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+{"text":"\\section{Introduction}\n\nRandom mixtures of several phases are common in nature. They appear, for example, near first order phase transitions,\nin percolating systems and when competing short and long-range interactions are present \\cite{Seul}. Here, we are concerned\nwith the situation where the inhomogeneity occurs on sufficiently large scales, such that each phase can be\ncharacterized by its own bulk kinetic properties, \\eg, its electrical and thermal conductivities. Typically,\nthe calculation of the response of the entire system is intractable due to the complicated distribution of\ncurrents in the sample. However, an exact analytical solution to this problem exists for two-dimensional systems\ncomprised of two phases with statistically identical distributions, as occurs at the percolation transition point.\nThe solution relies on the existence of a self-duality transformation, which interchanges the roles of the currents\nand the driving fields. The duality can be realized provided that the fields are potential gradients and the currents\nare conserved.\n\nThe aforementioned approach is originally due to Dykhne, who applied it to the case of a single (electrical) current\n\\cite{Dykhne71}. Subsequently, it was generalized to include the effects of a magnetic field\n\\cite{Dykhnemag,Shklovskii,Milton,DR1994,Balagurov-galvano95},\nand to the case of two conserved currents (electrical and heat) but without a magnetic field \\cite{Balagurov-reciprocity}.\n\nIn this paper we present several new exact solutions to the problem. First, we study a time-reversal symmetric system with\nthree conserved currents. Such a setting applies when measuring drag in a bilayer of electrically isolated films that are\nstrongly coupled thermally. Second, we provide a solution to the case of two conserved currents in the presence of a magnetic\nfield, $H$. Finally, we return to the bilayer problem and solve it perturbatively in $H$.\n\n\n\\section{Statement of the Problem}\n\n\nConsider a two-dimensional system composed of two isotropic phases whose random\nspatial distributions are statistically equivalent. Assume that there is a set of $N$\nconserved current densities in the problem ${\\cal J}=(\\bJ_1,\\cdots,\\bJ_N)^T$, with\n\\begin{equation}\n\\label{eq:contJ}\n\\bnabla\\cdot\\bJ_\\alpha=0,\n\\end{equation}\nthat can be induced by a set of $N$ external forces, ${\\cal F}=(\\bF_1,\\cdots,\\bF_N)^T$,\nsatisfying\n\\begin{equation}\n\\label{eq:Fcond}\n\\bnabla\\times\\bF_\\alpha=0.\n\\end{equation}\nWithin linear response the currents and forces are\nrelated by\n\\begin{equation}\n\\label{linearres}\n{\\cal J}(\\br)=\\hat L(\\br){\\cal F}(\\br),\n\\end{equation}\nwhere $\\hat L(\\br)$ is a matrix containing the kinetic coefficients, which takes\none of two values $\\hat L_1$ or $\\hat L_2$ corresponding to the phase present at point $\\br$.\nWe choose the forces such that the local entropy production rate is given by ${\\cal J}(\\br)\\cdot{\\cal F}(\\br)\/T(\\br)$.\nConsequently, Onsager's relations and the isotropy of the constituent phases assure that in the absence\nof a magnetic field $\\hat L$ is symmetric.\n\nOur goal is to calculate the macroscopic response of the system, as given by $\\hat L_{\\rm eff}$\nrelating the spatial averages of ${\\cal J}$ and ${\\cal F}$\n\\begin{equation}\n\\label{Leffdef}\n\\langle{\\cal J}\\rangle = \\hat L_{\\rm eff}\\langle {\\cal F}\\rangle.\n\\end{equation}\nTo this end, we follow Dykhne \\cite{Dykhne71} and introduce an auxiliary transport problem\ndefined in terms of\n\\begin{eqnarray}\n\\label{dualtrans}\n&&{\\bf J}'_\\alpha= U_{\\alpha\\beta}\\, \\hat{\\bf n} \\times {\\bf F}_\\beta,~~~~~{\\bf F}'_\\alpha =U^{-1}_{\\alpha\\beta}\\,\n\\hat{\\bf n} \\times {\\bf J}_\\beta,\n\\end{eqnarray}\nwhere $\\hat U$ is a symmetric matrix whose components are of the same physical dimensions as the components of $\\hat L$,\n$\\hat{\\bf n}$ is a unit vector perpendicular to the plane and where repeated indices are summed over.\nOne can easily check that the new currents and forces satisfy the requirements $\\bnabla\\cdot\\bJ'_\\alpha=0$ and\n$\\bnabla\\times\\bF'_\\alpha=0$. Furthermore, owing to the linear nature of the transformation (\\ref{dualtrans})\nthey are related by ${\\cal J}'(\\br)=\\hat L'(\\br){\\cal F}'(\\br)$. From Eqs. (\\ref{linearres})-(\\ref{dualtrans})\nit follows that $\\hat L'(\\br)=\\hat U \\hat L^{-1}(\\br) \\hat U$ and\n\\be\n\\label{Leffrelation}\n\\hat L'_{\\rm eff}=\\hat U \\hat L_{\\rm eff}^{-1} \\hat U.\n\\ee\nA particularly useful choice for $\\hat U$ is the one that interchanges the two components, {\\it i.e.},\n\\be\n\\hat L_2 = \\hat U \\hat L_1^{-1} \\hat U. \\label{rotation}\n\\ee\nFor such a duality transformation the auxiliary problem corresponds to a system that is obtained from the original\none by replacing one phase by the other. Consequently, their statistical equivalence implies that\n$\\hat L'_{\\rm eff}=\\hat L_{\\rm eff}$, which together with Eq. (\\ref{Leffrelation}) leads to $\\hat L_{\\rm eff}=\\hat U$,\nand thus\n\\be\n\\label{selfdualrel}\n\\hat L_2 = \\hat L_{\\rm eff}\\hat L_1^{-1} \\hat L_{\\rm eff}.\n\\ee\nThis algebraic equation determines the macroscopic response of the system.\n\n\\subsection{Examples}\nIn the case of a single conserved (electrical) current the matrices $\\hat L_{1,2}$ become the conductivities\nof the two phases, $\\sigma_{1,2}$, and Eq. (\\ref{selfdualrel}) reduces to $\\sigma_2=\\sigma_{\\rm eff}^2\\sigma_1^{-1}$\nwith the solution \\cite{Dykhne71}\n\\be\n\\sigma_{\\rm eff} = \\sqrt{\\sigma_1 \\sigma_2}.\n\\ee\n\nFor two or more conserved currents the non-commutativity of $\\hat L_1$ and $\\hat L_2$ plays an essential role.\nAs an example, consider the thermoelectric response of a two-dimensional film described by\n\\be\n\\label{thermolinear}\n\\left(\\begin{array}{c} {\\bf j}(\\br) \\\\{\\bf q}(\\br)  \\end{array}\\right)\n= \\hat L({\\bf r}) \\left(\\begin{array}{c} {\\bf E}(\\br) \\\\-\\frac{\\boldsymbol{\\nabla} T(\\br)}{T(\\br)} \\end{array} \\right),\n\\ee\nwhere ${\\bf j}$ and ${\\bf q}$ are the electrical and heat current densities, respectively, while ${\\bf E}$ and $T$\nare the electric field and the temperature. Note that $\\bf q$ is conserved within linear response that neglects\nsecond order Joule heating effects. The solution to Eq. (\\ref{selfdualrel}) in this case is \\cite{Balagurov-reciprocity,Snarskii-arxiv}\n\\be\n\\label{LefflayerB0}\n\\hat L_{\\rm eff}=c\\left( \\frac{\\hat L_1}{\\sqrt{d_1}}+\\frac{\\hat L_2}{\\sqrt{d_2}}\\right),\n\\ee\nwhere $d_j=\\det \\hat L_j$ and the constant $c$ is\n\\be\n\\label{cdef}\nc= (d_1 d_2)^{1\/4}\\left[\\det\\left( \\frac{\\hat L_1}{\\sqrt{d_1}}+\\frac{\\hat L_2}{\\sqrt{d_2}}\\right)\\right]^{-1\/2}.\n\\ee\nFor completeness, we present the derivation of this result in the Appendix.