connes conjecture.\n\n\n\\medskip\n\nLet us now describe the BGG complex in more detail.  In fact, knowledge of the cohomological version of the BGG complex is unnecessary for the present paper, since our $K$-homological version will be produced from scratch. \nBut it is such a strong motivation that it is worth spending some time explaining it.\n\nFinite dimensional holomorphic representations of $\\mathsf{G}$ are parameterized by their highest weights.  Let $V^\\lambda$ denote the representation with highest weight $\\lambda$.  Any weight $\\mu$ of $\\mathsf{G}$ extends to a holomorphic character of $\\mathsf{B}$ (see Section \\ref{sec:homogeneous_vector_bundles}), and we denote by $\\Lhol{\\mu}$ the corresponding induced holomorphic line bundle over $\\scrX:=\\mathsf{G\/B}$.  The Borel-Weil Theorem states that $V^\\lambda$ is equivariantly isomorphic to the space of global holomorphic sections of $\\Lhol{\\lambda}$.  \n\nRecall that the Weyl group $\\Lie{W}$ is a group of reflections on the weight space.  It is generated by the {\\em simple reflections}---reflections in the walls orthogonal to a choice of simple roots for $\\mathsf{G}$.  Word length in these generators defines a length function $l:\\Lie{W}\\to\\mathbb{N}$.  We need the {\\em shifted action} of the Weyl group defined by the formula $w\\star \\mu := w(\\mu+\\rho) - \\rho$, where $\\rho$ is the half-sum of the positive roots.  Bernstein, Gelfand and Gelfand \\cite{BGG} showed that there is a holomorphic $\\mathsf{G}$-equivariant differential operator from $\\Lhol{\\mu}$ to $\\Lhol{\\nu}$ if and only if $\\mu=w\\star\\lambda$ and $\\nu = w'\\star\\lambda$ for some dominant weight $\\lambda$ and some $w,w'\\in\\Lie{W}$ with $l(w')\\geql(w)$.  What is more, these operators can be assembled into an exact complex as follows\\footnote{Strictly speaking, Bernstein, Gelfand and Gelfand made a homological complex by assembling intertwiners between Verma modules.  What we are calling the BGG complex here is a dual cohomological complex.  See the appendix of \\cite{CSS} for an explanation of this. }.  One defines the degree $p$ cocycle space $C^p := \\bigoplus_{l(w) = p} C^\\infty(\\scrX, \\Lhol{w\\star \\lambda})$.   The collection of equivariant differential operators between any $\\Lhol{w\\star\\lambda}$ and $\\Lhol{w'\\star\\lambda}$ with $l(w)=p$, $l(w')=p+1$ defines a matrix of operators $C^p \\to C^{p+1}$.  With an appropriate choice of signs these operators resolve the Borel-Weil inclusion $V^\\lambda \\hookrightarrow C^\\infty(\\scrX;\\Lhol{\\lambda})$.\n\n\n\n\nIn the case of $\\SL(3,\\mathbb{C})$, we get a complex\n\\begin{equation}\n\\label{eq:BGG_resolution}\n \\xymatrix@!C=7.2ex{\n & C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_1}\\star\\lambda}) \\ar[rr]\\ar[ddrr]\n   \\ar@{.}[dd]|-{\\bigoplus}\n  && C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\star\\lambda}) \\ar[dr]\n   \\ar@{.}[dd]|-{\\bigoplus}\n \\\\\n V^\\lambda \\hookrightarrow C^\\infty(\\scrX;\\Lhol{\\lambda})\\ar[ur]\\ar[dr]\\quad\\quad\n &&&&\\quad\\quad C^\\infty(\\scrX;\\Lhol{w_\\rho\\star\\lambda}) \\\\\n & C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_2}\\star\\lambda})\\ar[rr]\\ar[uurr]\n  && C^\\infty(\\scrX;\\Lhol{\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\star\\lambda})\\ar[ur] \\\n  }\n\\end{equation}\nwhere $\\alpha_1$, $\\alpha_2$ and $\\rho = \\alpha_1+\\alpha_2$ are the positive roots, and $\\reflection{\\alpha}$ denotes the reflection in the wall orthogonal to $\\alpha$.\n\nIn this paper, we define a `normalized', {\\em i.e.}, $L^2$-bounded, version of this complex which is analogous to the equivariant Fredholm module constructed above from the Dolbeault complex of $\\CC \\mathrm{P}^1$.\n\n\\medskip\n\nTo complete this overview, we give a very brief description of the harmonic analysis upon which our $K$-homological BGG construction is based.  The space $\\scrX:= \\mathsf{G\/B}$ is the complete flag variety of $\\mathbb{C}^3$.   Corresponding to the simple roots $\\alpha_1$ and $\\alpha_2$, there are $\\mathsf{G}$-equivariant fibrations $\\scrX\\to\\scrX[i]$ $(i=1,2)$ where $\\scrX[1]$ and $\\scrX[2]$ are the Grassmannians of lines and planes in $\\mathbb{C}^3$.  As described in \\cite{Yuncken:PsiDOs}, associated to each of these fibrations is a $C^*$-algebra $\\scrK[\\alpha_i]$ of operators on the $L^2$-section space of any homogeneous line bundle over $\\scrX$.  This algebra contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the given fibration.  A key property is that the intersection $\\scrK[\\alpha_1]\\cap \\scrK[\\alpha_2]$ consists of compact operators.  Ultimately, this allows us to apply the Kasparov Technical Theorem to construct a Fredholm module from the normalized BGG operators.\n\n\n\n\n\n\n\n\\medskip\n\nThe structure of the paper is as follows.  Section \\ref{sec:preliminaries} gives the background on the structure theory of the semisimple Lie group $\\mathsf{G}=\\SL(3,\\mathbb{C})$, the flag variety $\\scrX$ and its homogeneous line bundles, mainly for the purpose of setting notation.  \n\nIn Section \\ref{sec:harmonic_analysis} we review the $C^*$-algebras $\\scrK[\\alpha_i]$ of \\cite{Yuncken:PsiDOs} and their relation to longitudinal pseudodifferential operators on the flag variety $\\scrX$.  We also prove two important new results concerning these algebras.  For the sake of stating these results elegantly, it is convenient to place the $C^*$-algebras $\\scrK[\\alpha_i]$ in the context of $C^*$-categories (see Section \\ref{sec:categories} for details).\n\n\n\\begin{theorem}\n\\label{thm:harmonic_analysis_results}\nLet $E$, $E'$ be $\\mathsf{G}$-homogeneous line bundles over $\\scrX$.  Let $\\scrA$ denote the simultaneous multiplier category of $\\scrK[\\alpha_1]$ and $\\scrK[\\alpha_2]$ (see Definition \\ref{def:A}).\n\\begin{enumerate}\n\\item  The translation operators $g: \\LXE \\to \\LXE$ belong to $\\scrA$, for all $g\\in\\mathsf{G}$.\n\\item  If $T:\\LXE \\to L^2(\\scrX;E')$ is a longitudinal pseudodifferential operator of order zero tangent to one of the fibrations $\\scrX\\to\\scrX[i]$ ($i=1,2$), then $T\\in\\scrA$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{thm:harmonic_analysis_results}(i) is proven in Section \\ref{sec:principal_series}.  Part (ii) is restated in Theorem \\ref{thm:PsiDOs_in_A}.  The proof requires some lengthy computations in $\\SU(3)$ harmonic analysis which are presented in Appendix \\ref{sec:PsiDOs_in_A}.  \n\nIn Section \\ref{sec:construction}, we combine the above results to construct an element of  $KK^\\mathsf{G}(C(\\scrX),\\mathbb{C})$ from the BGG complex.  We also explain why this yields the splitting of the restriction morphism $R(\\mathsf{G}) \\to R(\\mathsf{K})$.\n\n\\medskip\n\nPart of this work appeared in the author's doctoral dissertation\n\\cite{Yuncken-thesis}. I would like to thank my thesis adviser, Nigel Higson.  I would also like to thank Erik Koelink for\nseveral informative conversations.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Notation and Preliminaries}\n\\label{sec:preliminaries}\n\n\\subsection{Lie groups}\n\\label{sec:Lie_groups}\n\nThroughout this paper $\\mathsf{G}$ will denote the group $\\SL(3,\\mathbb{C})$.  We fix notation for the following subgroups:   $\\mathsf{K}=\\SU(3)$, its maximal compact subgroup; $\\mathsf{H}$, the Cartan subgroup of diagonal matrices; $\\mathsf{A}$, the subgroup of diagonal matrices with positive real entries; $\\mathsf{M}  = \\mathsf{H}\\cap\\mathsf{K}$, the maximal torus of $\\mathsf{K}$; $\\mathsf{N}$, the subgroup of upper triangular unipotent matrices; and $\\mathsf{B}=\\mathsf{MAN}$ the subgroup of upper triangular matrices.  Their Lie algebras are denoted  $\\mathfrak{g}$, $\\mathfrak{k}$, $\\mathfrak{h}$, $\\mathfrak{a}$, $\\mathfrak{m}$, $\\mathfrak{n}$ and $\\mathfrak{b}$.   \n\nWe use $V^\\dagger$ to denote the dual of a complex vector space $V$.   \nWe make the usual identifications of the complexifications $\\mathfrak{m}_\\mathbb{C}$ and $\\mathfrak{a}_\\mathbb{C}$ with $\\mathfrak{h}$ by extending the inclusions $\\mathfrak{a},\\mathfrak{m}\\hookrightarrow\\mathfrak{h}$ to $\\mathbb{C}$-linear maps.  We thereby identify characters of $\\mathsf{A}$ and $\\mathsf{M}$ with elements of $\\mathfrak{h}^\\dagger$.  Characters of $\\mathfrak{h}$ will be denoted by $\\chi = \\chi_{\\Lie{M}} \\oplus \\chi_{\\Lie{A}}$, where $\\chi_{\\Lie{M}}$ and $\\chi_{\\Lie{A}}$ are the restrictions of $\\chi$ to $\\mathfrak{m}$ and $\\mathfrak{a}$, respectively.  The corresponding group character of $\\mathsf{H}$ will be denoted $e^\\chi$.  The weight lattice in $\\mathfrak{m}_\\mathbb{C}^\\dagger \\cong \\mathfrak{h}^\\dagger$ will be denoted by $\\Lambda_W$.\n\nThe set of roots of $\\mathsf{K}$ is denoted $\\Delta$.  We fix the notation\n$$\nX_{\\alpha_1} = \\left(\\begin{array}{ccc} 0&1&0\\\\0&0&0\\\\0&0&0 \\end{array}\\right),~\nX_{\\alpha_2} = \\left(\\begin{array}{ccc} 0&0&0\\\\0&0&1\\\\0&0&0 \\end{array}\\right),~\nX_\\rho = \\left(\\begin{array}{ccc} 0&0&1\\\\0&0&0\\\\0&0&0 \\end{array}\\right)~\n\\in \\mathfrak{k}_\\mathbb{C} \\cong \\mathfrak{g},\n$$\nwhich are root vectors for the roots $\\alpha_1$, $\\alpha_2$ and $\\rho:=\\alpha_1+\\alpha_2$.  We fix these as our set of positive roots $\\Delta^+$, so $\\Sigma := \\{\\alpha_1, \\alpha_2\\}$ is the set of simple roots.  For each $\\alpha\\in\\Delta^+$, $Y_\\alpha$ will denote the transpose of $X_\\alpha$.  We abbreviate $X_{\\alpha_i}$ and $Y_{\\alpha_i}$ to $X_i$ and $Y_i$, whenever convenient.\n\nWe put $H_i := [X_i, Y_i] \\in \\mathfrak{m}_\\mathbb{C}$.  The elements $X_i, Y_i, H_i$ span a Lie subalgebra isomorphic to $\\mathfrak{sl}(2,\\mathbb{C})$, which we denote by $\\mathfrak{s}_i$.  We also put\n$$\n  H_1' := \\left(\\begin{array}{ccc} 1&0&0\\\\0&1&0\\\\0&0&-2 \\end{array}\\right), \\qquad\n  H_2' := \\left(\\begin{array}{ccc} -2&0&0\\\\0&1&0\\\\0&0&1 \\end{array}\\right) \\qquad \n  \\in \\mathfrak{m}_\\mathbb{C} \\cong \\mathfrak{h},\n$$\nso that for fixed $i=1,2$, $H_i$ and $H_i'$ span $\\mathfrak{h}$ and $H_i'$ commutes with $\\mathfrak{s}_i$.\n\n\nThe Weyl group of $\\mathsf{G}$ is $\\mathsf{W}\\cong S_3$.  We let $\\reflection{\\alpha}$ denote the reflection in the wall orthogonal to the root $\\alpha$.  The {simple reflections} $w_{\\alpha_1}$ and $w_{\\alpha_2}$ are generators of $W$, and the minimal word length in these generators defines the length function $l$ on $W$.   \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Homogeneous vector bundles}\n\\label{sec:homogeneous_vector_bundles}\n\nThroughout, $\\scrX$ will denote the homogeneous space $\\scrX=\\mathsf{G}\/\\mathsf{B} = \\mathsf{K}\/\\mathsf{M}$.\n\nLet $\\chi = \\chi_{\\Lie{M}}\\oplus\\chi_{\\Lie{A}}$ be a character of $\\mathfrak{h}$.  As usual, we extend it trivially on $\\mathfrak{n}$ to a character of $\\mathfrak{b}$.  We use $L_\\chi$ to denote the $\\mathsf{G}$-homogeneous line bundle over $\\scrX$ which is induced from $\\chi$.  That is, continuous sections of $L_\\chi$ are identified with $\\mathsf{B}$-equivariant functions on $\\mathsf{G}$ as follows:\n\\begin{multline}\n  \\CXL[\\chi] = \\{ s:\\mathsf{G}\\to\\mathbb{C} \\text{ continuous }\\; | \\; s (gman) = e^{\\chi_{\\Lie{M}}}(m^{-1})e^{\\chi_{\\Lie{A}}}(a^{-1}) s (g) \n     \\\\\n   \\forall g\\in\\mathsf{G}, m\\in\\mathsf{M}, a\\in\\mathsf{A}, n\\in\\mathsf{N} \\}.\n\\label{eq:B-equivariance}\n\\end{multline}\nThe $\\mathsf{G}$-action on sections is by left translation:  $g'\\cdots(g) := s({g'}^{-1}g)$.\nRestricting to $\\mathsf{K}$, we have the `compact picture' of $\\CXL[\\chi]$:\n\\begin{multline}\n  \\CXL[\\chi] \\cong \\{ s:\\mathsf{K} \\to \\mathbb{C} \\text{ continuous } \\; | \\; s(km) = e^{\\chi_{\\Lie{M}}}(m^{-1})s(k) \n     \\\\\n   \\forall k\\in\\mathsf{K}, m\\in\\mathsf{M} \\}.\n\\label{eq:M-equivariance}\n\\end{multline}\nNote that, as a $\\mathsf{K}$-homogeneous bundle, $L_\\chi$ depends only on $\\chi_{\\Lie{M}}$.\n\nThe compact picture gives a Hermitian metric on $L_\\chi$.  Specifically, the pointwise inner product of sections is given by\n$$\n  \\ip{s_1(k),s_2(k)} = \\overline{s_1(k)} s_2(k) \\quad \\in C(\\scrX).\n$$\nThe $L^2$-section space $\\LXL[\\chi]$ is the completion of $\\CXL[\\chi]$ with respect to the inner product\n\\begin{equation}\n\\label{eq:inner_product}\n  \\ip{s_1,s_2} = \\int_\\mathsf{K} \\overline{s_1(k)} s_2(k) \\, dk.\n\\end{equation}\n\n\n\nSome cases warrant special notation.  If $\\mu$ is a weight for $\\mathsf{K}$, we let $\\Lhol{\\mu}$ denote the holomorphic line bundle $L_{\\mu\\oplus\\mu}$.  We also let $E_\\mu$ denote the `unitarily induced' bundle $L_{\\mu\\oplus\\rho}$. On $E_\\mu$ the translation action $U_\\mu:\\mathsf{G} \\to \\scr{L}(\\LXE[\\mu])$ is a {unitary} representation.  These will be the main focus of our attention.\n\nRestricting $U_\\mu$ to $\\mathsf{K}$, $\\LXE[\\mu]$ becomes a subrepresentation of the left regular representation $\\mathsf{K}$.  If $R$ denotes the \\emph{right} regular representation, then the equivariance condition of Equation \\eref{eq:M-equivariance} becomes $R(m) s = e^{-\\mu}(m) s$ for all $m\\in\\mathsf{M}$.  Infinitesimally,\n\\begin{eqnarray}\n  \\LXE[\\mu] &=& \\{s\\in L^2(\\mathsf{K}) \\; | \\; R(M) s = -\\mu(M) s \\text{ for all } M\\in\\mathfrak{m} \\} \\nonumber \\\\\n    &=& p_{-\\mu} L^2(\\mathsf{K}),\n\\label{eq:m-equivariance}\n\\end{eqnarray}\nwhere $p_{-\\mu}$ denotes the orthogonal projection onto the $(-\\mu)$-weight space of the {\\em right} regular representation of $\\mathsf{K}$ on $L^2(\\mathsf{K})$.\n\nLet $\\chi$, $\\chi'$ be characters of $\\mathsf{B}$.  If $f\\in\\CXL[\\chi'-\\chi]$ then pointwise multiplication by $f$, denoted $\\multop{f}$, maps $\\CXL[\\chi]$ to $\\CXL[\\chi']$.  This gives a $\\mathsf{G}$-equivariant bundle isomorphism $\\End(L_\\chi,L_{\\chi'}) \\cong L_{\\chi'-\\chi}$.  In particular, $\\End(E_\\mu, E_{\\mu'}) \\cong L_{(\\mu'-\\mu)\\oplus0}$ for any weights $\\mu$, $\\mu'$.  Moreover, for any $f\\in\\CXL[(\\mu'-\\mu)\\oplus0]$,\n\\begin{equation}\n\\label{eq:covariance_of_multops}\n  U_{\\mu'}(g) \\multop{f} U_\\mu(g^{-1}) = \\multop{g\\cdot f}.\n\\end{equation}\nIn this picture, a locally trivializing partition of unity on $E_\\mu$ takes the following form.\n\n\\begin{lemma}\n\\label{lem:partition_of_unity}\n\nFor any weight $\\mu$, there exists a finite collection of continuous sections $\\varphi_1,\\ldots,\\varphi_n \\in \\CXL[\\mu\\oplus0]$ such that $\\sum_{j=1}^n \\multop{\\varphi_j} \\multop{\\overline{\\varphi_j}} = 1$.\n\n\\end{lemma}\n\n\\begin{proof}\nLet $f_1,\\ldots,f_n\\in \\CX$ be a partition of unity subordinate to a locally trivializing cover of $E_\\mu$.   Composing $f_j^{\\frac{1}{2}}$ with the corresponding local trivialization $L_0 \\xrightarrow{\\cong} L_{\\mu\\oplus0}$ gives the sections $\\varphi_j$.\n\\end{proof}\n\n\n\n\n\n\\subsection{Parabolic subgroups and equivariant fibrations}\n\\label{sec:parabolic_subgroups}\n\nLet $\\mathsf{P}$ be a parabolic subgroup, $\\mathsf{B}\\leq \\mathsf{P} \\leq \\mathsf{G}$, with Lie algebra $\\mathfrak{p}$.  Let $S\\subseteq \\Sigma$ be the set of simple roots $\\alpha$ such that the root space $\\mathfrak{g}_{-\\alpha}$ is contained in $\\mathfrak{p}$.  This set classifies $\\mathsf{P}$, and we therefore introduce the notation \n$$\n  \\mathsf{P}_{\\Sigma} := \\mathsf{G},  \\quad\n  \\mathsf{P}_{\\{\\alpha_1\\}} := \\left\\{ \\smatrix{ *&*&*\\\\ *&*&*\\\\0&0&* } \\right\\}, \\quad\n  \\mathsf{P}_{\\{\\alpha_2\\}} := \\left\\{ \\smatrix{ *&*&*\\\\ 0&*&*\\\\0&*&* } \\right\\}, \\quad\n  \\mathsf{P}_{\\emptyset} := \\mathsf{B}.\n$$\n(Here $*$ denotes possibly nonzero entries.)  We will simplify this by writing $\\mathsf{P}_i := \\mathsf{P}_{\\{\\alpha_i\\}}$ whenever convenient.\n\nFor $i=1,2$, let $\\scrX[i] := \\mathsf{G}\/\\mathsf{P}_i$.  The natural maps $\\fibration[i] : \\scrX \\to \\scrX[i]$ are equivariant fibrations with fibres $\\mathsf{P}_i\/\\mathsf{B} \\cong \\CC \\mathrm{P}^1$.  We will denote the corresponding foliations of $\\scrX$ by $\\foliation[i]:= \\ker D\\fibration[i]$.\n\n\nDenote the compact part of $\\mathsf{P}_S$ by $\\mathsf{K}_S := \\mathsf{P}_S \\cap \\mathsf{K}$. Explicitly,\n\\begin{eqnarray*}\n  \\mathsf{K}_\\Sigma &:=& \\mathsf{K}, \\\\\n  \\mathsf{K}_1 &:=&  \\mathsf{P}_1 \\cap \\mathsf{K}  \n    = \\left\\{ \\ulmatrix{A}{0}{0}{0&0&z} \\bigg| \\quad\n     A\\in\\mathrm{U}(2), ~  z = (\\det A)^{-1}  \\right\\}, \\\\\n  \\mathsf{K}_2 &:=&  \\mathsf{P}_2 \\cap \\mathsf{K}  \n    = \\left\\{ \\drmatrix{z&0&0}{0}{0}{A} \\bigg| \\quad\n     A\\in\\mathrm{U}(2), ~  z = (\\det A)^{-1}  \\right\\}, \\\\\n  \\mathsf{K}_\\emptyset &:=& \\mathsf{M}.\n\\end{eqnarray*}\nThen $\\scrX[i]=\\mathsf{K}\/\\mathsf{K}_i$ ($i=1,2$).\n\nThe complexified Lie algebra $(\\mathfrak{k}_i)_\\mathbb{C}$ of $\\mathsf{K}_i$ decomposes as $\\mathfrak{s}_i\\oplus\\mathfrak{z}_i$, where $\\mathfrak{s}_i := \\vspan{X_i, H_i, Y_i} \\cong \\mathfrak{sl}(2,\\mathbb{C})$ and $\\mathfrak{z}_i := \\vspan{H_i'}\\subset \\mathfrak{m}_\\mathbb{C}$.  (Notation as in Section \\ref{sec:Lie_groups}.)    For the sake of fixing notation, we recall the representation theory of $\\mathfrak{s}_i \\cong \\mathfrak{sl}(2,\\mathbb{C})$.  The weights of $\\mathfrak{sl}(2,\\mathbb{C})$ are parameterized by the integers.  The restriction of a weight $\\mu$ of $\\mathsf{K}$ to a weight of $\\mathfrak{s}_i$ is $\\mu_i := \\mu(H_i) \\in \\mathbb{Z}$.  The dominant weights are the nonnegative integers $\\mathbb{N}$.  \n\nLet $X,H,Y\\in\\mathfrak{sl}(2,\\mathbb{C})$ be the basis elements corresponding to $X_i, H_i. Y_i\\in\\mathfrak{s}_i$.  The irreducible representation of $\\mathfrak{sl}(2,\\mathbb{C})$ with highest weight $\\delta\\in\\mathbb{N}$ will be denoted $V^\\delta$.  It has an orthonormal basis of weight vectors $\\{ e_\\delta, e_{\\delta-2}, \\ldots, e_{-\\delta+2}, e_{-\\delta} \\}$, such that\n\\begin{eqnarray}\n\\label{eq:X-formula}\n  X\\cdot e_j &=& \\frac{1}{2} \\sqrt{(\\delta-j)(\\delta+j+2)} \\,e_{j+2} \\\\\n\\label{eq:H-formula}\n  H\\cdot e_j &=&  j\\, e_{j} \\\\\n\\label{eq:Y-formula}\n  Y\\cdot e_j &=& \\frac{1}{2} \\sqrt{(\\delta-j+2)(\\delta+j)} \\, e_{j-2} \n\\end{eqnarray}  \n\n\n\n\\subsection{Harmonic analysis}\n\\label{sec:harmonic_notation}\n\nFor any compact group $\\mathsf{C}$, we will use $\\irrep{C}$ to denote the set of irreducible representations of $\\mathsf{C}$, often referred as {\\em $\\mathsf{C}$-types}.  For any unitary representation $\\pi$ of $\\mathsf{C}$, we use $V^\\pi$ to denote its representation space, and $\\pi^\\dagger$ to denote its contragredient representation.\n\n\nFor a representation $\\pi$ of $\\mathsf{K}=\\SU(3)$ and elements $\\xi\\in V^\\pi$, $\\eta^\\dagger\\in V^{\\pi\\dagger}$, we use $\\matrixunit{\\eta^\\dagger}{\\xi}$ to denote the matrix unit $\\matrixunit{\\eta^\\dagger}{\\xi}(k) := (\\eta^\\dagger, \\pi(k)\\xi)$.  \nRecall the Peter-Weyl isomorphism\n\\begin{eqnarray*}\n  \\bigoplus_{\\pi\\in\\irrep{K}} V^{\\pi\\dagger} \\otimes V^\\pi & \\cong & L^2(\\mathsf{K}) \\\\\n    \\eta^\\dagger \\otimes \\xi & \\mapsto & (\\dim V^\\pi)^\\frac{1}{2} \\matrixunit{\\eta^\\dagger}{\\xi}.\n\\end{eqnarray*}\nwhich intertwines $\\bigoplus \\pi$ and $\\bigoplus\\pi^\\dagger$ with the left and right regular representations, respectively.\nIf $p_\\mu$ denotes the projection onto the $\\mu$-weight space of a representation then from Equation \\ref{eq:m-equivariance},\n$$\n  \\LXE[\\mu] \\cong  \\bigoplus_{\\pi\\in\\irrep{K}} V^{\\pi\\dagger} \\otimes p_{-\\mu}V^\\pi  .\n$$\n\n\n\n\n\n\\section{Harmonic analysis on the flag variety}\n\\label{sec:harmonic_analysis}\n\n\\subsection{Harmonic $C^*$-categories}\n\\label{sec:harmonic_decompositions}\n\\label{sec:categories}\n\n\nWe will make much use of the results of \\cite{Yuncken:PsiDOs} regarding harmonic analysis on flag manifolds for $\\SL(n,\\mathbb{C})$.  In this section, we review the major definitions and results of that paper.  Because we are only interested in $n=3$ here, we will simplify the notation somewhat.  \n\nLet $\\mathsf{K}'$ be a closed subgroup of $\\mathsf{K}=\\SU(3)$.  Let $\\scr{H}$ be a Hilbert space equipped with a unitary representation of $\\mathsf{K}$.  For $\\sigma\\in\\irrep{K}'$, we let $p_\\sigma$ denote the orthogonal projection onto the $\\sigma$-isotypical subspace of $\\mathcal{H}$ (with representation restricted to $\\mathsf{K}'$). \nIf $F\\subseteq \\irrep{K}'$ is a set of $\\mathsf{K}'$-types, we let $p_F := \\sum_{\\sigma\\inF} p_\\sigma$. \n\nWe are particularly interested in the four subgroups $\\mathsf{K} \\geq \\mathsf{K}_1, \\mathsf{K}_2, \\geq \\mathsf{M}$ above.   Note that the isotypical subspaces of $\\mathsf{M}$ are the weight spaces.\n\nIf $\\mathsf{K}''$ is a subgroup of $\\mathsf{K}'$, then the isotypical projections of $\\mathsf{K}'$ and $\\mathsf{K}''$ commute.  In particular, the isotypical projections of $\\mathsf{K}$, $\\mathsf{K}_1$ and $\\mathsf{K}_2$ commute with the weight-space projections.  