\n\n\n\\section{Thermoelectric response of a two-phase double-layer}\n\nNext, we consider the case with three conserved currents, motivated by drag experiments in double-layer\ngraphene \\cite{exp1,exp2,exp3,exp4}. In these experiments the two graphene layers are electrically isolated\nfrom each other, but they are in sufficient proximity such that they may be considered as a single layer\nfrom a thermal point of view. This thermalization is primarily due to intra and inter-layer\nelectron-electron inelastic scattering, while the electron-phonon coupling is much weaker and does not\nlead to a significant violation of the presumed conservation of heat in the system.\nUnder such conditions the linear response is described by\n\\be\n\\left(\\begin{array}{c} {\\bf j}_u(\\br)\\\\  {\\bf j}_d(\\br) \\\\{\\bf q}(\\br)  \\end{array}\\right)= \\hat L(\\br)\n\\left(\\begin{array}{c} {\\bf E}_u(\\br)\\\\  {\\bf E}_d(\\br)\\\\-\\frac{\\boldsymbol{\\nabla}T(\\br)}{T(\\br)} \\end{array} \\right),\n\\ee\nwhere ${\\bf j}_{u,d}$ are the electrical current densities in the upper and lower layers, respectively,\nand ${\\bf q}$ is the total heat current density through the system.  Similarly, ${\\bf E}_{u,d}$ are the electric fields\nin the two layers, and $T$ is the temperature field, which is identical in both.\nFurthermore, we will study the case where a domain of a particular phase in the upper layer appears above a similar\ndomain in the lower layer. As a result, $\\hat L(\\br)$ can take two values\n\\be\n\\label{eq:doublelayer}\n\\hat L_j= \\left(\\begin{array}{ccc}\n\\sigma_j & \\eta_j & \\alpha_j  \\\\\n\\eta_j & \\sigma_j &  \\alpha_j \\\\\n\\alpha_j &  \\alpha_j  & 2\\kappa_j  \\end{array} \\right), ~~~~~~ j=1,2\n\\ee\nwhere $\\sigma$ is the electrical conductivity, $\\alpha\/T\\sigma$ is the thermopower and $\\kappa\/T$ is the\nthermal conductivity (provided it is much larger than $\\alpha^2\/T\\sigma$, as in metals \\cite{Abrikosovbook}).\nFinally, $\\eta$ is the drag conductivity due to interaction-induced momentum transfer between the layers.\n\nBeyond momentum transfer there is another drag mechanism which originates\nfrom inter-layer energy transfer \\cite{lev1,lev2,lev3}. Interfaces between phases act as thermocouples.\nA current driven through one layer generates local temperature gradients by the Peltier effect. The strong\nthermal coupling in the system causes these temperature gradients to propagate to the second layer,\nwhich is under open circuit conditions, and generate thermopower there by the Seebeck effect.\n\nEquation (\\ref{Leffrelation}) can be exactly solved for the model defined by Eq. (\\ref{eq:doublelayer}),\nas described in the Appendix. Here, we focus on\ntwo relevant special cases and begin by considering a system for which $\\eta_1=\\eta_2=0$, where any resulting\ndrag is due to inter-layer energy transfer alone.\nSolving Eq. (\\ref{selfdualrel}) we obtain the effective conductivity\n\\be\n\\sigma_{\\rm eff}= \\frac{1}{2} \\sqrt{\\sigma_1 \\sigma_2} \\left( 1+ \\frac{ \\nu^{1\/2}\\cos\\theta_1+ \\nu^{-1\/2}\\cos\\theta_2}\n{\\sqrt{\\nu+ \\nu^{-1}+2\\cos(\\theta_1 + \\theta_2)}}\\right),\n\\ee\nwith\n\\be\n\\sin \\theta_j=\\frac{ \\alpha_j}{\\sqrt{\\kappa_j \\sigma_j}},~~~~~~ \\nu= \\frac{\\sigma_1 \\kappa_2}{\\sigma_2 \\kappa_1},\n\\ee\nwhere the angles $\\theta_j$ are defined in the range $[-\\pi\/2,\\pi\/2]$. The other effective transport coefficients\nare given in terms of $\\sigma_{\\rm eff}$ according to\n\\begin{eqnarray}\n&&\\eta_{\\rm eff} = \\sigma_{\\rm eff} -\\sqrt{\\sigma_1 \\sigma_2}, \\\\\n&&\\alpha_{\\rm eff}= \\frac{\\alpha_1 \\sqrt{d_2}+\\alpha_2 \\sqrt{d_1}}{\\sigma_1\\sqrt{d_2}+ \\sigma_2\\sqrt{d_1}}\n\\left(2 \\sigma_{\\rm eff}-\\sqrt{\\sigma_1 \\sigma_2} \\right), \\\\\n&&\\kappa_{\\rm eff} = \\frac{\\kappa_1 \\sqrt{d_2}+\\kappa_2 \\sqrt{d_1}}{\\sigma_1\\sqrt{d_2}+ \\sigma_2\\sqrt{d_1}}\n\\left(2 \\sigma_{\\rm eff}-\\sqrt{\\sigma_1 \\sigma_2} \\right),\n\\end{eqnarray}\nwhere, as before, $d_{1,2}=\\det \\hat L_{1,2}$.\n\nNext, we treat the case with inter-layer momentum transfer but where the two phases differ only by the sign\nof their charge carriers, while all other characteristics such as mobility and mass are identical, \\ie,\n\\be\n\\label{Lphmodelbi}\n\\hat L_{1,2}= \\left(\\begin{array}{ccc}\n\\sigma & \\eta & \\pm\\alpha  \\\\\n\\eta & \\sigma & \\pm\\alpha \\\\\n\\pm\\alpha & \\pm\\alpha & 2\\kappa  \\end{array} \\right).\n\\ee\nFor this model we find\n\\begin{eqnarray}\n\\label{Leff0i}\n&&\\sigma_{\\rm eff}= \\sigma \\cos^2 \\left( \\frac{\\theta}{2} \\right) - \\eta \\sin^2 \\left( \\frac{\\theta}{2} \\right) , \\\\\n&&\\eta_{\\rm eff}=-\\sigma \\sin^2 \\left( \\frac{\\theta}{2} \\right) + \\eta \\cos^2 \\left( \\frac{\\theta}{2} \\right), \\\\\n&&\\alpha_{\\rm eff}=0, \\\\\n\\label{Leff0f}\n&&\\kappa_{\\rm eff}=\\kappa \\cos\\theta,\n\\end{eqnarray}\nwhere\n\\be\n\\sin\\theta=\\frac{\\alpha}{\\sqrt{\\kappa (\\sigma+\\eta)}}.\n\\ee\n\n\n\\section{A system in a magnetic field}\n\nOur next goal is to extend the discussion to include a magnetic field, $H$, along the $\\hat z$ direction.\nIn the presence of the field the components of $\\hat L(\\br)$ and $\\hat L_{\\rm eff}$ in\nEqs. (\\ref{linearres},\\ref{Leffdef}) become $2\\times 2$ tensors, whose off-diagonal terms are antisymmetric owing to the\nassumption of isotropic phases. For example, the electrical conductivity tensor of the $j=1,2$ phases is\n\\be\n\\label{eq:sigform}\n\\hat\\sigma_j =\n\\left(\\begin{array}{cc}\n\\sigma_j & \\sigma_{Hj}\\\\\n-\\sigma_{Hj} & \\sigma_j\\\\ \\end{array}\\right),\n\\ee\nwhere $\\sigma_H$ is the Hall conductivity.\n\nEquations (\\ref{linearres}) and (\\ref{Leffdef}) may be condensed by representing two-dimensional vectors\nas complex numbers, {\\it e.g.}, ${\\bf E}=E_x+iE_y$, \\cite{Balagurov-galvano95} leading to current densities\nand forces of the form\n\\ba\n\\label{eq:vecsdef}\n&&{\\cal J}=\\left( J_{1x}+iJ_{1y},\\cdots,J_{Nx}+iJ_{Ny} \\right)^T, \\\\\n&&{\\cal F}=\\left( F_{1x}+iF_{1y},\\cdots,F_{Nx}+iF_{Ny} \\right)^T.\n\\ea\nIn this representation Eq. (\\ref{linearres}) becomes\n\\be\n\\label{eq:complexnot}\n{\\cal J}(\\br)=\\hat{\\cL}(\\br){\\cal F}(\\br),\n\\ee\nwhere the complex response matrix $\\hat\\cL(\\br)$ takes one of two values in the corresponding phases $j=1,2$,\n\\be\n\\label{eq:Ldef}\n\\hat \\cL_j = \\hat L_j-i\\hat L_{Hj},\n\\ee\nexpressed in terms of $N\\times N$ symmetric matrices $\\hat L_j$ and $\\hat L_{Hj}$\nholding the longitudinal and Hall parts, respectively, of the kinetic coefficients.