These isotypical projections can therefore be restricted to any weight-space of a unitary $\\mathsf{K}$-representation.\n\n\\begin{definition}\nA {\\em harmonic $\\mathsf{K}$-space} $H$ is a direct sum of weight spaces of unitary $\\mathsf{K}$-representations:  $H = \\bigoplus_{k} p_{\\mu_k} \\mathcal{H}_k$ for some weights $\\mu_k$ and unitary $\\mathsf{K}$-representations on $\\mathcal{H}_k$.\n\nA harmonic $\\mathsf{K}$-space $H$ is called {\\em finite multiplicity} if for every $\\pi\\in\\irrep{K}$, $p_\\pi H$ is finite dimensional.\n\\end{definition}\n\n\\begin{example}\n\\label{ex:finite_multiplicities}\nThe (right) regular representation is a finite multiplicity harmonic $\\mathsf{K}$-space by the Peter-Weyl Theorem, as is $\\LXE[\\mu]$ for any weight $\\mu$.  More generally, any homogeneous vector bundle $E$ over $\\scrX$ decomposes equivariantly into line bundles, so $\\LXE$ is a harmonic $\\mathsf{K}$-space.  \n\\end{example}\n\n\n\n\n\n\\begin{definition}\n\\label{def:A_K}\nLet $S\\subseteq\\Sigma$.  Let $A:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces.  For $\\sigma',\\sigma\\in\\irrep{K}_S$, let  $A_{\\sigma'\\sigma} := p_\\sigma' A p_\\sigma$, so that $(A_{\\sigma'\\sigma})$ is the matrix decomposition of $A$ with respect to the decompositions of $H,H'$ into $\\mathsf{K}_S$-types.\n\\begin{enumerate}\n\\item We say $A$ is {\\em $\\mathsf{K}_S$-harmonically proper} if the matrix $(A_{\\sigma'\\sigma})$ is row- and column-finite, {\\em i.e.}, if for every $\\sigma \\in \\irrep{K}_S$, there are only finitely many $\\sigma'\\in \\irrep{K}_S$ for which either $A_{\\sigma'\\sigma}$ or $A_{\\sigma\\sigma'}$ is nonzero.\n\\item We say $A$ is {\\em $\\mathsf{K}_S$-harmonically finite} if the matrix $(A_{\\sigma'\\sigma})$ has only finitely many nonzero entries.\n\\end{enumerate}\n\nDefine $\\scrA[S](H,H')$, resp.~$\\scrK[S](H,H')$, to be the operator-norm closure of the $\\mathsf{K}_S$-harmonically proper, resp.~$\\mathsf{K}_S$-harmonically finite, operators from $H$ to $H'$.  \n\\end{definition}\n\n\n\nIf $H=H'$, we write $\\scrA[S](H)$  and $\\scrK[S](H)$ for $\\scrA[S](H,H)$  and $\\scrK[S](H,H)$, respectively.  These are $C^*$-subalgebras of the algebras $\\scr{L}(H)$ of bounded operators on $H$.  Letting $H$ and $H'$ vary, we consider $\\scrA[S]$ and $\\scrK[S]$ as defining $C^*$-categories of operators between harmonic $\\mathsf{K}$-spaces.\nWe also use $\\scrK$ and $\\scr{L}$ to denote the $C^*$-categories of compact operators and bounded operators, respectively, between Hilbert spaces.\n\n\n\\begin{lemma}[\\thmcitemore{Yuncken:PsiDOs}{Lemma 3.2}]\n\nIf $S\\subseteq S' \\subseteq \\Sigma$ then $\\scrK[S'] \\subseteq \\scrK[S]$.\n\n\\end{lemma}\n\n\nThe following two results are restatements of Lemmas 3.4 and 3.5 of \\cite{Yuncken:PsiDOs}.\n\n\\begin{proposition}\n\\label{prop:KS_equivalent_conditions}\nLet $K:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces.  The following are equivalent:\n\\begin{enumerate}\n\\item  $K \\in \\scrK[S]$,\n\\item  For any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|p_F^\\perp K\\|<\\epsilon$ and $\\|K p_F^\\perp \\| < \\epsilon$.\n\\item  For any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|K - p_F K p_F \\|<\\epsilon$.\n\\end{enumerate}\n\\end{proposition}\n\n\nIf $A$ and $K$ are bounded linear operators, we say $K$ is {\\em right-composable} for $A$ if the codomain of $K$ is the domain of $A$.  {\\em Left-composability} is defined similarly.\n\n\\begin{proposition}\n\\label{prop:AS_equivalent_conditions}\nLet $A:H \\to H'$ be a bounded linear operator between harmonic $\\mathsf{K}$-spaces.  The following are equivalent:\n\\begin{enumerate}\n\\item  $A \\in \\scrA[S]$,\n\\item  For any $\\sigma\\in\\irrep{K}_S$, and any $\\epsilon>0$, there is a finite set $F\\subset\\irrep{K}_S$ of $\\mathsf{K}_S$-types such that $\\|p_F^\\perp A p_\\sigma\\|<\\epsilon$ and $\\| p_\\sigma A p_F^\\perp \\| < \\epsilon$.\n\\item For any $\\sigma\\in\\irrep{K}_S$, $Ap_\\sigma$ and $p_\\sigma A$ are in $\\scrK[S]$.\n\\item  $A$ is a two-sided multiplier of $\\scrK[S]$, meaning that $AK\\in\\scrK[S]$ for all right-composable $K\\in\\scrK[S]$, and $KA\\in\\scrK[S]$ for all left-composable $K\\in \\scrK[S]$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\n\nWe now describe some considerable simplifications from \\cite{Yuncken:PsiDOs} in the case of homogeneous vector bundles for $\\SU(3)$.\n\n\\begin{lemma}\n\\label{lem:degeneracy}\nLet $E$, $E'$ be $\\mathsf{K}$-homogeneous vector bundles over $\\scrX$, and put $H=\\LXE$, $H'= \\LXEprime$.  Then $\\scrK[\\Sigma](H,H') = \\scrK(H,H')$ and $\\scrA[\\Sigma](H,H') = \\scrK[\\emptyset](H,H') = \\scrA[\\emptyset](H,H') = \\scr{L}(H,H')$.\n\n\\end{lemma}\n\n\\begin{proof}\nSince $H$ and $H'$ are direct sums of finitely many weight spaces for the right regular representation of $\\mathsf{K}$, any bounded operator from $H$ to $H'$ is $\\mathsf{M}$-harmonically finite.  Hence, $\\scrK[\\emptyset](H, H') = \\scrA[\\emptyset](H,H') = \\scr{L}(H,H')$.\n\nLemma 3.3 of \\cite{Yuncken:PsiDOs} shows that $\\scrK[\\Sigma](H,H')= \\scrK(H,H')$.  By Proposition \\ref{prop:AS_equivalent_conditions} above, any bounded operator $A:H\\to H'$ is in $\\scrA[\\Sigma]$.\n\\end{proof}\n\n\nThe only nontrivial cases, then, are $\\scrK[\\{\\alpha_i\\}]$ and $\\scrA[\\{\\alpha_i\\}]$, which we abbreviate as $\\scrK[\\alpha_i]$ and $\\scrA[\\alpha_i]$.\n\n\n\\begin{definition}\n\\label{def:A}\nAs in \\cite{Yuncken:PsiDOs}, we put $\\scrA := \\cap_{S\\subseteq\\Sigma} \\scrA[S]$, the simultaneous multiplier category of all $\\scrK[S]$ ($S\\subseteq\\Sigma$). \nNote, though, that by Lemma \\ref{lem:degeneracy} this reduces to $\\scrA(H,H') = \\scrA[\\alpha_1](H,H') \\cap \\scrA[\\alpha_2](H,H')$ when $H$, $H'$ are $L^2$-section spaces of homogeneous vector bundles.\n\\end{definition}\n\n\nIn the generality of \\cite{Yuncken:PsiDOs}, it is necessary to adjust the operator spaces $\\scrK[S]$ by defining $\\scrJ[S] := \\scrK[S] \\cap \\scrA$.  The next lemma shows that this is not necessary for the current application.\n\n\n\n\\begin{lemma}\n\\label{lem:Ji_is_Ki}\n\nWith $H,H'$ as in Lemma \\ref{lem:degeneracy}, $\\scrK[\\alpha_i](H,H') \\subseteq \\scrA(H,H')$, for $i=1,2$.\nThus, $\\scrJ[\\alpha_i](H,H') = \\scrK[\\alpha_i](H,H')$.\n\n\n\\end{lemma}\n\n\n\\begin{proof}\nLet $i=1$.  \nIt is immediate that $\\scrK[\\alpha_1](H,H')\\subseteq \\scrA[\\alpha_1](H,H')$.  Lemma 5.4 of \\cite{Yuncken:PsiDOs} implies that on $H$ and $H'$, $p_{\\sigma_1} p_{\\sigma_2}$ is compact for any $\\sigma_1\\in\\irrep{K}_1$ and $\\sigma_2\\in\\irrep{K}_2$.  Thus, if $K:H\\to H'$ is $\\mathsf{K}_1$-harmonically finite, then $K p_{\\sigma_2}\\in\\scrK(H,H') \\subseteq \\scrK[\\alpha_2](H,H')$.  By Proposition \\ref{prop:AS_equivalent_conditions}, $K\\in\\scrA[\\alpha_2](H,H')$.  Taking the norm-closure, $\\scrK[\\alpha_1](H,H')\\subseteq \\scrA[\\alpha_2](H,H')$, which proves the result.  The case $i=2$ is analogous.\n\\end{proof}\n\nWe therefore avoid the notation $\\scrJ[\\alpha_i]$ altogether.\n\n\n\\begin{theorem}[\\thmcitemore{Yuncken:PsiDOs}{Theorem 1.11}]\n\\label{thm:lattice_of_ideals}\nLet $E$ be a $\\mathsf{K}$-homogeneous vector bundle over $\\scrX$, and $H:=\\LXE$.  Then\n\\begin{enumerate}\n\\item $\\scrK[\\alpha_i](H)$ is an ideal in $\\scrA(H)$, for $i=1,2$.\n\\item $\\scrK[\\alpha_1](H) \\cap \\scrK[\\alpha_2](H) = \\scrK(H)$.\n\\end{enumerate}\n\n\\end{theorem}\n\n\\begin{lemma}[\\thmcitemore{Yuncken:PsiDOs}{Lemma 8.1}]\n\\label{lem:mult_ops_in_A}\n  Let $\\mu$, $\\nu$ be weights.  For any $f\\in\\CXE[\\mu-\\nu]$, the multiplication operator $\\multop{f}:\\LXE[\\nu] \\to \\LXE[\\mu]$ is in $\\scrA$.\n\\end{lemma}\n\n\\begin{remark}\n\\label{rmk:mult_ops_in_compact_picture}\nLemma \\ref{lem:mult_ops_in_A} depends on $\\mathsf{K}$-equivariant structure only, so that $f$ may be (the restriction to $\\mathsf{K}$ of) a section of $L_{(\\mu-\\nu)\\oplus\\chi_{\\Lie{A}}}$ for any $\\chi_{\\Lie{A}}\\in\\mathfrak{m}_\\mathbb{C}^\\dagger$.\n\\end{remark}\n\n\n\\subsection{Principal series representations}\n\\label{sec:principal_series}\n\n\n\n\nThe purpose of this section is to prove the following important fact, the first of two rather technical harmonic analysis results.  \n\n\n\\begin{proposition}\n\\label{prop:U(g)_in_A}\nLet $\\mu\\in\\Lambda_W$.  For any $g\\in \\mathsf{G}$, $U_\\mu(g) \\in\\scrA(\\LXE[\\mu])$.\n\\end{proposition}\n\n\nWe will use the notation for the elements of $\\mathfrak{k}_\\mathbb{C}$ from Section \\ref{sec:Lie_groups}, noting that the elements $X_\\alpha$, $Y_\\alpha$ ($\\alpha\\in\\Delta^+$) and $H_i$, $H_i'$ (for either $i=1$ or $2$) form a basis for $\\mathfrak{g}$.  We let $X_\\alpha^\\dagger, Y_\\alpha^\\dagger, H_i^\\dagger, {H_i'}^\\dagger$ denote the dual basis elements of $\\mathfrak{g}^\\dagger$.  We also recall the notation $\\matrixunit{\\eta^\\dagger}{\\xi}$ for matrix units.\n\n\n\\begin{lemma}\n\\label{lem:a-action}\nLet $A\\in\\mathfrak{a}$.  Let $\\pi\\in\\irrep{K}$ and $\\eta^\\dagger \\in V^{\\pi\\dagger}$, $\\xi \\in (V^\\pi)_{-\\mu}$.  Then $U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi} = \\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}$, where\n$$\n  \\Xi(\\xi) := \\rho(H_i)\\xi\\otimes H_i^\\dagger + \\rho(H_i')\\xi \\otimes {H_i'}^\\dagger \n    + \\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\pi(X_{\\alpha})\\xi\\otimes X_\\alpha^\\dagger \n    \\quad \\in V^\\pi \\otimes \\mathfrak{g}^\\dagger.\n$$\n\\end{lemma}\n\nNote that $\\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}$ is a matrix unit for the \\emph{non-irreducible} representation $\\pi\\otimes\\Ad^\\dagger$, hence a sum of matrix units for the irreducible components of $\\pi\\otimes\\Ad^\\dagger$.\n\n\\begin{proof}\nDefine functions $\\kappa$, $\\mathsf{a}$, $\\mathsf{n}$ on $\\mathsf{G}$ using the Iwasawa decomposition:\n$$\n  g =: \\kappa(g) \\mathsf{a}(g) \\mathsf{n}(g) \\in \\mathsf{KAN}, \\qquad \\text{for $g\\in\\mathsf{G}$}.\n$$\nThe derivatives ${D} \\kappa_e$, ${D} \\mathsf{a}_e$ and ${D} \\mathsf{n}_e$ at the identity are the ($\\mathbb{R}$-linear) projections of $\\mathfrak{g}$  onto the components of the decomposition $\\mathfrak{g}=\\mathfrak{k\\oplus a\\oplus n}$.  If $P\\in\\mathfrak{g}$, let us write $P= P_+ +  P_0 +  P_-$ where $P_+$, $P_0$, $P_-$ are strictly upper-triangular, diagonal, and strictly lower-triangular, respectively.  If $P$ is self-adjoint, the $\\mathfrak{k\\oplus a\\oplus n}$ decomposition of $P$ is $P=(-P_+ + P_-) \\oplus P_0 \\oplus 2P_+$.  Thus,\n\\begin{eqnarray}\n\\label{eq:k-derivative}\n  {D} \\kappa_e (P) &=& \\left(-\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) X_\\alpha\\otimes X_\\alpha^\\dagger \\right) P, \\\\\n\\label{eq:a-derivative}\n  {D} \\mathsf{a}_e (P) &=& \\big( H_i \\otimes H_i^\\dagger + H_i'\\otimes {H_i'}^\\dagger \\big) P.\n\\end{eqnarray}\n\n\nFor $a\\in\\mathsf{A}$, $k\\in\\mathsf{K}$, \n$$\n  a^{-1}k = kk^{-1}a^{-1}k = k\\,\\kappa(k^{-1}a^{-1}k) \\mathsf{a}(k^{-1}a^{-1}k) \\mathsf{n}(k^{-1}a^{-1}k).\n$$\nIn order to describe the $\\mathsf{G}$-action on a $\\mathsf{K}$-matrix unit, one must extend $\\matrixunit{\\eta^\\dagger}{\\xi}$ to a $\\mathsf{B}$-equivariant function on $\\mathsf{G}$.  Equation \\eref{eq:B-equivariance}) gives\n\\begin{eqnarray}\n  U_{\\mu}(a)\\matrixunit{\\eta^\\dagger}{\\xi}(k)\n    &:=& \\matrixunit{\\eta^\\dagger}{\\xi}(a^{-1} k) \\nonumber\\\\\n    &=& e^{\\rho}(\\mathsf{a}(k^{-1}ak)) \\matrixunit{\\eta^\\dagger}{\\xi}(k\\, \\kappa(k^{-1}a^{-1}k)) \\nonumber\\\\\n    &=& e^{\\rho}(\\mathsf{a}(k^{-1}ak)) \\big( \\eta^\\dagger, \\pi(k) \\pi(\\kappa(k^{-1}a^{-1}k)) \\xi\\big).\n    \\label{eq:extension_to_G}\n\\end{eqnarray}\nLet $a=\\exp(tA)$, and take the derivative with respect to $t$ at $t=0$:\n\\begin{equation*}\n\\label{eq:UA1}\n  U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi}(k)\n    = \\rho({D} \\mathsf{a}_e(\\Ad k^{-1}(A))) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big)\n      - \\big( \\eta^\\dagger, \\pi(k) \\pi({D} \\kappa_e(\\Ad k^{-1}(A)) \\xi\\big).\n\\end{equation*}\nSince $\\Ad k^{-1} (A)$ is self-adjoint, Equations \\eref{eq:k-derivative} and \\eref{eq:a-derivative} give\n\\begin{eqnarray*}\n  \\lefteqn{U_{\\mu}(A)\\matrixunit{\\eta^\\dagger}{\\xi}(k) } \\quad \\\\\n    &=& \\rho(H_i) \\big(H_i^\\dagger, \\Ad k^{-1}(A)\\big) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big)\n      + \\rho(H_i') \\big({H_i'}^\\dagger, \\Ad k^{-1}(A)\\big) \\big( \\eta^\\dagger, \\pi(k) \\xi\\big) \\\\\n    &&\\qquad +\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\big( \\eta^\\dagger, \\pi(k) \\pi(X_\\alpha) \\big(X_\\alpha^\\dagger, \\Ad k^{-1}(A)\\big) \\xi\\big) \\\\\n    &=& \\big( A, \\Ad^\\dagger k (H_i^\\dagger) \\big) \\big( \\eta^\\dagger, \\pi(k) \\rho(H_i)\\xi\\big)\n      + \\big( A, \\Ad^\\dagger k ({H_i'}^\\dagger) \\big) \\big( \\eta^\\dagger, \\pi(k) \\rho(H_i')\\xi\\big) \\\\\n    && \\qquad  +\\sum_{\\alpha\\in\\Delta} \\sign(\\alpha) \\big(A,  \\Ad^\\dagger k (X_\\alpha^\\dagger)\\big)\\big( \\eta^\\dagger, \\pi(k) \\pi(X_\\alpha) \\xi\\big) \\\\\n    &=& \\matrixunit{\\eta^\\dagger\\otimes A}{\\Xi(\\xi)}(k).\n\\end{eqnarray*}\n\n\\end{proof}\n\nRecall the decomposition $(\\mathfrak{k}_i)_\\mathbb{C} = \\mathfrak{s}_i \\oplus \\mathfrak{z}_i$ of Section \\ref{sec:parabolic_subgroups}.  Let $\\mu\\in\\Lambda_W$.  Since $\\mathfrak{z}_i\\subseteq\\mathfrak{h}$, the action of $\\mathfrak{z}_i$ on the $(-\\mu)$-weight space of any $\\mathsf{K}$-representation is completely determined by $\\mu$.  Thus, the $\\mathsf{K}_i$-isotypical subspaces of $\\LXE[\\mu] $ are the $\\mathfrak{s}_i$-isotypical subspaces.  Moreover, since $\\LXE[\\mu]$ has $\\mathfrak{s}_i$-weight $-\\mu_i := -\\mu(H_i)$, the $\\mathsf{s}_i$-types which occur must have highest weights $|\\mu_i|, |\\mu_i|+2, \\ldots$\n\nIn what follows, we fix $i=1$ or $2$ and let $\\sigma_l$ denote the $\\mathfrak{s}_i$-type with highest weight $l\\in\\mathbb{N}$.  We abbreviate $p_l := p_{\\sigma_l}$.  Note that $p_l=0$ on $\\LXE[\\mu]$ if $l \\not\\equiv \\mu_i \\pmod{2}$ or $l < |\\mu_i|$.\nThe next lemma shows that $U_\\mu(A)$ is tridiagonal with respect to $\\mathsf{K}_i$-types, and that the off-diagonal entries have at most linear growth.\n\n\n\\begin{lemma}\n\\label{lem:tridiagonal}\nFix $\\mu\\in\\Lambda_W$ and let $A\\in \\mathfrak{a}$.  There exists a constant $C>0$ such that for any $m,l \\in \\mathbb{N}$,\n\\begin{equation*}\n  \\begin{array}{rcll}\n  \\| p_m U_\\mu(A) p_l \\|  &=&    0 &\\text{if $|m-l|>2$}, \\\\\n  \\| p_m U_\\mu(A) p_l \\|  &\\leq& C(l+1) \\qquad &\\text{if $|m-l|=2$}.\n  \\end{array}\n\\end{equation*}\n\n\\end{lemma}\n\n\n\\begin{proof}\nLet us take $i=1$, with the case of $i=2$ being entirely analogous.\nSuppose $\\matrixunit{\\eta^\\dagger}{\\xi} \\in p_l \\LXE[\\mu]$, which is to say that $\\eta^\\dagger\\in V^{\\pi\\dagger}$, $\\xi \\in (V^\\pi)_{\\sigma_l}$ for some $\\pi\\in\\irrep{K}$.  By Lemma \\ref{lem:a-action}, we need to understand the decomposition of $\\Xi(\\xi)$ into $\\mathfrak{s}_1$-types.\n\nThe adjoint representation of $\\mathfrak{g}$ decomposes into the $\\mathfrak{s}_1$-representations\n$$\n   \\vspan{X_1, H_1, Y_1}, \\quad \\vspan{H_1'}, \\quad \\vspan{X_2, X_3}, \\quad \\vspan{Y_2, Y_3},\n$$\nand $\\mathfrak{g}^\\dagger$ decomposes dually.  We break up the expression for $\\Xi(\\xi)$ into corresponding parts.\n\nFirstly, $H_1'$ has trivial $\\mathfrak{s}_1$-type, so $\\rho(H_1')\\xi\\otimes{H_1'}^\\dagger$ has $\\mathfrak{s}_i$-type $l$.  Next, note that the vector $X_2 \\otimes X_2^\\dagger + X_3 \\otimes X_3^\\dagger \\in \\mathfrak{g}\\otimes\\mathfrak{g}^\\dagger$ also has trivial $\\mathfrak{s}_1$-type, since it corresponds to the identity map on the subrepresentation $\\vspan{X_2, X_3}$.  \nThe map\n\\begin{eqnarray*}\n  V^\\pi \\otimes \\mathfrak{g}\\otimes \\mathfrak{g}^\\dagger &\\to& V^\\pi \\otimes \\mathfrak{g}^\\dagger \\\\\n  \\zeta \\otimes Z \\otimes Z^\\dagger &\\mapsto& \\pi(Z)\\zeta \\otimes Z^\\dagger\n\\end{eqnarray*}\nis a morphism of $\\mathsf{K}$-representations, in particular of $\\mathsf{s}_i$-representations, so $\\pi(X_2)\\xi \\otimes X_2^\\dagger + \\pi(X_3) \\xi \\otimes X_3^\\dagger$ also has $\\mathfrak{s}_1$-type $l$.  Similarly, $-\\pi(Y_2)\\xi \\otimes Y_2^\\dagger - \\pi(Y_3) \\xi \\otimes Y_3^\\dagger$ has $\\mathfrak{s}_1$-type $l$.\n\nThus, all the off-diagonal components of $U_\\mu(A)$ are due to the components\n\\begin{equation}\n\\label{eq:off-diagonal_terms}\n  \\Xi_1(\\xi) := \\rho(H_1)\\xi \\otimes H_1^\\dagger + \\pi(X_1)\\xi \\otimes X_1^\\dagger - \\pi(Y_1) \\xi \\otimes Y_1^\\dagger\n\\end{equation}\nof $\\Xi(\\xi)$.  The coadjoint representation of $\\mathfrak{s}_1$ on $\\vspan{X_1^\\dagger, H_1^\\dagger, Y_1^\\dagger}$ has highest weight $2$, so the fusion rules for $\\SU(2)$-representations imply that \\eref{eq:off-diagonal_terms} contains $\\mathfrak{s}_i$-types $l-2, l, l+2$ only.  \n\nIt remains to prove the norm estimate on the off-diagonal terms.  By Equations \\eref{eq:X-formula}--\\eref{eq:Y-formula},\n\\begin{equation*}\n\\begin{array}{rclcl}\n  \\|\\rho(H_1) \\xi\\| &=& 2\\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|, \\\\ \\\\\n  \\|\\pi(X_1) \\xi \\| &=& \\frac{1}{2} \\sqrt{(l-\\mu_i)(l+\\mu_i+2)} \\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|, \\\\ \\\\\n  \\|\\pi(Y_1) \\xi \\| &=& \\frac{1}{2} \\sqrt{(l-\\mu_i+2)(l+\\mu_i)} \\|\\xi\\| &\\leq& (l+1)\\|\\xi\\|,\n\\end{array}\n\\end{equation*}\nso the norm of $\\Xi_1(\\xi)$ is bounded by $C_0(l+1) \\|\\xi\\|$ for some constant $C_0$.\nWe need to convert this into a bound on the norm of the matrix units.  \n\nDecompose $\\pi\\otimes \\Ad^\\dagger$ into irreducible $\\mathsf{K}$-subrepresentations.  Suppose $\\pi'$ is an irreducible subrepresentation of $\\pi\\otimes\\Ad^\\dagger$.  By orthogonality of characters, $\\pi$ is a subrepresentation of $\\pi'\\otimes \\Ad$.  Therefore $\\dim \\pi \\leq \\dim (\\pi'\\otimes\\Ad) = 8\\dim \\pi' $, so that $\\dim \\pi' \\geq \\frac{1}{8} \\dim \\pi$.  This also shows that the number of irreducible components of $\\pi\\otimes\\Ad^\\dagger$ is at most $64$.  \n\nFor each irreducible subrepresentation $\\pi'$ of $\\pi\\otimes\\Ad^\\dagger$, let  $y_{\\pi'}^\\dagger$ denote the ${\\pi'}^\\dagger$-component of $\\eta^\\dagger\\otimes A$,  and $x_{\\pi'}$ the $\\pi'$-component of $\\Xi_1(\\xi)$.\nWe get\n\\begin{eqnarray*}\n  \\| p_{l\\pm2} U_\\mu(A) p_l \\matrixunit{\\eta^\\dagger}{\\xi} \\|^2\n    & \\leq & \\| \\matrixunit{\\eta\\otimes A}{\\Xi_1(\\xi)} \\|^2 \\\\\n    & = & \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\|y_{\\pi'}^\\dagger\\|^2   \\|x_{\\pi'}\\|^2 \\\\\n    &\\leq& \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\| \\eta^\\dagger \\otimes A \\|^2 \\| \\Xi_1(\\xi) \\|^2 \\\\\n    &\\leq& \\sum_{\\pi'} \\frac{1}{\\dim \\pi'} \\|\\eta^\\dagger\\|^2 \\|A\\|^2 C_0^2(l+1)^2 \\|\\xi\\|^2 \\\\\n    &\\leq& \\|A\\|^2 C_0^2 (l+1)^2   \\sum_{\\pi'} \\frac{8}{\\dim \\pi} \\| \\eta^\\dagger\\|^2 \\| \\xi \\|^2 \\\\\n    &\\leq& 8.64.\\|A\\|^2 C_0^2 (l+1)^2 \\| \\matrixunit{\\eta^\\dagger}{\\xi} \\|^2.\n\\end{eqnarray*}\nPutting $C=  \\sqrt{512}\\, \\|A\\| \\,C_0$ gives the result.\n\n\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:U(g)_in_A}]\nWe need to show $U_\\mu(g)\\in\\scrA[\\alpha_i]$ for $i=1,2$.\nFor $k\\in\\mathsf{K}$, the left translation action $U_\\mu(k)$ commutes with the decomposition into right $\\mathsf{K}_i$-types, so that $U_\\mu(k) \\in \\scrA[\\alpha_i]$ trivially.  By the $\\mathsf{KAK}$-decomposition, it suffices to prove the proposition for $g=a\\in\\mathsf{A}$. \n\n\nWe continue with the notation of the previous lemma.  Put $P_m:= \\sum_{j=0}^m p_j$.  We will show that for any $l\\in\\mathbb{N}$ and any $\\epsilon>0$, there exists $m\\in\\mathbb{N}$ such that\n$\\| P_m^\\perp U_\\mu(a) p_l\\| <\\epsilon$ and $\\|  p_l U_\\mu(a) P_m^\\perp\\| <\\epsilon$, from which Lemma \\ref{prop:AS_equivalent_conditions} gives $U_\\mu(a)\\in\\scrA[\\alpha_i]$. \n\n\nLet $A\\in\\mathfrak{a}$ such that $e^A = a$.   Define $\\phi:\\mathbb{N}\\to [0,1]$ by\n$$\n  \\phi(n) := \\begin{cases}\n    1, & n\\leq l, \\\\\n    \\max\\,\\{0, 1-\\frac{\\epsilon^2}{4C} \\log(n+3) \\}, & n>l,\n  \\end{cases}\n$$\nwhere $C$ is the constant of the previous lemma.  Define  $\\Phi:= \\sum_{n\\in\\mathbb{N}} \\phi(n) p_n$, an operator on $\\LXE[\\mu]$ which is scalar on each $\\mathsf{K}_i$-type.\n\nWe now decompose $U_\\mu(A)$ into its diagonal and off-diagonal components.  For convenience of notation, we put $U:= U_\\mu(A)$, then write $U=U_-+U_0+U_+$, where\n$$\n  U_- = \\sum_{n=2}^\\infty p_{n-2}U p_n, \\qquad\n  U_0 = \\sum_{n=0}^\\infty p_n U p_n, \\qquad\n  U_+ = \\sum_{n=0}^\\infty p_{n+2} U p_n.\n$$\nThe diagonal component $U_0$ commutes with $\\Phi$.  