\n\nThe magnetic field also requires that the duality transformation, Eq. (\\ref{dualtrans}), be generalized to include\na component proportional to the original fields \\cite{Dykhnemag,DR1994}. In the complex notation it reads\n\\be\n\\label{eq:comtran1}\n{\\cal J}'(\\br)=\\hat A {\\cal J}(\\br) -i\\hat B {\\cal F}(\\br),~~~~~~\n{\\cal F}'(\\br)=\\hat C {\\cal F}(\\br) -i\\hat D {\\cal J}(\\br),\n\\ee\nwhere $\\hat A,\\hat B,\\hat C, \\hat D$ are $N\\times N$ matrices that have to be real in order for the transformed\nfields to still satisfy the conditions $\\bnabla\\cdot\\bJ'_\\alpha=0$ and $\\bnabla\\times\\bF'_\\alpha=0$.\nSubstituting Eqs. (\\ref{eq:complexnot},\\ref{eq:comtran1})\ninto ${\\cal J}'(\\br)=\\hat{\\cL}'(\\br){\\cal F}'(\\br)$ yields the relation\n\\be\n\\label{eq:rel1}\n\\hat A\\hat\\cL(\\br) - i\\hat B=\\hat\\cL'(\\br)\\left[\\hat C-i\\hat D\\hat\\cL(\\br)\\right].\n\\ee\nA similar relation\n\\be\n\\label{eq:rel1eff}\n\\hat A\\hat\\cL_{\\rm eff} - i\\hat B=\\hat\\cL'_{\\rm eff}\\left(\\hat C-i\\hat D\\hat\\cL_{\\rm eff}\\right).\n\\ee\nholds for the effective response matrix $\\hat\\cL_{\\rm eff}$ connecting the spatially averaged fields\n$\\langle{\\cal J}\\rangle=\\hat\\cL_{\\rm eff}\\langle{\\cal F}\\rangle$.\n\nAs before, we are interested in the transformation that fulfills Eq. (\\ref{eq:rel1}) for the cases $\\cL=\\cL_1$, $\\cL'=\\cL_2$\nand $\\cL=\\cL_2$, $\\cL'=\\cL_1$. Because $\\cL_{1,2}$ are symmetric matrices one can verify that a solution to the first case\nis also a solution to the second, provided that we take $\\hat B$ and $\\hat D$ to be symmetric and set $\\hat C=-\\hat A^T$.\nEmploying this choice and separating the real and imaginary parts of Eq. (\\ref{eq:rel1}) we arrive at the defining\nrelations for the required duality transformation\n\\begin{eqnarray}\n\\label{eq:isorelr}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\hat A \\hat L_1 + \\hat L_2 \\hat A^T + \\hat L_2 \\hat D \\hat L_{H1} + \\hat L_{H2} \\hat D \\hat L_1 = 0, \\\\\n\\label{eq:isoreli}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\hat B + \\hat A \\hat L_{H1} + \\hat L_{H2} \\hat A^T - \\hat L_2 \\hat D \\hat L_1 + \\hat L_{H2} \\hat D \\hat L_{H1} = 0.\n\\end{eqnarray}\nWe note that these equations constitute only $2N^2$ conditions for the $2N^2+N$ independent entries in\n$\\hat A$, $\\hat B$ and $\\hat D$, thereby leaving $N$ of them undetermined. Nevertheless, we find that\nthe resulting freedom in choosing the duality transformation does not manifest itself in the effective response\nmatrix. The latter is obtained, once $\\hat A$, $\\hat B$ and $\\hat D$ have been established, by solving\nEq. (\\ref{eq:rel1eff}) with $\\hat\\cL'_{\\rm eff}=\\hat\\cL_{\\rm eff}=\\hat L_{\\rm eff}-i\\hat L_{H{\\rm eff}}$, which\ntranslates to solving\n\\begin{eqnarray}\n\\label{eq:isorelreff}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\hat A \\hat L_{\\rm eff} + \\hat L_{\\rm eff} \\hat A^T + \\hat L_{\\rm eff} \\hat D \\hat L_{H{\\rm eff}}\n+ \\hat L_{H{\\rm eff}} \\hat D \\hat L_{\\rm eff} = 0, \\\\\n\\label{eq:isorelieff}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\hat B + \\hat A \\hat L_{H{\\rm eff}} + \\hat L_{H{\\rm eff}} \\hat A^T - \\hat L_{\\rm eff} \\hat D \\hat L_{\\rm eff}\n+ \\hat L_{H{\\rm eff}} \\hat D \\hat L_{H{\\rm eff}} = 0.\n\\end{eqnarray}\n\n\n\\subsection{The thermoelectric response of a two-phase single-layer in a magnetic field}\n\nThe above scheme can be used to numerically calculate $\\hat \\cL_{\\rm eff}$ for a general two-phase layer.\nWe were able to obtain closed analytical results for the thermoelectric response in the simple case where\nthe two phases differ only by the sign of the charge carriers, implying\n\\be\n\\label{phmodel}\n\\hat L_{1,2}= \\left( \\begin{array}{cc} \\sigma & \\pm\\alpha \\\\ \\pm\\alpha & \\kappa \\end{array} \\right),~~~\n\\hat L_{H1,H2}= \\left( \\begin{array}{cc} \\pm\\sigma_H & \\alpha_H \\\\ \\alpha_H & \\pm\\kappa_H \\end{array} \\right).\n\\ee\nIn this case $\\hat B$ and $\\hat D$ are diagonal matrices, and the latter can be chosen as\n\\be\n\\hat D=\\left(\\begin{array}{cc} a & 0 \\\\\n0 & b \\end{array} \\right),\n\\ee\nwith $a$ and $b$ arbitrary constants of dimensions $[\\sigma^{-1}]$ and $[\\kappa^{-1}]$, respectively, reflecting\nthe aforementioned freedom. In terms of them and the Hall angles\n\\be\n\\label{Hallangles}\n\\tan\\theta_\\sigma=\\frac{\\sigma_H}{\\sigma},~~~\\tan\\theta_\\kappa=\\frac{\\kappa_H}{\\kappa},~~~\n\\tan\\theta_\\alpha=\\frac{\\alpha_H}{\\alpha},\n\\ee\ndefined in the range $[-\\pi\/2,\\pi\/2]$, the matrix $\\hat A$ takes the form\n\\be\n\\hat A=\\frac{\\alpha}{\\sin(\\theta_\\sigma+\\theta_\\kappa)}\n\\left( \\begin{array}{cc}0 & \\hspace{0cm}\\frac{\\sigma}{\\kappa}\\frac{\\cos\\theta_\\kappa}{\\cos\\theta_\\sigma}a\n-\\frac{\\cos(\\theta_\\sigma+\\theta_\\kappa-\\theta_\\alpha)}{\\cos\\theta_\\alpha} b \\\\\n\\frac{\\kappa}{\\sigma}\\frac{\\cos\\theta_\\sigma}{\\cos\\theta_\\kappa}b\n-\\frac{\\cos(\\theta_\\sigma+\\theta_\\kappa-\\theta_\\alpha)}{\\cos\\theta_\\alpha} a & \\hspace{0cm} 0 \\end{array} \\right),\n\\ee\nwhile $\\hat B$ is given by Eq. (\\ref{eq:isoreli}). Substituting the transformation back into Eqs. (\\ref{eq:isorelreff},\\ref{eq:isorelieff})\nand solving for $\\hat \\cL_{\\rm eff}$ yields 16 solutions. All are independent of $a$ and $b$ but only one produces real\ntransport coefficients that are also consistent with the physical requirements $\\sigma_{\\rm eff}\\geq 0$ and $\\kappa_{\\rm eff}\\geq 0$.\nThis solution is\n\\begin{eqnarray}\n&&\\sigma_{\\rm eff}=\\frac{\\sigma}{\\cos\\theta_\\sigma}\\sqrt{1-\\frac{\\alpha^2}{\\sigma\\kappa}\n\\frac{\\cos\\theta_\\sigma\\cos\\theta_\\kappa}{\\cos^2\\left(\\frac{\\theta_\\sigma+\\theta_\\kappa}{2}\\right)}},\\\\\n&&\\kappa_{\\rm eff}=\\frac{\\kappa}{\\cos\\theta_\\kappa}\\sqrt{1-\\frac{\\alpha^2}{\\sigma\\kappa}\n\\frac{\\cos\\theta_\\sigma\\cos\\theta_\\kappa}{\\cos^2\\left(\\frac{\\theta_\\sigma+\\theta_\\kappa}{2}\\right)}},\\\\\n&&\\alpha_{H{\\rm eff}}=\\alpha\\left[\\tan\\theta_\\alpha-\\tan\\left(\\frac{\\theta_\\sigma+\\theta_\\kappa}{2}\\right)\\right], \\\\\n&&\\alpha_{\\rm eff}=\\sigma_{H{\\rm eff}}=\\kappa_{H{\\rm eff}}=0,\n\\end{eqnarray}\nwhich agrees with Eq. (\\ref{LefflayerB0}) when applied to model (\\ref{phmodel}) in the limit of vanishing Hall angles.\n\n\\subsection{The thermoelectric response of a two-phase double-layer in a magnetic field}\n\nObtaining a complete analytical solution to Eqs. (\\ref{eq:isorelr})-(\\ref{eq:isorelieff}) becomes difficult\neven for the simplest models when the number of conserved currents is increased beyond two. However, one can\ntreat the problem perturbatively in the magnetic field, $H$, as we next demonstrate for the double-layer model\ndefined in Eq. (\\ref{Lphmodelbi}) and augmented by the Hall part of the thermoelectric response of the two phases\n\\be\n\\label{LphmodelbiH}\n\\hat L_{H1,H2}= \\left(\\begin{array}{ccc}\n\\pm\\sigma_H & \\pm\\eta_H & \\alpha_H  \\\\\n\\pm\\eta_H & \\pm\\sigma_H & \\alpha_H \\\\\n\\alpha_H & \\alpha_H & \\pm 2\\kappa_H  \\end{array} \\right).