On the other hand,\n\\begin{eqnarray*}\n  \\| [p_{n-2} U p_n, \\Phi] \\| \n    &=& \\| (\\phi(n) - \\phi(n-2))\\, p_{n-2} U p_n \\| \\\\\n    &\\leq& \\frac{\\epsilon^2}{4C}(\\log(n+3)-\\log(n+1)) \\\\\n    &\\leq& \\frac{\\epsilon^2}{2C} \\frac{1}{(n+1)} \\\\\n    &\\leq& \\frac{\\epsilon^2}{2},\n\\end{eqnarray*}\nby Lemma \\ref{lem:tridiagonal}.  Thus,\n$$\n  \\| [U_-, \\Phi] \\| = \\sup_{n\\in\\mathbb{N}} \\| [U_{n-2,n}, \\Phi] \\|  \\leq \\frac{1}{2} \\epsilon^2.\n$$\nSimilarly, $\\| [U_+, \\Phi] \\| \\leq \\frac{1}{2} \\epsilon^2$.  Therefore, $\\| [U_\\mu(A), \\Phi] \\| \\leq \\epsilon^2$.\n\nLet $s \\in p_l \\LXE[\\mu]$ have norm one.  Put $s_t := U_\\mu(e^{tA})s$ for $0\\leq t \\leq 1$.  Then\n$$\n  | \\frac{d}{dt} \\ip{ \\Phi s_t, s_t } |\n    = | \\ip{ \\Phi U_\\mu(A) s_t, s_t } + \\ip{ \\Phi s_t, U_\\mu(A) s_t } |\n    = | \\ip{ [\\Phi, U_\\mu(A)] s_t, s_t } |\n    \\leq \\epsilon^2,\n$$\nfor all $t$.  Therefore,\n\\begin{eqnarray*}\n  |\\ip{\\Phi s_1, s_1}| &=& \\left| \\ip{\\Phi s_0, s_0} + \\int_{t'=0}^1 \\frac{d}{dt} \\ip{\\Phi s_t, s_t} \\, dt'  \\right|\\\\\n    &\\geq& 1 - \\epsilon^2. \n\\end{eqnarray*}\nLet $m$ be the smallest integer for which $\\phi(m)=0$.   Put $v := P_m s_1$ and $w:= P_m^\\perp s_1$.   Then $\\|v\\|^2 + \\|w\\|^2 =1$, but also\n$$\n  \\| v\\|^2  > \\ip{\\Phi v,v} =  \\ip{\\Phi v,v} + \\ip{\\Phi w,w} = \\ip{\\Phi s_1, s_1} \\geq 1 -\\epsilon^2.\n$$\nIt follows that $ \\|w\\| < \\epsilon$, {\\em ie}, $\\|P_m^\\perp U_\\mu(a) s \\| < \\epsilon$.  Since $s\\in p_l\\LXE[\\mu]$ was arbitrary, $\\|P_m^\\perp U_\\mu(a) p_l\\| < \\epsilon$.  \n\nReplacing $a$ with $a^{-1}$, there exists $m'\\in\\mathbb{N}$ such that $\\| P_{m'}^\\perp U_\\mu(a^{-1}) p_l \\| < \\epsilon$.  Thus, after enlarging $m$ to be at least $m'$, we have\n$$\n  \\| p_l U_\\mu(a) P_{m}^\\perp \\| = \\| P_{m}^\\perp U_\\mu(a^{-1}) p_l \\| < \\epsilon.\n$$\n\n\\end{proof}\n\n\n\nIn fact, Proposition \\ref{prop:U(g)_in_A} holds for any generalized principal series representation.  Although we don't actually need this here, it is now trivial to prove.\n\n\\begin{corollary}\nFor any $\\mathsf{G}$-homogeneous line bundle $\\LXL[\\chi]$ over $\\scrX$, the translation operators $s \\mapsto g\\cdot s$ belong to $\\scrA$.\n\\end{corollary}\n\n\\begin{proof}\nLet $\\chi=\\chi_{\\Lie{M}}\\oplus\\chi_{\\Lie{A}}$.  A computation of the form of Eq.~\\eref{eq:extension_to_G}\ngives \n$$\n  g\\cdots(k) = e^{\\chi_{\\Lie{A}}}(\\mathsf{a}(k^{-1}gk)) s (k\\, \\kappa(k^{-1}g^{-1}k)),\n$$\nfor any $k\\in\\mathsf{K}$, while\n$$\n  U_{\\chi_{\\Lie{M}}}(g)s(k) = e^{\\rho}(\\mathsf{a}(k^{-1}gk)) s (k\\, \\kappa(k^{-1}g^{-1}k)).\n$$\nNote that $\\mathsf{a}(m^{-1}gm) = \\mathsf{a}(g)$ for any $m\\in\\mathsf{M}$, $g\\in\\mathsf{G}$.\nTherefore, $g\\cdots = \\multop{f} U_{\\chi_{\\Lie{M}}}(g) s$, where $f(k) := e^{\\chi_{\\Lie{A}}-\\rho}(\\mathsf{a}(k^{-1}gk))$ is in $C(\\mathsf{K\/M}) = C(\\scrX)$.  Since $\\multop{f}$ and $U_{\\chi_{\\Lie{M}}}(g)$ are in $\\scrA$, we are done.\n\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Longitudinal pseudodifferential operators}\n\\label{sec:PsiDOs}\n\n\nLet $X\\in\\mathfrak{k}_\\mathbb{C}$ be a root vector, of weight $\\alpha$.  Via the right regular representation, $X$ defines a left $\\mathsf{K}$-invariant differential operator on $C^\\infty(\\mathsf{K})$.  \nFor each weight $\\mu$, $X$ maps $p_{-\\mu} L^2(\\mathsf{K})$ to $p_{-\\mu+\\alpha} L^2(\\mathsf{K})$, so it defines a $\\mathsf{K}$-invariant differential operator\n$$\n  X : \\LXE[\\mu] \\to \\LXE[\\mu-\\alpha].\n$$\nThe principal symbol of this differential operator is a $\\mathsf{K}$-equivariant linear map from the cotangent bundle $T^*\\scrX \\cong K\\times_M (\\mathfrak{k\/m})^*$ to $\\End( E_\\mu, E_{\\mu-\\alpha}) \\cong E_{-\\alpha}$.  (Here $(\\mathfrak{k\/m})^*$ denotes the {\\em real} dual of $\\mathfrak{k\/m}$.)  By equivariance, this map is determined by its value on the cotangent fibre at the identity coset $\\coset{e}\\in\\scrX$, which is\n\\begin{eqnarray}\n\\label{eq:diff_op_symbol}\n  \\Symbol (X) :  T^*_{\\coset{e}}\\scrX = (\\mathfrak{k\/m})^*  \n    & \\to & \\mathbb{C} \\\\\n  \\xi & \\mapsto & \\xi(X). \\nonumber\n\\end{eqnarray}\n\n\n\nIf $X\\in(\\mathfrak{k}_i)_\\mathbb{C}$ ($i=1$ or $2$), then the differential operator $X:\\CinftyXE[\\mu] \\to \\CinftyXE[\\mu-\\alpha]$ is tangential to the foliation $\\foliation[i]$ of Section \\ref{sec:parabolic_subgroups}.  We will refer to such an operator as an $\\foliation[i]$-longitudinal differential operator.  Its {longitudinal principal symbol} is the $\\mathsf{K}$-equivariant map $\\Symbol[i]: \\foliation[i]^* \\to E_{-\\alpha}$ which, at the identity coset, is given by\n\\begin{eqnarray*}\n  \\Symbol[i] :  (\\foliation[i]^*)_{\\coset{e}}  = (\\mathfrak{k}_i \/ \\mathfrak{m})^* &\\to& \\mathbb{C} \\\\\n  \\xi &\\mapsto & \\xi(X).\n\\end{eqnarray*}\n\nAn $\\foliation[i]$-longitudinal differential operator is \\emph{longitudinally elliptic} if its longitudinal principal symbol is invertible off the zero section of $T^*\\foliation[i]$.  Note that $X_i = -\\frac{1}{2}(X_i' + \\sqrt{-1} \\, X_i'') \\in (\\mathfrak{k}_i)_\\mathbb{C}$ where\n$$\n  X_i' = \\smatrix{ 0 & -1 \\\\ 1 & 0 }, \\quad X_i'' = \\smatrix{ 0 & \\sqrt{-1} \\\\ \\sqrt{-1} & 0 }\n$$\nspan $\\mathfrak{k}_i\/\\mathfrak{m}$, so that $X_i$ is $\\foliation[i]$-longitudinally elliptic.  Similarly, $Y_i$ is $\\foliation[i]$-longitudinally elliptic.  Moreover, $X_i$ and $Y_i$ are formal adjoints.  We shall use $X_i$, $Y_i$ also to denote their closures as unbounded operators on the $L^2$-section spaces.\n\nFix $\\mu\\in\\Lambda_W$.  Let $E := E_\\mu \\oplus E_{\\mu-\\alpha_i}$, and define  $D_i := \\smatrix{0&Y_i\\\\X_i&0}$ on $\\LXE$.  The $\\mathfrak{s}_i$-isotypical subspaces of $\\LXE$ are eigenspaces for $D_i$, and by the representation theory of $\\mathfrak{s}_i$---specifically Equations \\eref{eq:X-formula} and \\eref{eq:Y-formula}---its spectrum is discrete.\n\n\n\\medskip\n\nFor the definition and basic properties of longitudinal pseudodifferential operators, we refer the reader to \\cite{MS-GAFS}\\footnote{In this reference, they are called {tangential} pseudodifferential operators.}.  If $E$, $E'$ are vector bundles over $\\scrX$, we denote the set of $\\foliation[i]$-longitudinal pseudodifferential operators of order at most $p$ by $\\PsiDO[i]^p(E,E')$.  If $E=E'$, we abbreviate this to $\\PsiDO[i]^p(E)$.\n\nLet $C(S^*\\foliation[i]; \\End(E))$ denote the algebra of continuous sections of the pullback of $\\End(E)$ to the cosphere bundle of the foliation $\\foliation[i]$.  The longitudinal principal symbol map $\\Symbol[i]:\\PsiDO[i]^0(E) \\to C(S^*\\foliation[i]; \\End(E))$ extends to the operator-norm closure $\\overline{\\PsiDO[i]^{0}}(E)$, and we have Connes' short exact sequence,\n\\begin{equation}\n\\label{eq:symbol-sequence}\n  \\xymatrix{\n    0 \\ar[r] &\n    \\overline{\\PsiDO[i]^{-1}}(E) \\ar[r] &\n    \\overline{\\PsiDO[i]^{0}}(E) \\ar[r]^-{\\Symbol[i]} &\n    C(S^*\\foliation[i]; \\End(E)) \\ar[r] &\n    0.\n  }\n\\end{equation}\n  \n\n\nFor any closed, densely defined, unbounded operator $T$ between Hilbert spaces, we let $\\Ph T$ denote the phase in the polar decomposition: $T = (\\Ph{T}) |T|$.  We also use $\\Ph{z}$ to denote the phase of a complex number $z\\in\\mathbb{C}^\\times$.\n\n\n\\begin{lemma}\n\\label{lem:F_in_PsiDO}\nFor any weight $\\mu$, $\\Ph{X_i}:\\LXE[\\mu] \\to \\LXE[\\mu-\\alpha_i]$ and $\\Ph{Y_i}:\\LXE[\\mu-\\alpha_i] \\to \\LXE[\\mu]$ are $\\foliation[i]$-longitudinal pseudodifferential operators.  Their longitudinal principal symbols at the identity coset are\n\\begin{eqnarray*}\n  \\Symbol[i] (\\Ph{X_i}) (\\xi) &=& \\Ph{(\\xi(X_i))}, \\\\\n  \\Symbol[i] (\\Ph{Y_i}) (\\xi) &=& \\Ph{(\\xi(Y_i))} \\;=\\; \\overline{\\Ph{(\\xi(X_i))}}.\n\\end{eqnarray*}\nfor $ \\xi$ in the unit sphere of $(\\mathfrak{k}_i\/\\mathfrak{m})^* \\cong (\\foliation[i]^*)_{\\coset{e}}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $E:= E_\\mu\\oplus E_{\\mu-\\alpha_i}$.\nFix $\\epsilon>0$ such that $\\Spec(D_i) \\cap (-\\epsilon,\\epsilon) = \\{0\\}$.  Let $f:\\mathbb{R}\\to[-1,1]$ be smooth with $f(0)=0$ and $f(x) = \\sign(x)$ for all $|x|\\geq\\epsilon$.  \nA fibrewise application of \\cite[Theorem 1.3]{Taylor} shows that $ f(D_i) = \\Ph{D_i} \\in \\PsiDO[i]^0(\\scrX;E)$.  Moreover the proof of the theorem shows that its full symbol has an asymptotic expansion with leading term $f(\\Symbol[i] D_i)$.  Note that\n$$\n  (\\Symbol[i] D_i)(\\xi) = \\smatrix{ 0 & \\xi(X_i) \\\\ \\overline{\\xi(X_i)} &0 }\n$$\nhas spectrum $\\{\\pm | \\xi(X_i) | \\}$, so if $\\xi$ is large enough that $|\\xi(X_i)|>\\epsilon$, then\n$$\n  f (\\Symbol[i] D_i)(\\xi) = \\Ph{(\\Symbol[i] D_i(\\xi))} \n    = \\smatrix{ 0 & \\Ph{(\\xi(X_i))} \\\\ \\Ph{(\\overline{\\xi(X_i)})} &0 }.\n$$\nThis is radially constant on $(\\mathfrak{k}_i\/\\mathfrak{m})^*$ for $|\\xi(X_i)|>\\epsilon$.  The principal symbol is the limit at the sphere at infinity.  \n\\end{proof}\n\n\n\\begin{theorem}\n\\label{prop:PsiDOs_in_K}\n\\label{thm:PsiDOs_in_A}\nLet $E, E'$ be $\\mathsf{K}$-homogeneous vector bundles over $\\scrX$.  Then \n\\begin{enumerate}\n\\item $\\PsiDO[i]^{-1}(E,E') \\subseteq \\scrK[\\alpha_i]$,\n\\item $\\PsiDO[i]^{0}(E,E') \\subseteq \\scrA$,\n\\end{enumerate}\n\\end{theorem}\n\nPart (i) is proven in Proposition 1.12 of \\cite{Yuncken:PsiDOs}.  It is also shown there that $\\PsiDO[i]^0(E,E') \\subseteq \\scrA[i]$.  The more difficult question of showing $\\PsiDO[i]^0(E,E') \\subseteq \\scrA[j]$ for $j\\neq i$ requires some lengthy computations in noncommutative harmonic analysis.  In order not to disrupt the flow of ideas too severely, we have presented the proof in Appendix \\ref{sec:PsiDOs_in_A}.\n\nAs an indication of the subtleties involved, we remark that the longitudinally elliptic differential operator $X_1$ is {\\em not} an unbounded multiplier of $\\scrK[\\alpha_2]$.  To see this, note that $(1+X_1^*X_1)^{-\\frac{1}{2}} \\in \\PsiDO[1]^{-1}(E_\\mu) \\subseteq \\scrK[\\alpha_1]$.  Since $\\scrK[\\alpha_1].\\scrK[\\alpha_2] \\subseteq \\scrK$, the range of $(1+X_1^*X_1)^{-\\frac{1}{2}}$ as a multiplier of $\\scrK[\\alpha_2]$ is not dense.  Thus, $X_1$ is not regular with respect to $\\scrK[\\alpha_2]$ (see \\cite[Chapter 10]{Lance}).  Hence, proving that $\\Ph{X_1}$ multiplies $\\scrK[2]$ can not be achieved by direct functional calculus.\n\n\\begin{lemma}\n\\label{lem:F_f_commute}\nLet $i=1,2$ and let $\\mu, \\nu$ be weights.  For any $f\\in\\CXE[\\nu-\\mu]$, the diagram\n$$ \n  \\xymatrix{\n    \\LXE[\\mu] \\ar[r]^{M_f} \\ar[d]_{\\Ph{X_i}} &\n      \\LXE[\\nu] \\ar[d]^{\\Ph{X_i}} \\\\\n    \\LXE[\\mu-\\alpha_i] \\ar[r]_{M_f} &\n      \\LXE[\\nu-\\alpha_i]\n  }\n$$\ncommutes modulo $\\scrK[\\alpha_i]$.\n\\end{lemma}\n\n\\begin{remark}\n\\label{rem:commutator_notation}\nWe abbreviate this result by writing $[\\Ph{X_i}, M_s] \\in \\scrK[\\alpha_i]$.  By taking adjoints, we also have $[\\Ph{Y_i}, M_s] \\in \\scrK[\\alpha_i]$. \n\\end{remark}\n\n\\begin{proof}\nAs an element of $C(S^*\\foliation[i];E_{\\alpha_i})$, the principal symbol of $\\Ph{X_i}:\\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is independent of the weight $\\mu$.  Thus, the above diagram commutes at the level of principal symbols.\n\\end{proof}\n\n\n\n\n\n\\section{The normalized BGG complex}\n\\label{sec:construction}\n\n\n\\subsection{$\\mathsf{G}$-continuity}\n\\label{sec:G-continuity}\n\nBefore embarking on the main construction, we need to make some remarks regarding the issue of $\\mathsf{G}$-continuity.  Recall that a bounded operator $A$ between unitary $\\mathsf{G}$-representations is $\\mathsf{G}$-continuous if the map $g \\mapsto g .A. g^{-1}$ is continuous in the operator-norm topology. \n\nRather than burden the notation with extra decorations, we choose to make the convention that {\\bf throughout this section, we use $\\scrK[\\alpha_i]$ ($i=1,2$) to denote its $C^*$-subcategory of $\\mathsf{G}$-continuous elements.} \n\nThis is reasonable, since almost every operator we deal with is $\\mathsf{G}$-continuous.  From \\cite{AS4}, we know that for any homogeneous vector bundles $E$, $E'$ over $\\scrX$, the set of longitudinal pseudodifferential operators $\\overline{\\PsiDO[i]^0}(E,E,')$ consists of $\\mathsf{G}$-continuous operators.  This includes continuous multiplication operators, in the sense of Section \\ref{sec:homogeneous_vector_bundles} (which are $\\mathsf{G}$-continuous for much simpler reasons).\nThe notable exceptions, of course, are the representations $U_\\mu(g)$ of the group elements themselves.\n\nIn the majority of instances, where $\\mathsf{G}$-continuity is a trivial consequence of the above remarks, we will not make specific mention of it in the proofs.\n\n\n\\subsection{Intertwining operators}\n\n\n\nLet $\\mu$, $\\mu'$ be weights for $\\mathsf{K}= \\SU(3)$.  It is well known that the principal series representations $U_\\mu$ and $U_{\\mu'}$ are unitarily equivalent if and only if $\\mu'=w\\cdot\\mu$ for some Weyl group element $w\\in W$.  \nWhen $w=\\reflection{\\alpha_i}$ is a simple reflection corresponding to the root $\\alpha_i$, there is a very concise formula for the intertwining operator.  \n\n\n\\begin{proposition}\n\\label{prop:intertwiner_formula}\nLet $\\mu$, $\\mu'$ be weights with $\\mu'=\\reflection{\\alpha_i}\\mu$, so that $\\mu-\\mu' = n\\alpha_i$ for some $n\\in\\mathbb{Z}$.  If $n>0$, the operator $(\\Ph{X_i})^n:  \\LXE[\\mu] \\to \\LXE[\\mu']$ intertwines $U_\\mu$ and $U_\\mu'$.  If $n<0$, then $(\\Ph{Y_i})^n: \\LXE[\\mu] \\to \\LXE[\\mu']$ is an intertwiner.\n\\end{proposition}\n\n\n\nThis is essentially the formula given by Duflo in \\cite[Ch.~III]{Duflo}.  However, Duflo's formulation is sufficiently different that we feel a brief comparison is worthwhile.  \n\n\\begin{proof}\nWe follow the notation for $\\mathfrak{sl}(2,\\mathbb{C})$-representations from the end of Section \\ref{sec:parabolic_subgroups}.  Note that Equations \\eref{eq:X-formula} and \\eref{eq:Y-formula} imply that\n$(\\Ph{X})e_j = e_{j+2}$ and $(\\Ph{Y}) e_j = e_{j-2}$.  Secondly, with $w=\\smatrix{0&-1\\\\1&0}$, \n\\begin{equation}\n\\label{eq:Phase(X)_vs_w}\n\\begin{array}{rcccl}\n  (\\Ph{X})^j \\cdot e_{-j} &=& e_j& =& (-1)^{\\frac{1}{2}(\\delta+j)} w\\cdot e_{-j}, \\\\\n  (\\Ph{Y})^j \\cdot e_{j} &= &e_{-j}& =& (-1)^{\\frac{1}{2}(\\delta-j)} w\\cdot e_{j},\n\\end{array}\n\\end{equation}\nfor any $j\\geq 0$. (See \\cite[\\S{}III.3.5]{Duflo}.)\n\nRecall that the restriction of $\\mu$ to a weight of $\\mathfrak{s}_i$ is $\\mu_i := \\mu(H_i)\\in\\mathbb{Z}$.  The hypotheses of the proposition are equivalent to saying $\\mu_i = -\\mu'_i = n$.\n\nFirst consider the case $n>0$.  Let $A=A(w_i, \\mu,0):\\LXE[\\mu] \\to \\LXE[\\mu']$ be the intertwiner of \\cite[\\S{}III.3.1]{Duflo}.  The action of $A$ upon matrix units is given in \\cite[\\S{}III.3.3 and \\S{}III.3.9]{Duflo} as follows.  Let $\\pi\\in\\irrep{K}$, $\\eta^\\dagger\\in V^{\\pi\\dagger}$, $\\xi\\in p_{-\\mu}(V^\\pi)$ and suppose that $\\xi$ lies in an irreducible $\\mathfrak{s}_i$-subrepresentation of $V^\\pi$ with highest weight $\\delta$.  Then, in the notation of Section \\ref{sec:harmonic_notation}, $A:\\matrixunit{\\eta^\\dagger}{\\xi} \\mapsto \\matrixunit{\\eta^\\dagger}{\\xi'}$ where\n\\begin{eqnarray*}\n\\label{eq:Duflo-formula}\n  \\xi' &=& (-1)^{\\frac{1}{2}(\\delta + |\\mu_i|)} |\\mu_i|^{-1} \\pi(w_i)\\xi \\\\\n    &=& |\\mu_i|^{-1} (\\Ph{X_i})^n \\xi.\n\\end{eqnarray*}\nHence, $A=|\\mu_i|^{-1} (\\Ph{X_i})^n:\\LXE[\\mu] \\to \\LXE[\\mu']$, where $X_i$ here denotes the right regular action.  Thus, $(\\Ph{X_i})^n$ differs from $A$ by the positive scalar $|\\mu_i|=n$.\n\nThe case $n<0$ follows since $\\Ph{Y_i} = \\Ph{X_i}^*$.\n\n\\end{proof}\n\n\n\n\\medskip\n\nWe now recap the directed graph structure which underlies the BGG complex.  For our $K$-homological purposes, it will be convenient to make an undirected graph, or more accurately, to include also the reversal of each edge.  \n\nAs before, if $\\alpha$ is a positive root, we use $\\reflection{\\alpha}\\in\\Lie{W}$ to denote the reflection in the wall orthogonal to $\\alpha$.  For $w,w'\\in\\Lie{W}$, we write $\\edge[\\alpha]{w}{w'}$ if $w'=\\reflection{\\alpha}w$ and $l(w') = l(w) \\pm 1$.  We will write $\\edge{w}{w'}$ if $\\edge[\\alpha]{w}{w'}$ for some $\\alpha\\in\\Delta^+$.  An edge $\\edge[\\alpha]{w}{w'}$ will be called  {\\em simple} if $\\alpha$ is a simple root.\n\nFor $\\mathsf{G}=\\SL(3,\\mathbb{C})$, this yields the graph\n\\begin{equation}\n\\label{eq:Weyl_graph}\n  \\xymatrix{\n    & \\stackrel{\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[rr]^{\\rho} \\ar@{<->}[ddrr]^(0.7){\\alpha_2} \n      && \\stackrel{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[dr]^{\\alpha_2} \\\\\n    \\stackrel{1}{\\bullet} \\ar@{<->}[ur]^{\\alpha_1} \\ar@{<->}[dr]_{\\alpha_2} \n      &&&& \\stackrel{w_\\rho}{\\bullet} \\\\\n    & \\stackrel{\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[rr]_{\\rho} \\ar@{<->}[uurr]_(0.7){\\alpha_1} \n      && \\stackrel{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[ur]_{\\alpha_1} \n  }\n\\end{equation}\n\n\n\n\n\n\\begin{definition}\n\\label{def:intertwiners}\nFix a dominant weight $\\lambda$.  If $\\edge[\\alpha_i]{w}{w'}$ is a simple edge, we denote by $\\intertwiner{\\lambda}{w}{w'}$ the intertwining operator of Lemma \\ref{prop:intertwiner_formula}:\n$$\n  \\intertwiner{\\lambda}{w}{w'} := \n    \\begin{cases} \n      (\\Ph{X_i})^n  & \\text{if $n\\geq0$},\\\\\n      (\\Ph{Y_i})^{-n}  & \\text{if $n\\leq0$},\n    \\end{cases}      \n$$\nwhere $w\\lambda - w'\\lambda = n\\alpha_i$.\nThese will be referred to as {\\em simple intertwiners}.  Note that $\\intertwiner{\\lambda}{w'}{w} = \\intertwiner{\\lambda}{w}{w'}^*$.\n\nFor the non-simple edges, we define intertwiners as compositions of simple intertwiners:\n\\begin{eqnarray}\n  \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} &:=& \n    \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n    \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n    \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\nonumber \\\\\n  \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}} &:=& \n    \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}.\n    \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_1}}.\n    \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{1}, \n  \\label{eq:non-simple_intertwiners}\n\\end{eqnarray}\nand \n$\\intertwiner{\\lambda}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}} := \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}^*$, $\\intertwiner{\\lambda}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}} := \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}^*$.\n\\end{definition}\n\n\\begin{remark}\n\\label{rmk:commuting_diagram_of_intertwiners}\nDuflo's intertwiners form a commuting diagram of the form\n\\begin{equation}\n\\label{eq:simple_intertwiners}\n  \\xymatrix@!C=5ex{\n    & \\LXE[\\reflection{\\alpha_1}\\lambda] \\ar[ddrr] && \\LXE[\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\lambda] \\ar[dr] \\\\\n   \\LXE[\\lambda] \\ar[ur] \\ar[dr] &&&& \\LXE[w_\\rho\\lambda] \\\\\n    & \\LXE[\\reflection{\\alpha_2}\\lambda] \\ar[uurr] && \\LXE[\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\lambda] \\ar[ur] \n  }\n\\end{equation}\nSince the simple intertwiners $\\intertwiner{\\lambda}{w}{w'}$ defined here are positive scalar multiples of Duflo's, \nthe corresponding diagram of intertwiners $\\intertwiner{\\lambda}{w}{w'}$ commutes up to some positive scalar. But $\\intertwiner{\\lambda}{w}{w'} = (\\Ph{X_i})^n$ is unitary, so that scalar is $1$.  The non-simple intertwiners defined by Equation \\eref{eq:non-simple_intertwiners} are precisely those that complete \\eref{eq:simple_intertwiners} to a commuting diagram of the form \\eref{eq:Weyl_graph}.\n\\end{remark}\n\n\n\\begin{definition}\nDefine $\\scrK[\\rho] := \\scrK[\\alpha_1] + \\scrK[\\alpha_2]$.  That is, $\\scrK[\\rho](H,H') := \\scrK[\\alpha_1] (H,H') + \\scrK[\\alpha_2] (H,H') $ for any harmonic $\\mathsf{K}$-spaces $H$, $H'$.  Following the convention of Section \\ref{sec:G-continuity}, we are including the condition of $\\mathsf{G}$-continuity in this definition.\n\n\\end{definition}\n\n\n\\begin{lemma}\n\\label{lem:intertwiners_in_A}\nLet $\\lambda$ be a dominant weight.\n\\begin{enumerate}\n\\item  For each $\\edge{w}{w'}$, $\\intertwiner{\\lambda}{w}{w'} \\in \\scrA$.\n\\item  If $\\edge[\\alpha]{w}{w'}$, then $[\\intertwiner{\\lambda}{w}{w'}, \\multop{f}] \\in \\scrK[\\alpha]$ for any $f\\in C(\\scrX)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nPart {(i)} is immediate from Theorem \\ref{thm:PsiDOs_in_A}.  If $\\alpha$ is a simple root, then {(ii)} follows from Lemma \\ref{lem:F_f_commute}.  For $\\alpha=\\rho$, there are four intertwiners to be checked.  The following calculation is representative of all of them:\n\\begin{eqnarray*}\n[\\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} , \\multop{f}] &=& \n  [\\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}, \\multop{f}].