\n\\ee\n\nTo proceed we assume that $\\hat L_{1,2}$ are\nindependent of $H$ and $\\hat L_{H1,H2}$ are linear in $H$. Based on our knowledge of the field-free case\nwe look for a solution where $\\hat B$, $\\hat D$ and $\\hat L_{\\rm eff}$ are even functions of $H$ while\n$\\hat A$ and $\\hat L_{H{\\rm eff}}$ are odd, implying the expansions $\\hat A= \\hat A^{(1)} + \\hat A^{(3)} + \\dots$,\nand $\\hat B= \\hat B^{(0)} + \\hat B^{(2)} + \\dots$, etc. Plugging these expansions into Eqs. (\\ref{eq:isorelr},\\ref{eq:isoreli})\none finds at zeroth order\n\\be\n\\label{eq:t01}\n\\hat B^{(0)}=\\hat L_2\\hat D^{(0)}\\hat L_1.\n\\ee\nThis equation leaves 3 parameters undetermined, which we denote by $a,b,z$ and incorporate into the form of $\\hat D^{(0)}$\n\\be\n\\label{D0}\n\\hat D^{(0)}= \\left(\\begin{array}{ccc}\na+b & a-b & z  \\\\\na-b & a+b & z \\\\\nz & z & \\frac{\\sigma+\\eta}{\\kappa}a  \\end{array} \\right).\n\\ee\n$\\hat B^{(0)}$ is then given by Eq. (\\ref{eq:t01}), and the zeroth order contribution to $\\hat L_{\\rm eff}$ is\ndetermined from the corresponding order of Eq. (\\ref{eq:isorelieff})\n\\be\n\\hat B^{(0)}=\\hat L^{(0)}_{\\rm eff}\\hat D^{(0)}\\hat L^{(0)}_{\\rm eff}.\n\\ee\nThis equation yields 8 solutions, of which one is physical and properly\nagrees with Eqs. (\\ref{Leff0i})-(\\ref{Leff0f}).\n\nThe first order equation, which determines $A^{(1)}$ and derived from Eq. (\\ref{eq:isorelr})\n\\be\n\\label{eq:t11}\n\\hat A^{(1)}\\hat L_1+\\hat L_2\\left.{\\hat A}^{(1)}\\right.^{T}\\!\\! +\\hat L_2 \\hat D^{(0)} \\hat L_{H1}\n+\\hat L_{H2} \\hat D^{(0)} \\hat L_1=0,\n\\ee\nhas a solution provided we set $z=0$ in $\\hat D^{(0)}$ (and consequently in $\\hat B^{(0)}$).\nThe solution includes a new undetermined constant, $c$, and can be written in the form\n\\be\n\\label{A1}\n\\hat A^{(1)}= \\left(\\begin{array}{ccc}\n0 & 0 & \\frac{\\sigma+\\eta}{\\kappa}\\left[\\left(\\frac{\\sigma_H+\\eta_H}{\\sigma+\\eta}+\\frac{\\kappa_H}{\\kappa}\n-2\\frac{\\alpha_H}{\\alpha}\\right)\\alpha a -\\frac{c}{2}\\right]  \\\\[5pt]\n0 & 0 & \\frac{\\sigma+\\eta}{\\kappa}\\left[\\left(\\frac{\\sigma_H+\\eta_H}{\\sigma+\\eta}+\\frac{\\kappa_H}{\\kappa}\n-2\\frac{\\alpha_H}{\\alpha}\\right)\\alpha a -\\frac{c}{2}\\right] \\\\[5pt]\nc & c & 0 \\end{array}\\right).\n\\ee\nSubstituting this result into Eq. (\\ref{eq:isorelreff}) and expanding to first order yields an equation\nthat is the same as Eq. (\\ref{eq:t11}) upon substituting $\\hat L_1=\\hat L_2=\\hat L^{(0)}_{\\rm eff}$ and\n$\\hat L_{H1}=\\hat L_{H2}=\\hat L^{(1)}_{H{\\rm eff}}$. Solving for the latter gives the first order contribution\nto the effective Hall components\n\\begin{eqnarray}\n\\label{LeffH1}\n&&\\sigma^{(1)}_{H{\\rm eff}}=0,~~~~\\eta^{(1)}_{H{\\rm eff}}=0,~~~~\\kappa^{(1)}_{H{\\rm eff}}=0, \\\\\n&&\\alpha^{(1)}_{H{\\rm eff}}=\\alpha_H-\\frac{\\alpha}{2}\\left(\\tan\\theta_{\\sigma_+}+\\tan\\theta_\\kappa \\right),\n\\end{eqnarray}\nwhere for brevity we have introduced\n\\be\n\\sigma_{\\pm}=\\sigma\\pm\\eta, ~~~~~~ \\tan\\theta_{\\sigma_\\pm}=\\frac{\\sigma_H\\pm\\eta_H}{\\sigma\\pm\\eta}.\n\\ee\n\nFinally, the second order equation derived from Eq. (\\ref{eq:isoreli})\n\\be\n\\label{eq:t21}\n\\hat B^{(2)} + \\hat A^{(1)} \\hat L_{H1} + \\hat L_{H2}\\!\\left.{\\hat A}^{(1)}\\right.^{\\!T}\n- \\hat L_2 \\hat D^{(2)} \\hat L_1 + \\hat L_{H2} \\hat D^{(0)} \\hat L_{H1} = 0,\n\\ee\nmay be solved for $\\hat B^{(2)}$ and $\\hat D^{(2)}$. The solution, which depends on 3 additional free parameters\nis then substituted into Eq. (\\ref{eq:isorelieff}) and results in\n\\begin{eqnarray}\n\\nonumber\n&&\\!\\!\\!\\hspace{-0.5cm} \\hat B^{(2)} + \\hat A^{(1)} L_{H{\\rm eff}}^{(1)} + L_{H{\\rm eff}}^{(1)}\\left.{\\hat A}^{(1)}\\right.^{\\!T}\n= \\hat L_{\\rm eff}^{(2)} \\hat D^{(0)} \\hat L_{\\rm eff}^{(0)} \\\\\n&&\\!\\!\\!\\hspace{-0.5cm} + \\hat L_{\\rm eff}^{(0)} \\hat D^{(2)} \\hat L_{\\rm eff}^{(0)}\n + \\hat L_{\\rm eff}^{(0)} \\hat D^{(0)} \\hat L_{\\rm eff}^{(2)}\n-  L_{H{\\rm eff}}^{(1)} \\hat D^{(0)} L_{H{\\rm eff}}^{(1)}.\n\\end{eqnarray}\nThis in turn yields the second order corrections contained in $\\hat L_{\\rm eff}^{(2)}$\n\\begin{eqnarray}\n&&\\hspace{-1.2cm}\\sigma_{\\rm eff}^{(2)}=\\frac{\\sigma_+}{4}\\cos\\theta\n\\left[\\tan^2\\theta_{\\sigma_+} + \\frac{\\tan^2\\theta}{4}\\left(\\tan\\theta_{\\sigma_+}-\\tan\\theta_\\kappa\\right)^2\n\\right]+\\frac{\\sigma_-}{4}\\tan^2\\theta_{\\sigma_-}, \\\\\n&&\\hspace{-1.2cm}\\eta_{\\rm eff}^{(2)}=\\sigma_{\\rm eff}^{(2)}-\\frac{\\sigma_-}{2}\\tan^2\\theta_{\\sigma_-}, \\\\\n&&\\hspace{-1.2cm}\\alpha_{\\rm eff}^{(2)}=0, \\\\\n&&\\hspace{-1.2cm}\\kappa_{\\rm eff}^{(2)}=\\kappa\\cos\\theta\n\\left[\\tan^2\\theta_{\\kappa} + \\frac{\\tan^2\\theta}{4}\\left(\\tan\\theta_{\\sigma_+}-\\tan\\theta_\\kappa\\right)^2\\right],\n\\end{eqnarray}\nwhich are again invariant with respect to the freedom in the duality transformation.\n\n\n\\section{Discussion}\n\nIn this work we have provided new exact solutions to the two-phase model in multi-layer systems with both electrical\nand heat currents. Our analysis relies on several assumptions. First, only the linear response regime is considered.\nSecond, that the systems is electrically and thermally isolated. Third, we assume strong thermal coupling between layers\nin a multi-layered system but no electrical current leakage.  Obviously, these assumptions are\nan idealization of reality, and cease to hold true beyond the relaxation lengths set by inter-layer electrical leakage\nand thermal coupling to the environment. Nevertheless, our results are relevant on scales shorter than these relaxation\nlengths, provided that they are much larger than the typical inhomogeneity scale.\n\nThe ability to obtain an exact solution to the problem crucially depends on the statistical equivalence between the\nspatial distributions of the two phases. Violating this condition, \\eg, by moving away from the percolation critical\npoint, necessitates a perturbative approach or a numerical solution. Our results provide benchmarks for the latter.\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis research was supported by the Israel Science Foundation (ISF) Grant No. 302\/14 (O.A.) and Grant No. 701\/17 (D.O.),\nby the Binational Science Foundation (BSF) Grant No. 2014265 (D.O.), and by the Simons Foundation (I.A.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIceCube Neutrino Observatory, the world's largest neutrino\ndetector, has detected 54  neutrino events within 1347 days with energy between 20 TeV to 2.3 PeV ~\\cite{aartsen2014observation,kop}. \nShower events, most likely due to $\\nu_e$ or\n$\\nu_\\tau$ charge current $\\nu N$ interactions and also due to neutral current $\\nu N$ interactions of all flavors, dominate the event\nlist (39 including 3 events with 1--2 PeV energy) while track events,\nmost likely due to $\\nu_\\mu$ charge current $\\nu N$ interactions,\nconstitute the rest. Among a total of 54 events about 21 could be due\nto atmospheric neutrino ($9.0^{+8.0}_{-2.2}$) and muon ($12.6\\pm 5.1$)\nbackgrounds.  