\n    \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n    \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n  &&\\quad + \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n   [ \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}, \\multop{f} ].\n    \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n  &&\\quad +  \\intertwiner{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n    \\intertwiner{\\lambda}{1}{\\reflection{\\alpha_2}}.\n    [ \\intertwiner{\\lambda}{\\reflection{\\alpha_1}}{1}, \\multop{f} ] . \\\\\n  &\\in& \\scrK[\\alpha_1] + \\scrK[\\alpha_2] + \\scrK[\\alpha_1] = \\scrK[\\rho].\n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\\subsection{Normalized BGG operators}\n\n\n\n\\begin{definition}\n\\label{def:shifted_action}\nDefine the {\\em shifted action} of the Weyl group on weights by $w\\star \\mu := w(\\mu+\\rho) -\\rho$.\n\\end{definition}\n\nFrom now on, $\\lambda$ will denote a dominant weight.\n\n\\begin{definition}\n\\label{def:BGG_operators}\nIf $\\edge[\\alpha_i]{w}{w'}$ is a simple edge, then $w\\star \\lambda - w'\\star \\lambda = n\\alpha_i$ for some $n\\in\\mathbb{Z}$.  We define the {\\em normalized BGG operator} $\\BGG{\\lambda}{w}{w'} : \\LXE[w\\star \\lambda] \\to \\LXE[w'\\star \\lambda]$ by\n$$\n  \\BGG{\\lambda}{w}{w'} := \n    \\begin{cases} \n      (\\Ph{X_i})^n  & \\text{if $n\\geq0$},\\\\\n      (\\Ph{Y_i})^{-n}  & \\text{if $n\\leq0$}.\n    \\end{cases}      \n$$\nwhere $w\\star\\lambda-w'\\star\\lambda = n \\alpha_i$.\n\nFor the non-simple arrows, define\n\\begin{eqnarray*}\n  \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}} &:=& \n    \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}.\n    \\BGG{\\lambda}{1}{\\reflection{\\alpha_2}}.\n    \\BGG{\\lambda}{\\reflection{\\alpha_1}}{1} \\\\\n  \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}} &:=& \n    \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}.\n    \\BGG{\\lambda}{1}{\\reflection{\\alpha_1}}.\n    \\BGG{\\lambda}{\\reflection{\\alpha_2}}{1} \\\\\n  \\BGG{\\lambda}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\reflection{\\alpha_1}} &:=& \\BGG{\\lambda}{\\reflection{\\alpha_1}}{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}^* \\\\  \n  \\BGG{\\lambda}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\reflection{\\alpha_2}} &:=& \\BGG{\\lambda}{\\reflection{\\alpha_2}}{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}^*.\n\\end{eqnarray*}\n\\end{definition}\n\nObviously, the definitions of the normalized BGG operators $\\BGG{\\lambda}{w}{w'}$ are identical to the definitions of the intertwining operators $\\intertwiner{\\lambda+\\rho}{w}{w'}$, except that the weights of the principal series representations on which they act differ by the shift of $\\rho$.  The next few lemmas describe the consequences of this.  To begin with, we have an exact analogue of Lemma \\ref{lem:intertwiners_in_A}, with essentially identical proof.\n\n\\begin{lemma}\n\\label{lem:BGG_in_A}\nLet $\\lambda$ be a dominant weight.\n\\begin{enumerate}\n\\item  For each arrow $\\edge{w}{w'}$, $\\BGG{\\lambda}{w}{w'} \\in \\scrA$.\n\\item  If $\\edge[\\alpha]{w}{w'}$, then $[\\BGG{\\lambda}{w}{w'}, \\multop{f}] \\in \\scrK[\\alpha]$ for any $f\\in C(\\scrX)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:po1_definition}\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be such that $\\sum_{j=1}^k |\\varphi_j|^2 =1$, as in Lemma \\ref{lem:partition_of_unity}.  If $\\edge[\\alpha]{w}{w'}$, then\n$$\n  \\BGG{\\lambda}{w}{w'}   \n    \\equiv \\sum_{j=1}^k \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_j} \n    \\pmod{\\scrK[\\alpha]}. \n$$\n\\end{lemma}\n\n\\begin{proof}\nThis is an immediate consequence of Lemma \\ref{lem:F_f_commute}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:F2-1_in_K}\nIf $\\edge[\\alpha]{w}{w'}$, then  $\\BGG{\\lambda}{w'}{w} \\BGG{\\lambda}{w}{w'}-1 \\in \\scrK[\\alpha]$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be as in the previous lemma.  By Lemmas \\ref{lem:po1_definition} and \\ref{lem:intertwiners_in_A},\n\\begin{eqnarray*}\n  \\BGG{\\lambda}{w'}{w} \\BGG{\\lambda}{w}{w'} \n   &\\equiv& \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w'}{w'} \\multop{\\varphi_j \\overline{\\varphi_{j'}}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_{j'}} \\pmod{\\scrK[\\alpha]}\\\\\n   &\\equiv& \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w'}{w} \\intertwiner{\\lambda+\\rho}{w}{w'}  \\multop{\\varphi_j \\overline{\\varphi_{j'}}}\\multop{\\varphi_{j'}} \\pmod{\\scrK[\\alpha]}\\\\\n   &=& \\sum_{j,j'} \\multop{\\overline{\\varphi_j} \\varphi_j \\overline{\\varphi_{j'}} \\varphi_{j'}} \\\\\n   &=& 1.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:diagram_commutes}\nThe diagram of normalized BGG operators\n\\begin{equation}\n\\label{eq:BGG_diagram}\n  \\xymatrix@!C=5ex{\n    & \\LXE[\\reflection{\\alpha_1}\\star\\lambda] \\ar@{<->}[ddrr] \\ar@{<->}[rr] && \\LXE[\\reflection{\\alpha_1}\\reflection{\\alpha_2}\\star\\lambda] \\ar@{<->}[dr] \\\\\n   \\LXE[\\lambda] \\ar@{<->}[ur] \\ar@{<->}[dr] &&&& \\LXE[w_\\rho\\star\\lambda] \\\\\n    & \\LXE[\\reflection{\\alpha_2}\\star\\lambda] \\ar@{<->}[uurr] \\ar@{<->}[rr] && \\LXE[\\reflection{\\alpha_2}\\reflection{\\alpha_1}\\star\\lambda] \\ar@{<->}[ur] \n  }\n\\end{equation}\ncommutes modulo $\\scrK[\\rho]$.\n\\end{lemma}\n\n\\begin{proof}\nFor adjacent edges $w\\xleftrightarrow{\\alpha} w' \\xleftrightarrow{\\alpha'} w''$, a calculation analogous to that of the previous proof gives\n\\begin{equation*}\n  \\BGG{\\lambda}{w'}{w''} \\BGG{\\lambda}{w}{w'}\n   \\equiv \\sum_{j,j'} \\multop{\\overline{\\varphi_j}} \\left(\\intertwiner{\\lambda+\\rho}{w'}{w''} \\intertwiner{\\lambda+\\rho}{w}{w'}\\right) \\multop{\\varphi_{j} \\overline{\\varphi_{j'}}}\\multop{\\varphi_{j'}} \n   \\pmod{\\scrK[\\alpha']}.\n\\end{equation*}\nNote that $\\scrK[\\alpha']\\subseteq\\scrK[\\rho]$.  The commutativity of \\eref{eq:BGG_diagram} modulo $\\scrK[\\rho]$ is therefore a consequence of the commutativity of the corresponding diagram of intertwiners $\\intertwiner{\\lambda+\\rho}{w}{w'}$ (Remark \\ref{rmk:commuting_diagram_of_intertwiners}).\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:F_g_commute}\nLet $\\edge[\\alpha]{w}{w'}$.  For any $g\\in G$, \n\\begin{equation}\n\\label{eq:F_g_commute}\n  U_{w'\\star\\lambda}(g) \\BGG{\\lambda}{w}{w'} U_{w\\star\\lambda}(g^{-1}) - \\BGG{\\lambda}{w}{w'} \\; \\in \\;\n      \\scrK[\\alpha] \n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{proof}\nWe first note that if $A$ is a $\\mathsf{G}$-continuous operator, then so is $g.A.g^{-1}$.\nLet $\\varphi_1,\\ldots,\\varphi_k\\in\\CXE[\\rho]$ be as in Lemma \\ref{lem:po1_definition}.  Then,\n\\begin{eqnarray}\n\\lefteqn{U_{w'\\star\\lambda}(g) \\BGG{\\lambda}{w}{w'} U_{w\\star\\lambda}(g^{-1})} \\qquad \\nonumber\\\\\n &\\equiv& \\sum_{j} U_{w'\\star\\lambda}(g) \\multop{\\overline{\\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{\\varphi_j} U_{w\\star\\lambda}(g^{-1}) \\nonumber \\pmod{\\scrK[\\alpha]}  \\\\\n &=& \\sum_{j} U_{w'\\star\\lambda}(g) \\multop{\\overline{\\varphi_j}}U_{w'(\\lambda+\\rho)}(g^{-1}) \\intertwiner{\\lambda+\\rho}{w}{w'} U_{w(\\lambda+\\rho)}(g) \\multop{\\varphi_j} U_{w\\star\\lambda}(g^{-1}) \\nonumber \\\\\n &=& \\sum_{j} \\multop{\\overline{g\\cdot \\varphi_j}} \\intertwiner{\\lambda+\\rho}{w}{w'} \\multop{g\\cdot \\varphi_j}.\n \\label{eq:F_g_commute_computation}\n\\end{eqnarray}\nSince $\\sum_{j=1}^k |g\\cdot \\varphi_j|^2 =  1$, Lemma \\ref{lem:po1_definition} shows that \\eref{eq:F_g_commute_computation} equals $\\BGG{\\lambda}{w}{w'}$ modulo $\\scrK[\\alpha]$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Construction of the gamma element}\n\\label{sec:gamma}\n\nFix a dominant weight $\\lambda$.  Let $H_\\lambda := \\bigoplus_{w\\in\\Lie{W}} \\LXE[w\\star \\lambda]$.  For each $w\\in\\Lie{W}$, let $\\component{w}$ denote the orthogonal projection onto the summand $\\LXE[w \\star \\lambda]$ of $H_\\lambda$.  We put a grading on $H_\\lambda$ by declaring $\\LXE[w\\star \\lambda]$ to be even or odd according to the parity of $l(w)$.  \n\nFor $f\\in C(\\scrX)$, $M_f$ will denote the multiplication operator on $H_\\lambda$, acting diagonally on the summands.  We let  $U$ denote the diagonal representation $\\oplus_{w\\in\\Lie{W}} U_{w\\star\\lambda}$ of $\\mathsf{G}$.  For each $\\edge{w}{w'}$, we extend the normalized BGG operator $\\BGG{\\lambda}{w}{w'}:\\LXE[w\\star\\lambda] \\to \\LXE[w'\\star\\lambda]$ to an operator $\\BGGextended{\\lambda}{w}{w'}:H_\\lambda \\to H_\\lambda$ by defining it to be zero on the components $\\LXE[w''\\star \\lambda]$ with $w''\\neq w$. \n\nFor the remainder of this section, we use $\\scrK[\\alpha]$, $\\scrA$, $\\scrK$, $\\scr{L}$ to denote $\\scrK[\\alpha](H_\\lambda)$, $\\scrA (H_\\lambda) $, $\\scrK (H_\\lambda) $, $\\scr{L} (H_\\lambda)$.\n\n\n\\begin{lemma}[Kasparov Technical Theorem]\n\\label{lem:KTT}\nThere exist positive $\\mathsf{G}$-continuous operators $N_1,N_2\\in \\scr{L}$ with the following properties:\n\\begin{enumerate}\n\\item $N_1^2+N_2^2 = 1$,\n\\item $N_i \\cdot\\scrK[\\alpha_i] \\subseteq \\scrK$ for each $i=1,2$,\n\\item $N_i$ commutes modulo compact operators with\n\\begin{itemize}\n  \\item $\\multop{f}$ for all $f\\in C(\\scrX)$,\n  \\item $U(g)$ for all $g\\in\\mathsf{G}$,\n  \\item the normalized BGG operators $\\BGGextended{\\lambda}{w}{w'}$, for all $\\edge{w}{w'}$,\n\\end{itemize}\n\\item $N_i$ commutes on the nose with $U(k)$ for all $k\\in\\mathsf{K}$,\n\\item $N_i$ commutes on the nose with the projections $\\component{w}$  for all $w\\in\\Lie{W}$, {\\em i.e.}, $N_i$ is diagonal with respect to the direct sum decomposition of $H_\\lambda$.\n\\end{enumerate}\n\\end{lemma}\n\nNote also that  $N_1$ and $N_2$ commute, by \\emph{(i)}.\n\n\\begin{proof}\nSee \\cite[Theorem 20.1.5]{Blackadar}.  The $\\mathsf{K}$-invariance of (iv) is obtained by averaging over the $\\mathsf{K}$-translates $U(k)\\, N_i\\, U(k^{-1})$ of $N_i$.  Also, the operators $\\sum_w \\pm Q_w$ (taking all possible choices of signs) form a finite group of unitaries, so that a similar averaging trick gives property (v).\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\label{lem:operator_po1}\nThere exist mutually commuting operators $\\poI{w}{w'} \\in \\scr{L}$, indexed by the edges of the graph \\eref{eq:Weyl_graph}, with the following properties:\n\\begin{enumerate}\n\\item $\\poI{w}{w'} = \\poI{w'}{w}$\n\\item If $\\edge[\\alpha]{w}{w'}$ for $\\alpha\\in\\{\\alpha_1,\\alpha_2,\\rho\\}$, then  $\\poI{w}{w'} \\scrK[\\alpha] \\subseteq \\scrK$.