A background-only origin of all 54 events has been\nrejected at 6.5-$\\sigma$ level ~\\cite{kop}. Therefore a\ncosmic origin of a number of neutrino events is robust. The track\nevents have on average $\\sim 1^\\circ$ angular resolution, but the\ndominant, shower events have much poorer angular resolution, $\\sim\n15^\\circ$ on average ~\\cite{kop}. Searching for sources of these events is now one of the major\nchallenges in astrophysics. Pinpointing the astrophysical sources\nwhere these neutrinos are coming from is difficult due to large\nuncertainty in their arrival directions.\n\nHigh energy cosmic rays (CRs)\ncan interact with low energy photons and\/or low energy protons to\nproduce neutrinos and high energy gamma rays inside the source or\nwhile propagating to earth. So a multi-messenger study of neutrinos, Cosmic Rays (CRs)\nand gamma-rays can identify the possible astrophysical sources. In our first attempt to search for sources we tried to see a correlation with Ultra-High Energy (UHE) CRs with the earlier 37 cosmic neutrino events ~\\cite{Moharana:2015nxa}. A detail analysis of IceCube neutrino events with the Pierre Auger Observa (PAO) and Telescope Array (TA) has been done in collaboration ~\\cite{Aartsen:2015dml}.\n\nHere we study correlation of IceCube neutrino events with TeVCat, {\\it Swift}-BAT 70 month X-ray source catalog ~\\cite{2013ApJS..207...19B} and 3LAC source catalog ~\\cite{Ackermann:2015yfk}. A similar study of correlation of IceCube neutrinos with the gamma ray sources has also been done ~\\cite{Resconi:2015nva} to find a correlation of $Fermi$-LAT source with only track HESE events is also done ~\\cite{anthony}. Recently a detail analysis of correlation showed at least 2 $\\sigma$ result with extreme blazars ~\\cite{Resconi2} and 3$\\sigma$ with the starforming regions ~\\cite{Emig:2015dma}. To do specific correlation study we use different cuts on the energy flux of these sources, and also different sets of source types, and showed the results of this study.\n\n\\begin{figure}[h]\n\\includegraphics[width=36pc]{allsource.png}\n\\caption{\\label{skymap}Sky map of the 52 IceCube cosmic neutrino events with error circles and sources from different catalogs in Galactic coordinate system.}\n\\end{figure}\n\n\n\\section{IceCube neutrino events and Source catalogs}\n\nFor our analysis we consider all 52 IceCube detected neutrino events. Two track events (event numbers 28\nand 32) are coincident hits in the IceTop surface array and are almost\ncertainly a pair of atmospheric muon background events\n~\\cite{aartsen2014observation}. Therefore we excluded them from our analysis.\nFig.~\\ref{skymap} shows sky map of the 52  events in Galactic \ncoordinates with reported angular errors.  \n\n\nFor the correlation analysis we have used 3 different source catalogs. \n{\\it Swift}-BAT 70 month X-ray source catalog ~\\cite{2013ApJS..207...19B}, $Fermi$ Third Catalog of Active Galactic Nuclei (3LAC) ~\\cite{Ackermann:2015yfk}, TeVCat ~\\cite{2008ICRC....3.1341W}. The sky map in Fig.~\\ref{skymap} shows the extragalactic sources from these catalogs.\n\n{\\it Swift}-BAT 70 month X-ray source catalog includes 1210 objects, and after excluding Galactic sources the number of sources become 785. In our previous study ~\\cite{Moharana:2015nxa} we found 18 sources from this catalog that are correlated simultaneously with UHECRs and IceCube neutrino events. PAO collaboration has also found an anisotropy at $\\sim 98.6\\%$ CL in\nUHECRs with energy $\\ge 58$ EeV and within $\\sim 18^\\circ$ circles\naround the AGNs in {\\em Swift}-BAT catalog at\ndistance $\\le 130$ Mpc ~\\cite{PierreAuger:2014yba} . These 18 sources mostly have an X-ray energy flux $\\ge10 ^{-11}$ ${\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$. So, in the present analysis we use all the sources from this catalog which have flux $\\ge10 ^{-11}$ ${\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$. This condition decreased the number of sources to 687. In the sky map of Fig.~\\ref{skymap} we have shown these 687 sources. \n\nTeVCat contains sources that are detected with very high energy (VHE) gamma rays with energy $\\ge 50$ GeV. It includes 161 sources, out of which 22 are unidentified sources. This is the highest energy source catalog, particularly interesting for $\\nu$ production. Sky map in Fig~\\ref{skymap} contains TeVCat sources that are not in the Galactic plane.\n\nThe {\\it Third Catalog of Active Galactic Nuclei (AGNs)} detected by Fermi LAT (3LAC) ~\\cite{Ackermann:2015yfk} is a subset of the {\\it Fermi} LAT {\\it Third Source Catalog (3FGL)} ~\\cite{2015ApJS..218...23A}. The 3FGL catalog includes 3033 sources detected above a 4$\\sigma$ significance (test statistic $>$ 25) on the whole sky, during the first 4 years of the Fermi mission (2008-2012). The original 3LAC sample includes 1591 AGNs from 3FGL, though 28 are duplicate associations. An additional cut had also been performed to exclude the Galactic plane region ($|b| \\leq  10^\\circ$) where the incompleteness of the counterpart catalogs significantly hinders AGN association. However, in this paper, we chose to study what we call the ``extended 3LAC\" sample of 1773 sources, that includes sources of the Galactic plane, and that could be associated to several neutrino events. In the extended 3LAC sample, 491 sources are flat spectrum radio quasars (FSRQs), 662 are BL Lacs, 585 are blazars of unknown type (BCU), and 35 are non-blazar AGNs. \n\n\\section{Statistical method for Correlation study}\nTo study correlation between cosmic neutrinos and sources from different catalogs separately, we map the\nRight Ascension and Declination $(RA, Dec)$ of the event directions and sources\ninto unit vectors on a sphere as\n$$\n{\\hat x} = (\\sin\\theta \\cos\\phi, \\sin\\theta \\sin\\phi, \\cos\\theta)^T,\n$$\nwhere $\\phi = RA$ and $\\theta = \\pi\/2 - Dec$. Scalar product of the\nneutrino and source vectors $({\\hat x}_{\\rm neutrino}\\cdot {\\hat\n  x}_{\\rm source})$ therefore is independent of the coordinate system.\nThe angle between the two vectors\n\\begin{equation}\n\\label{gamma}\n\\gamma = \\cos^{-1} ({\\hat x}_{\\rm neutrino}\\cdot \n{\\hat x}_{\\rm source}),\n\\end{equation}\nis an invariant measure of the angular correlation between the\nneutrino event and source directions ~\\cite{Virmani:2002xk,Moharana:2015nxa}. Following ref.~\\cite{Virmani:2002xk} we use a\nstatistic made from invariant $\\gamma$ for each neutrino direction\n${\\hat x}_i$ and source direction ${\\hat x}_j$ pair as\n\\begin{equation}\n\\label{delta}\n\\delta\\chi^2_i = {\\rm min}_j (\\gamma_{ij}^2\/\\delta\\gamma_i^2),\n\\end{equation}\nwhich is minimized for all $j$. Here $\\delta\\gamma_i$ is the\n1-$\\sigma$ angular resolution of the neutrino events. We use the exact\nresolutions reported by the IceCube collaboration for each event\n~\\cite{aartsen2014observation}.\n\nA value $\\delta \\chi^2_i \\le 1$ is considered a ``good match'' between\nthe $i$-th neutrino and a source directions. We exploit\ndistributions of all $\\delta\\chi^2_i$ statistics to study angular\ncorrelation between IceCube neutrino events and sources in catalog.  The\ndistribution with observed data giving a number of ``hits'' or $N_{\\rm\n  hits}$ with $\\delta\\chi^2 \\le 1$ therefore forms a basis to claim\ncorrelation. Note that in case more than one source direction from the catalog are\nwithin the error circle of a neutrino event, the $\\delta\\chi^2$ value\nfor UHECR closest to the neutrino direction is chosen in this method.\n\nWe estimate the significance of any correlation in data by comparing\n$N_{\\rm hits}$ with corresponding number from null distributions.\nWe construct null distributions by randomizing only the $RA$ of the sources, keeping their $Dec$ the same as\n  their direction in the catalog. This {\\it semi-isotropic null} is a quick-way to check\n  significance. We perform 100,000 realizations of drawing random numbers to\n  assign new $RA$ and $Dec$ values for each event to construct\n  $\\delta\\chi^2$ distributions in the same way as done with real data.\n\nWe calculate statistical significance of correlation\n  in real data or $p$-value (chance probability) using frequentists'\n  approach. We count the number of times we get a random data set that\n  gives equal or more hits than the $N_{\\rm hits}$ in real data within\n  $\\delta\\chi^2 \\le 1$ bin.  Dividing this number with the total\n  number of random data sets generated (100,000) gives us the\n  $p$-value. We cross-check this $p$-value by calculating the Poisson\n  probability of obtaining $N_{\\rm hits}$ within $\\delta\\chi^2 \\le 1$\n  bin given the corresponding average hits expected from the null\n  distribution. We found the $N_{\\rm hits}$ distribution in $\\delta\\chi^2 \\le 1$ does not follow the Poisson distribution.\n\\section{Results and Discussions}\n We used all 45 HBL (high-frequency peaked BL Lacs) type source listed in TeVCat for our first  correlation study with neutrino events.  A similar correlation study was carried out in ~\\cite{Sahu:2014fua} using HBLs and neutrino data. Our study showed a $p$-value 0.58 with frequentists method while with Poisson distribution probability is 0.1, with 16 neutrinos correlating with different HBLs, almost the same as the null distribution. The distribution is shown in Fig.~\\ref{hbl}.\n  \n {\\it Swift}-BAT 70 month X-ray source included 657 sources with energy flux $10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$. The study of correlation with neutrino events showed a $p$-value 0.825 with 39 $N_{\\rm hits}$ for the real data and nearly 40 for null distribution, as shown in Fig.~\\ref{swift}.\n \n \\begin{figure}[ht]\n\\begin{minipage}{16pc}\n\\includegraphics[width=16pc]{hbl}\n\\caption{\\label{hbl}Correlation Study for all 45 HBL sources from TeVCat.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\includegraphics[width=16pc]{swift_11}\n\\caption{\\label{swift}Correlation Study for {\\it Swift} BAT X-ray catalog sources with energy flux more than $10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$ also shown in ~\\cite{rewin}.}\n\\end{minipage} \n\\end{figure}\n \nThe correlation study of all 1773 sources in the extended 3LAC catalog gives a $p$-value 0.806 with 41 $N_{\\rm hits}$ in for real data, as shown in Fig.~\\ref{3lac_all}. Most of the 3LAC sources are populated in the region of energy flux $10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$, and the population decreases abruptly at higher flux. So, we took a set of sources with energy flux  $\\ge 10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$. It decreased the number of sources in the set to 652, and the correlation study has a $p$-value 0.763 , with $N_{\\rm hits}$ in $\\delta\\chi^2 \\le 1$, 39, shown in Fig.~\\ref{3lac_f}.\n\n\\begin{figure}[h]\n\\begin{minipage}{16pc}\n\\includegraphics[width=16pc]{3lac}\n\\caption{\\label{3lac_all}Correlation Study for all 1773 sources of extended 3LAC catalog also shown in ~\\cite{rewin}.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{16pc}\n\\includegraphics[width=16pc]{3lac_11}\n\\caption{\\label{3lac_f}Correlation Study for sources from extended 3LAC catalog with energy flux $\\ge$ $10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$.}\n\\end{minipage} \n\\end{figure}\n\nIn order to do further study for different type of sources we used the 662 BL Lac source set from the extended 3LAC catalog. The correlation $p$-value for these sources is  0.764, shown in Fig. ~\\ref{bllac}. Similarly for the 491 FSRQ sources  from the extended 3LAC catalog the $p$-value is 0.784, shown in Fig.~\\ref{fsrq}. For BL Lac and FSRQ sources we found 39 and 38 $N_{\\rm hits}$ respectively.\n\n\\begin{figure}[h]\n\\begin{minipage}{16pc}\n\\includegraphics[width=16pc]{3lac_bll}\n\\caption{\\label{bllac}Correlation Study of BL Lac sources from extended 3LAC catalog.}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{16pc}\n\\includegraphics[width=16pc]{3lac_fsrq}\n\\caption{\\label{fsrq}Correlation Study of FSRQ sources from extended 3LAC catalog.}\n\\end{minipage} \n\\end{figure}\n\nThe correlation study of IceCube neutrino events with different type of sources as TeVCat HBL, 3LAC BL Lac and FSRQ is done but we have not found any statistically significant result for these sets. We have also put constraints on the energy flux of 3LAC catalog and the sources observed by {\\it Swift} in 70 months of its observation, and the result is not significant. However with this type of study we can discard different type of extragalactic sources for IceCube neutrino events.