\n\\item If $w\\leftrightarrow w' \\leftrightarrow w''$ with $w\\neq w''$ then $\\poI{w'}{w''}\\poI{w}{w'} \\scrK[\\rho] \\subseteq \\scrK$.\n\\item For any $w,w''\\in\\Lie{W}$, $\\sum_{w'} \\poI{w'}{w''}\\poI{w}{w'} =\\delta_{w,w''}$, where the sum is over $w'$ such that $w \\leftrightarrow w' \\leftrightarrow w''$.\n\\item $\\poI{w}{w'}$ satisfies {(iii)}, {(iv)} and{(v)} of Lemma \\ref{lem:KTT}.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{remark}\nTo clarify a possibly misleading notational point, $\\poI{w}{w'}$ does not designate an operator between $\\LXE[w\\star\\lambda]$ and $\\LXE[w'\\star\\lambda]$.  Rather it is an operator on $H_\\lambda$ which we will use to modify the operator $\\BGG{\\lambda}{w}{w'}$.\n\\end{remark}\n\n\n\\begin{proof}\nWith $N_1, N_2$ as in the previous lemma, assign operators $\\poI{w}{w'}$ to each arrow as follows:\n$$\n  \\xymatrix@!C{\n    & \\stackrel{\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[rr]^{-N_1N_2} \\ar@{<->}[ddrr]^(0.7){-N_2^2} \n      && \\stackrel{\\reflection{\\alpha_1}\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[dr]^{-N_2} \\\\\n    \\stackrel{1}{\\bullet} \\ar@{<->}[ur]^{N_1} \\ar@{<->}[dr]_{N_2} \n      &&&& \\stackrel{w_\\rho}{\\bullet} \\\\\n    & \\stackrel{\\reflection{\\alpha_2}}{\\bullet} \\ar@{<->}[rr]_{N_1N_2} \\ar@{<->}[uurr]_(0.7){N_1^2} \n      && \\stackrel{\\reflection{\\alpha_2}\\reflection{\\alpha_1}}{\\bullet} \\ar@{<->}[ur]_{N_1} \n  }\n$$\nThe asserted properties can be easily checked using the properties of $N_1$ and $N_2$ from Lemma \\ref{lem:KTT} and the diagram \\eref{eq:Weyl_graph}.  It is worth noting particularly that $N_1N_2$ multiplies $\\scrK[\\rho]$ into the compact operators. \n\\end{proof}\n\n\\begin{definition}\n\\label{def:Fredholm_components}\nDefine $\\poIBGG{\\lambda}{w}{w'} := \\poI{w}{w'} \\BGGextended{\\lambda}{w}{w'}$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:compact_commutators}\nFor any $\\edge{w}{w'}$,\n\\begin{enumerate}\n\\item $\\poIBGG{\\lambda}{w}{w'} - \\poIBGG{\\lambda}{w'}{w}^*\\in \\scrK$.\n\\item $[\\poIBGG{\\lambda}{w}{w'}, M_f] \\in \\scrK$, for any $f\\in C(\\scrX)$,\n\\item $U(g)\\,\\poIBGG{\\lambda}{w}{w'}\\,U(g^{-1}) - \\poIBGG{\\lambda}{w}{w'} \\in \\scrK$, for any $g\\in\\mathsf{G}$,\n\\item $\\poIBGG{\\lambda}{w}{w'}$ is $\\mathsf{K}$-invariant, {\\em ie}, $[\\poIBGG{\\lambda}{w}{w'}, U(k)] =0$, for any $k\\in\\mathsf{K}$,\n\\item $\\poIBGG{\\lambda}{w}{w'}$ is $\\mathsf{G}$-continuous.\n\\end{enumerate}\nAlso,\n\\begin{enumerate}\n\\setcounter{enumi}{5}\n\\item For any $w,w''\\in\\Lie{W}$, $\\left( \\sum_{w'} \\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w}{w'} \\right) \\equiv \\delta_{w,w''} \\component{w} \\pmod{\\scrK}$, where the sum is over $w'\\in\\Lie{W}$ such that $w\\leftrightarrow w' \\leftrightarrow w''$.\n\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\edge[\\alpha]{w}{w'}$. By definition, $\\BGG{\\lambda}{w}{w'} = \\BGG{\\lambda}{w'}{w}^*$, so $\\poIBGG{\\lambda}{w}{w'} - \\poIBGG{\\lambda}{w'}{w}^* = [\\poI{w}{w'},\\BGGextended{\\lambda}{w}{w'}]$, which proves (i).  \n\nSince $\\poI{w}{w'}$ commutes modulo compacts with multiplication operators,\n$$\n  [ \\poIBGG{\\lambda}{w}{w'} , \\multop{f}]\n   \\equiv \\poI{w}{w'} [\\BGGextended{\\lambda}{w}{w'}, \\multop{f}] \\pmod{\\scrK} \n$$\nBy Lemma \\ref{lem:BGG_in_A}, the latter is in $\\poI{w}{w'}\\scrK[\\alpha] \\subseteq \\scrK$, which proves (ii).  Similarly, for (iii),\n\\begin{multline*}\n  U(g)\\,\\poIBGG{\\lambda}{w}{w'}\\,U(g^{-1}) - \\poIBGG{\\lambda}{w}{w'} \\\\\n    \\equiv \\poI{w}{w'} \\big( U(g)\\,\\BGGextended{\\lambda}{w}{w'}\\,U(g^{-1}) - \\BGGextended{\\lambda}{w}{w'} \\big)\\pmod{\\scrK}\n\\end{multline*}\nand the latter is in $\\poI{w}{w'} \\scrK[\\alpha] \\subseteq \\scrK$ by Lemma \\ref{lem:F_g_commute}.\n\n\nFor any weight $\\mu$, the differential operator $X_i: \\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is $\\mathsf{K}$-invariant.  Likewise for its essential adjoint $Y_i:\\LXE[\\mu-\\alpha_i] \\to \\LXE[\\mu]$. Hence, $\\Ph{X_i}:\\LXE[\\mu]\\to\\LXE[\\mu-\\alpha_i]$ is $\\mathsf{K}$-equivariant.  The normalized BGG operators $\\BGG{\\lambda}{w}{w'}$ are compositions of such operators, and $\\poI{\\lambda}{w}{w'}$ is $\\mathsf{K}$-invariant by definition.  This proves (iv).\n\nOnce again, $\\mathsf{G}$-continuity is trivial.\n\nWe prove (vi) in two separate cases.  Firstly, suppose $w=w''$.  For any $w'$ with $\\edge{w}{w'}$, Lemma \\ref{lem:F2-1_in_K} implies that $\\BGGextended{\\lambda}{w'}{w}\\BGGextended{\\lambda}{w}{w'} \\equiv \\component{w} \\pmod{\\scrK[\\alpha]}$.  By Lemma \\ref{lem:operator_po1}(iv),\n\\begin{eqnarray*}\n  \\sum_{w'} \\poIBGG{\\lambda}{w'}{w} \\poIBGG{\\lambda}{w}{w'}\n    &\\equiv& \\sum_{w'} \\poI{w'}{w}\\poI{w}{w'} \\BGGextended{\\lambda}{w'}{w}\\BGGextended{\\lambda}{w}{w'}\n      \\pmod{\\scrK} \\\\\n    &\\equiv& \\sum_{w'} \\poI{w'}{w}\\poI{w}{w'} \\component{w} \\pmod{\\scrK} \\\\\n    &=& \\component{w}.\n\\end{eqnarray*}\nIf $w\\neq w'$, the result is trivial unless there exists at least one $w'$ such that $w \\leftrightarrow w' \\leftrightarrow w''$.  If such a $w'$ exists, Lemma \\ref{lem:diagram_commutes} implies that the products $\\BGG{\\lambda}{w'}{w''} \\BGG{\\lambda}{w}{w'}$ are independent of this intermediate vertex $w'$, modulo $\\scrK[\\rho]$.  Let us fix one such product and denote it temporarily by $\\BGG{\\lambda}{w\\to\\cdot}{w''}$.  Then by Lemma \\ref{lem:operator_po1}(iv),\n\\begin{eqnarray*}\n  \\sum_{w'} \\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w}{w'}\n    &\\equiv& \\sum_{w'} \\poI{w'}{w''}\\poI{w}{w'} \\BGGextended{\\lambda}{w'}{w''}\\BGGextended{\\lambda}{w}{w'}\n      \\pmod{\\scrK} \\\\\n    &\\equiv& \\left(\\sum_{w'} \\poI{w'}{w}\\poI{w}{w'}\\right) \\BGGextended{\\lambda}{w\\to \\cdot}{w''}  \\pmod{\\scrK} \\\\\n    &=& 0.\n\\end{eqnarray*}\n\n\n\\end{proof}\n\n\n\\begin{definition}\nDefine $\\Fredholm{\\lambda} :=\\sum \\poIBGG{\\lambda}{w}{w'}$, where the sum is over all directed edges in the graph \\eref{eq:Weyl_graph}.\n\\end{definition}\n\n\n\\begin{theorem}\n\\label{thm:main_theorem}\nThe operator $\\Fredholm{\\lambda}\\in \\scr{L}$ defines an element $\\Kcycle{\\lambda} \\in K^\\mathsf{G}(C(\\scrX),\\mathbb{C})$.  That is, \n\\begin{enumerate}\n\\item $\\Fredholm{\\lambda}$ is {\\em odd} with respect to the grading of $H_\\lambda$,\n\\item $\\Fredholm{\\lambda} - \\Fredholm{\\lambda}^* \\in \\scrK$,\n\\item $\\Fredholm{\\lambda}^2 - 1 \\in \\scrK$,\n\\item $[\\Fredholm{\\lambda}, M_f] \\in \\scrK$, for any $f\\in C(\\scrX)$,\n\\item $[\\Fredholm{\\lambda}, U(g)] \\in \\scrK$, for any $g\\in\\mathsf{G}$,\n\\item $\\Fredholm{\\lambda}$ is $\\mathsf{G}$-continuous,\n\\end{enumerate}\nMoreover, $\\Fredholm{\\lambda}$ is $\\mathsf{K}$-invariant: $[\\Fredholm{\\lambda},U(k)] = 0$ for all $k\\in\\mathsf{K}$.\n\\end{theorem}\n\n\\begin{proof}\nThis is mostly immediate from the previous lemma.  To be explicit about the proof of (iii), Lemma \\ref{lem:compact_commutators}(vi) gives\n\\begin{eqnarray*}\n  \\Fredholm{\\lambda}^2 \n    &=& \\!\\!\\sum_{w,w',w''\\in\\Lie{W} \\atop w\\leftrightarrow w' \\leftrightarrow w''} \\!\\!\\poIBGG{\\lambda}{w'}{w''} \\poIBGG{\\lambda}{w'}{w''} \\\\\n    &\\equiv& \\sum_{w} \\component{w}  \\pmod{\\scrK} \\\\\n    &=& 1.\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{definition}\nLet $\\pi_\\lambda$ denote the irreducible representation of $\\mathsf{K}$ with highest weight $\\lambda$.  Define a homomorphism of abelian groups \n\\begin{eqnarray*}\n  \\theta:R(\\mathsf{K}) &\\to& KK^\\mathsf{G}(C(\\scrX),\\mathbb{C}) \\\\ \n     ~ [\\pi_\\lambda] &\\mapsto& \\Kcycle{\\lambda}.\n\\end{eqnarray*}\n\\end{definition}\n\nLet $\\iota:\\mathbb{C}\\to C(\\scrX)$ denote the $G$-equivariant $C^*$-morphism induced by the map of $\\scrX$ to a point.\n\n\\begin{theorem}\n\\label{thm:splitting}\nThe map $\\iota^*\\circ\\theta:R(\\mathsf{K})\\to  R(\\mathsf{G})$ is a ring homomorphism which splits the restriction homomorphism $\\Res_\\mathsf{K}^\\mathsf{G}:R(\\mathsf{G}) \\to R(\\mathsf{K})$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\lambda$ be a dominant weight.  We have that $\\Res^\\mathsf{G}_\\mathsf{K}\\iota^*\\circ\\theta([\\pi_\\lambda])$ is the $\\mathsf{K}$-index of $\\Fredholm{\\lambda}$.  Since $\\Fredholm{\\lambda}$ is $\\mathsf{K}$-equivariant, it decomposes as a direct sum of operators on the $\\mathsf{K}$-isotypical subspaces of $H_\\lambda$, each of which is finite dimensional (Example \\ref{ex:finite_multiplicities}).  The $\\mathsf{K}$-index of $\\Fredholm{\\lambda}$ is the sum of the indices of each component.\n\nTo compute this index, we compare with the classical BGG complex.  Let $\\mu:= w\\star\\lambda$ be in the shifted Weyl orbit of $\\lambda$.  The induced bundle $E_\\mu$ of our normalized $BGG$-complex and the holomorphic bundle $\\Lhol{\\mu}$ of the classical BGG complex \\eref{eq:BGG_resolution} are identical as $\\mathsf{K}$-homogeneous line bundles.  The classical BGG resolution  is exact and $\\mathsf{K}$-equivariant, so exact in each $\\mathsf{K}$-type.  It follows that the index of $\\Fredholm{\\lambda}$ is $[\\pi_\\lambda]$.  Thus the composition $\\Res_\\mathsf{K}^\\mathsf{G}\\circ\\iota^*\\circ\\theta$ is the identity on $R(\\mathsf{K})$.\n\nBy Theorem \\ref{thm:split_surjection}, $\\Res_\\mathsf{K}^\\mathsf{G}: \\gamma R(\\mathsf{G}) \\to R(\\mathsf{K})$ is a ring isomorphism, so it suffices to show that the image of $\\iota^*\\theta$ is in $\\gamma R(\\mathsf{G})$.  Using \\cite[Theorem 3.6(1)]{Kas88}, we have\n\\begin{eqnarray*}\n\\gamma \\cdot (\\iota^*\\Kcycle{\\lambda}) \n  &=& \\iota^* \\otimes_{C(\\mathsf{G\/B})} (1_{C(\\mathsf{G\/B})}\\otimes\\gamma)\n    \\otimes_{C(\\mathsf{G\/B})} \\Kcycle{\\lambda} \\\\\n  &=& \\iota^* \\otimes_{C(\\mathsf{G\/B})} (\\Ind_\\mathsf{B}^\\mathsf{G} \\Res_\\mathsf{B}^\\mathsf{G}\\gamma) \\otimes_{C(\\mathsf{G\/B})} \\Kcycle{\\lambda}. \n\\end{eqnarray*}\nSince $\\mathsf{B}$ is amenable, $\\gamma$ restricts to the unit in $R(\\mathsf{B})$, so  $\\gamma \\cdot (\\iota^* \\Kcycle{\\lambda}) = \\iota^* \\Kcycle{\\lambda}$.\n\n\\end{proof}\n\n\n\n\n\\begin{corollary}\n\\label{cor:gamma}\n $\\gamma = [(H_0, U, \\Fredholm{0})] \\in R(\\mathsf{G})$.\n\\end{corollary}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}