\n\n\\begin{table*}\\centering\n\\begin{tabular}{|c|c|c|c|} \\hline\n{Catalog Name} & {Source type} & {\\# of sources} & {p-value}\n\\\\  \\hline\nTeVCAT              &   HBL                                                                  & 45      & 0.58    \\\\\n$Swift$ Bat X-ray   & energy flux > $10^{-11} \\,{\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$             & 657     & 0.825    \\\\\n3LAC (Extended)               & All                                                                    & 1773    & 0.806   \\\\\n3LAC (Extended)              & energy flux > $10 ^{-11} \\,{\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$             & 652     & 0.763             \\\\\n3LAC (Extended)              & BL Lac                                                                 & 662     & 0.764    \\\\\n3LAC (Extended)              & FSRQ                                                                   & 491     & 0.786   \\\\\n \\hline\n \\end{tabular}\n\\caption{Results of correlation study. }\n\\label{tab:res}\n\n\\end{table*}\n\n\\section{Summary}\nIceCube neutrino observatory has detected at least 54 neutrino events within energy 30 TeV-2 PeV. Sources for these events is still a puzzle for both particle physics and astrophysics. In our project we have tried to find correlation of the arrival direction of these events with direction of sources from TeVCat, {\\it Swift} and 3LAC catalogs. In order to test correlation we have used invariant statistics, called the minimum $\\delta \\chi^2$, as in ~\\cite{Virmani:2002xk,Moharana:2015nxa}. Out of 52 neutrino events, 16 were correlated with HBLs from TeVCat but the statistical significance of this correlation $p-$value is 0.58. Similarly we study correlation of neutrino events with sources from {\\it Swift} and 3LAC having energy flux $\\ge$ $10^{-11} {\\rm{erg} \\, \\rm{cm^{-2}} \\, \\rm{sec}^ {-1}}$, for which we also found a poor statistical significance. The FSRQ and BL Lacs from 3LAC catalog also showed less significant statistics for the correlation study.  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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h}\n\\define\\ii{\\bold i}\n\\define\\jj{\\bold j}\n\\define\\kk{\\bold k}\n\n\\define\\mm{\\bold m}\n\\define\\nn{\\bold n}\n\\define\\oo{\\bold o}\n\\define\\pp{\\bold p}\n\\define\\qq{\\bold q}\n\\define\\rr{\\bold r}\n\\redefine\\ss{\\bold s}\n\\redefine\\tt{\\bold t}\n\\define\\uu{\\bold u}\n\\define\\vv{\\bold v}\n\\define\\ww{\\bold w}\n\\define\\zz{\\bold z}\n\\redefine\\xx{\\bold x}\n\\define\\yy{\\bold y}\n\n\\redefine\\AA{\\bold A}\n\\define\\BB{\\bold B}\n\\define\\CC{\\bold C}\n\\define\\DD{\\bold D}\n\\define\\EE{\\bold E}\n\\define\\FF{\\bold F}\n\\define\\GG{\\bold G}\n\\define\\HH{\\bold H}\n\\define\\II{\\bold I}\n\\define\\JJ{\\bold J}\n\\define\\KK{\\bold K}\n\\define\\LL{\\bold L}\n\\define\\MM{\\bold M}\n\\define\\NN{\\bold N}\n\\define\\OO{\\bold O}\n\\define\\PP{\\bold P}\n\\define\\QQ{\\bold Q}\n\\define\\RR{\\bold R}\n\\define\\SS{\\bold S}\n\\define\\TT{\\bold T}\n\\define\\UU{\\bold U}\n\\define\\VV{\\bold V}\n\\define\\WW{\\bold W}\n\\define\\ZZ{\\bold Z}\n\\define\\XX{\\bold X}\n\\define\\YY{\\bold Y}\n\n\\define\\ca{\\Cal A}\n\\define\\cb{\\Cal B}\n\\define\\cc{\\Cal C}\n\\define\\cd{\\Cal D}\n\\define\\ce{\\Cal E}\n\\define\\cf{\\Cal F}\n\\define\\cg{\\Cal G}\n\\define\\ch{\\Cal H}\n\\define\\ci{\\Cal I}\n\\define\\cj{\\Cal J}\n\\define\\ck{\\Cal K}\n\\define\\cl{\\Cal L}\n\\define\\cm{\\Cal M}\n\\define\\cn{\\Cal N}\n\\define\\co{\\Cal O}\n\\define\\cp{\\Cal P}\n\\define\\cq{\\Cal Q}\n\\define\\car{\\Cal R}\n\\define\\cs{\\Cal S}\n\\define\\ct{\\Cal T}\n\\define\\cu{\\Cal U}\n\\define\\cv{\\Cal V}\n\\define\\cw{\\Cal W}\n\\define\\cz{\\Cal Z}\n\\define\\cx{\\Cal X}\n\\define\\cy{\\Cal Y}\n\n\\define\\fa{\\frak a}\n\\define\\fb{\\frak b}\n\\define\\fc{\\frak c}\n\\define\\fd{\\frak d}\n\\define\\fe{\\frak e}\n\\define\\ff{\\frak f}\n\\define\\fg{\\frak g}\n\\define\\fh{\\frak h}\n\\define\\fii{\\frak i}\n\\define\\fj{\\frak j}\n\\define\\fk{\\frak k}\n\\define\\fl{\\frak l}\n\\define\\fm{\\frak m}\n\\define\\fn{\\frak n}\n\\define\\fo{\\frak o}\n\\define\\fp{\\frak p}\n\\define\\fq{\\frak q}\n\\define\\fr{\\frak r}\n\\define\\fs{\\frak s}\n\\define\\ft{\\frak t}\n\\define\\fu{\\frak u}\n\\define\\fv{\\frak v}\n\\define\\fz{\\frak z}\n\\define\\fx{\\frak x}\n\\define\\fy{\\frak y}\n\n\\define\\fA{\\frak A}\n\\define\\fB{\\frak B}\n\\define\\fC{\\frak C}\n\\define\\fD{\\frak D}\n\\define\\fE{\\frak E}\n\\define\\fF{\\frak F}\n\\define\\fG{\\frak G}\n\\define\\fH{\\frak H}\n\\define\\fI{\\frak I}\n\\define\\fJ{\\frak J}\n\\define\\fK{\\frak K}\n\\define\\fL{\\frak L}\n\\define\\fM{\\frak M}\n\\define\\fN{\\frak N}\n\\define\\fO{\\frak O}\n\\define\\fP{\\frak P}\n\\define\\fQ{\\frak Q}\n\\define\\fR{\\frak R}\n\\define\\fS{\\frak S}\n\\define\\fT{\\frak T}\n\\define\\fU{\\frak U}\n\\define\\fV{\\frak V}\n\\define\\fZ{\\frak Z}\n\\define\\fX{\\frak X}\n\\define\\fY{\\frak Y}\n\n\\define\\ta{\\ti a}\n\\define\\tb{\\ti b}\n\\define\\tc{\\ti c}\n\\define\\td{\\ti d}\n\\define\\te{\\ti e}\n\\define\\tf{\\ti f}\n\\define\\tg{\\ti g}\n\\define\\tih{\\ti h}\n\\define\\tj{\\ti j}\n\\define\\tk{\\ti k}\n\\define\\tl{\\ti l}\n\\define\\tm{\\ti m}\n\\define\\tn{\\ti n}\n\\define\\tio{\\ti\\o}\n\\define\\tp{\\ti p}\n\\define\\tq{\\ti q}\n\\define\\ts{\\ti s}\n\\define\\tit{\\ti t}\n\\define\\tu{\\ti u}\n\\define\\tv{\\ti v}\n\\define\\tw{\\ti w}\n\\define\\tz{\\ti z}\n\\define\\tx{\\ti x}\n\\define\\ty{\\ti y}\n\\define\\tA{\\ti A}\n\\define\\tB{\\ti B}\n\\define\\tC{\\ti C}\n\\define\\tD{\\ti D}\n\\define\\tE{\\ti E}\n\\define\\tF{\\ti F}\n\\define\\tG{\\ti G}\n\\define\\tH{\\ti H}\n\\define\\tI{\\ti I}\n\\define\\tJ{\\ti J}\n\\define\\tK{\\ti K}\n\\define\\tL{\\ti L}\n\\define\\tM{\\ti M}\n\\define\\tN{\\ti N}\n\\define\\tO{\\ti O}\n\\define\\tP{\\ti P}  \n\\define\\tQ{\\ti Q}\n\\define\\tR{\\ti R}\n\\define\\tS{\\ti S}\n\\define\\tT{\\ti T}\n\\define\\tU{\\ti U}\n\\define\\tV{\\ti V}\n\\define\\tW{\\ti W}\n\\define\\tX{\\ti X}\n\\define\\tY{\\ti Y}\n\\define\\tZ{\\ti Z}\n\\define\\tiO{\\ti\\Om}\n\\define\\sh{\\sharp}\n\\define\\Mod{\\text{\\rm Mod}}\n\\define\\Irr{\\text{\\rm Irr}}\n\\define\\sps{\\supset}\n\\define\\app{\\asymp}\n\\define\\uP{\\un P}\n\\define\\bnu{\\bar\\nu}\n\\define\\bc{\\bar c}\n\\define\\bp{\\bar p}\n\\define\\br{\\bar r}\n\\define\\bg{\\bar g}\n\\define\\hC{\\hat C}\n\\define\\bE{\\bar E}\n\\define\\bS{\\bar S}\n\\define\\bP{\\bar P}\n\\define\\bce{\\bar\\ce}\n\\define\\tce{\\ti\\ce}\n\\define\\bul{\\bullet}\n\\define\\uZ{\\un Z}\n\\define\\che{\\check}\n\\define\\cha{\\che{\\a}}\n\\define\\bfU{\\bar\\fU}\n\\define\\Rf{\\text{\\rm R}}\n\\define\\tfK{\\ti{\\fK}}\n\\define\\tfD{\\ti{\\fD}}\n\\define\\tfC{\\ti{\\fC}}\n\\define\\bfK{\\bar{\\fK}}\n\\define\\ufK{\\un{\\fK}}\n\\define\\bat{\\bar\\t}\n\\define\\ucl{\\un\\cl}\n\\define\\ucm{\\un\\cm}\n\\define\\dcl{\\dot{\\cl}}\n\\define\\udcl{\\un{\\dot{\\cl}}}\n\\define\\udcm{\\un{\\dot{\\cm}}}\n\\define\\cir{\\circ}\n\\define\\ndsv{\\not\\dsv}\n\\define\\prq{\\preceq}\n\n\\define\\tss{\\ti{\\ss}}\n\\define\\tSS{\\ti{\\SS}}\n\\define\\ciss{\\ci_\\ss}\n\\define\\cits{\\ci_{\\tss}}\n\\define\\Upss{\\Up^\\ss}\n\\define\\Uptss{\\Up^{\\tss}}\n\\define\\tcj{\\ti{\\cj}}\n\\define\\tcp{\\ti{\\cp}}\n\\define\\tcf{\\ti{\\cf}}\n\\define\\tcb{\\ti{\\cb}}\n\\define\\tcy{\\ti{\\cy}}\n\\define\\y{\\ti r}\n\\define\\ttir{\\ti t_{\\y}}\n\\define\\tgtir{\\tg{\\y}}\n\\define\\sscj{\\ss_\\cj}\n\\define\\tsstcj{\\tss_{\\tcj}}\n\\define\\ccl{\\check{\\cl}}\n\\define\\uucl{\\un{\\ucl}}\n\\define\\bUp{\\bar{\\Up}}\n\\define\\tip{\\ti\\p}\n\\define\\chR{\\check R}\n\\define\\tib{\\ti\\b}\n\\define\\tir{\\ti r}\n\\define\\uzz{\\un\\zz}\n\\define\\tzz{\\ti\\zz}\n\\define\\bcs{\\bar{\\cs}}\n\\define\\tcs{\\ti\\cs}\n\\define\\ddu{\\ddot u}\n\\define\\AL{Al}\n\\define\\ALL{AL}\n\\define\\SPE{L1}\n\\define\\CL{L2}\n\\define\\UC{L3}\n\\define\\OR{L4}\n\\define\\ICC{L5}\n\\define\\USU{L6}\n\\define\\SH{S1}\n\\define\\SHO{S2}\n\\define\\SP{Sp}\n\n\\head Introduction \\endhead\n\\subhead 0.1\\endsubhead\nLet $G$ be a simple adjoint algebraic group over $\\CC$ and let $\\cx$ be the set of unipotent conjugacy classes in\n$G$. Let $C\\in\\cx$ and let $u\\in C$. The following invariants of $C$ are important in representation theory:\n\n-the dimension $\\bb_C$ of the fixed point set of $\\Ad(u)$ on the flag manifold of $G$;\n\n-the number $\\zz_C$ of connected components of the centralizer of $u$ in $G$;\n\n-the number $\\tzz_C$ of connected components of the centralizer of a unipotent element in the simply connected\ncovering of $G$ which projects to $u$;\n\n-the irreducible representation $\\r_C$ of the Weyl group $\\WW$ of $G$ corresponding to $C$ and the constant local\nsystem under the Springer correspondence \\cite{\\SP}.\n\\nl\nLet $\\tcs_\\WW$ be the set of isomorphism classes of irreducible representations of $\\WW$ of the form $\\r_C$ for\nsome $C\\in\\cx$. It is known \\cite{\\SP} \nthat $C\\m\\r_C$ is a bijection $\\cx@>\\si>>\\tcs_\\WW$.\n\nNote that the definition of each of $\\bb_C$, $\\zz_C$, $\\tzz_C$ is based on considerations of algebraic geometry \nand in the case of $\\tcs_\\WW$, also on considerations of \\'etale cohomology.\n\nIn \\cite{\\SPE, Sec.9} I conjectured that $\\tcs_\\WW$, $C\\m\\bb_C$ and $C\\m\\zz_C$ can be determined purely in terms \nof data involving the Weyl group $\\WW$ (more precisely, the \"special representations\" of the \"parahoric\" subgroups\nof $\\WW$, see 1.1, 1.2). At that time I could only prove this conjecture for $\\tcs_\\WW$ and for $C\\m\\bb_C$ assuming that $G$ is\nof classical type (my proof was based on \\cite{\\SH}) and a little later for $G$ of type $F_4$ (based on \n\\cite{\\SHO}). In \\cite{\\ALL} the conjecture for $\\tcs_\\WW$ and $C\\m\\bb_C$ was established for $G$ of type \n$E_6,E_7,E_8$. At the time \\cite{\\OR} was written, I proved the remaining \nconjecture of \\cite{\\SPE} (concerning $C\\m\\zz_C$); this was stated in \n\\cite{\\OR, 13.3}. For\nclassical groups the proof involved a new description (in terms of \"symbols\") of the Springer correspondence for \nclassical groups (given in \\cite{\\ICC}) while for exceptional groups this was a purely mechanical verification \nbased on the tables \\cite{\\AL}. The conjecture of \\cite{\\SPE} is restated and proved here as Theorem \n1.5(a),(b1),(b2). At the same time we state and prove a complement to that \nconjecture, namely that $C\\m\\tzz_C$ is \ndetermined purely in terms of data involving $\\WW$ (see Theorem \n1.5(b3)). Note that for classical groups this involves some combinatorial considerations while for exceptional \ngroups this involves only a purely mechanical verification based on the known tables.\n\n{\\it Notation.} For a finite set $F$ let $|F|$ be the cardinal of $F$. For $i,j$ in $\\ZZ$ we set \n$[i,j]=\\{n\\in\\ZZ;i\\le n\\le j\\}$. For $x,y$ in $\\ZZ$ we write $x\\ll y$ if $x\\le y-2$.\n\n\\head Contents\\endhead\n1. Statement of the main result.\n\n2. Combinatorics.\n\n3. Type $A_{n-1}$.\n\n4. Type $B_n$.\n\n5. Type $C_n$.\n\n6. Type $D_n$.\n\n7. Exceptional types.\n\nIndex.\n\n\\head 1. Statement of the main result\\endhead\n\\subhead 1.1\\endsubhead\nLet $W$ be a finite crystallographic Coxeter group. Let $\\Irr(W)$ be the set of isomorphism classes of irreducible\nrepresentations of $W$ over $\\QQ$. If $E\\in\\Irr(W)$ and $E'$ is a finite dimensional $\\QQ[W]$-module, let \n$[E:E']_W$ be the multiplicity of $E$ in $E'$. Let $S^i_W$ be the $i$-th symmetric power of the reflection \nrepresentation of $W$. For any $E\\in\\Irr(W)$ we define integers $f_E\\ge1$, $a_E\\ge0$ by the requirement that the \ngeneric degree of the Hecke algebra representation corresponding to $E$ is of the form \n$\\fra{1}{f_E}\\qq^{a_E}+$higher powers of $\\qq$ ($\\qq$ is an indeterminate); let $b_E$ be the smallest integer \n$i\\ge0$ such that $[E:S^i_W]_W\\ge1$. As observed in \\cite{\\SPE, Sec.2}, we have $a_E\\le b_E$ for any $E\\in\\Irr(W)$; \nfollowing \\cite{\\SPE, Sec.2} we set $\\cs_W=\\{E\\in\\Irr(W);a_E=b_E\\}$; this is the set of \"special representations\" of\n$W$. Let $\\Irr(W)^\\da=\\{E\\in\\Irr(W);[E:S^{b_E}_W]_W=1\\}$. We have $\\cs_W\\sub\\Irr(W)^\\da$. \n\n\\subhead 1.2\\endsubhead\nIn this paper we fix a root datum of finite type $\\car=(Y,X,\\cha_i,\\a_i(i\\in I),\\la,\\ra)$. (Here $Y,X$ are free \nabelian groups of finite rank, $\\la,\\ra:Y\\T X@>>>\\ZZ$ is a perfect pairing, $\\cha_i\\in Y$ are the simple coroots \nand $\\a_i\\in X$ are the simple roots.) We assume that $I\\ne\\em$ and that $\\car$ is of adjoint type that is, \n$\\{\\a_i;i\\in I\\}$ is a $\\ZZ$-basis of $X$. Let $R\\sub X$ (resp. $\\che R\\sub Y$) be the set of roots (resp. \ncoroots); let $\\cha\\lra\\a$ be the canonical bijection $\\che R\\lra R$. We assume that $\\car$ is irreducible that \nis, there is a unique $\\a_0\\in R$ such that $\\cha_0-\\cha_i\\n\\che R$ for any $i\\in I$. Let $\\tI=I\\sqc\\{0\\}$. For \n$i\\in\\tI$ let $s_i:X@>>>X$ be the reflection determined by $\\a_i,\\cha_i$. Let $\\WW$ be the subgroup of $GL(X)$ \ngenerated by $\\{s_i;i\\in I\\}$, a finite crystallographic Coxeter group containing $s_0$. The elements \n$s_i(i\\in\\tI)$ in $\\WW$ satisfy the relations of the affine Weyl group of type dual to that of $\\car$. Let \n$\\tca=\\{J;J\\subsetneqq\\tI\\}$. For any $J\\in\\tca$, let $\\WW_J$ be the subgroup of $\\WW$ generated by \n$\\{s_i;i\\in J\\}$, a finite crystallographic Coxeter group with set of generators $\\{s_i;i\\in J\\}$, said to be a \n{\\it parahoric subgroup} of $\\WW$. \n\nLet $\\Om$ be the (commutative) subgroup of $\\WW$ consisting of all $\\o\\in\\WW$ \nsuch that $\\o(\\a_i)=\\a_{\\un\\o(i)}$ ($i\\in\\tI$) for some (necessarily unique) \npermutation $\\un\\o:\\tI@>\\si>>\\tI$.\n\n\\subhead 1.3\\endsubhead\nIf $J\\in\\tca$ and $E_1\\in\\Irr(\\WW_J)^\\da$, there is a unique $E\\in\\Irr(\\WW)$ such that $b_E=b_{E_1}$ and \n$[E:\\Ind_{\\WW_J}^\\WW(E_1)]_\\WW\\ge1$. (Then $[E:\\Ind_{\\WW_J}^\\WW(E_1)]_\\WW=1$ and $E\\in\\Irr(\\WW)^\\da$.) We write \n$E=j_{\\WW_J}^\\WW(E_1)$. Let $E\\in\\Irr(\\WW)$ and let\n$$\\cz_E=\\{(J,E_1);J\\in\\tca,E_1\\in\\cs_{\\WW_J},E=j_{\\WW_J}^\\WW(E_1)\\}.$$\nLet \n$$\\bcs_\\WW=\\{E\\in\\Irr(\\WW);\\cz_E\\ne\\em\\}.$$\nLet $E\\in\\bcs_\\WW$. We set \n$$\\fa_E=\\max_{(J,E_1)\\in\\cz_E}f_{E_1}.$$\nLet $\\cz_E^\\sp=\\{(J,E_1)\\in\\cz_E;f_{E_1}=\\fa_E\\}$. We have $\\cz_E^\\sp\\ne\\em$. \n\nIf $(J,E_1)\\in\\cz_E$ and $\\o\\in\\Om$ then $\\Ad(\\o):\\WW_J@>\\si>>\\WW_{\\un\\o(J)}$ carries $E_1$ to a representation \n${}^\\o E_1\\in\\cs_{\\WW_{\\un\\o(J)}}$ such that $\\Ind_{\\WW_J^\\WW}(E_1)=\\Ind_{\\WW_{\\un\\o(J)}^\\WW}({}^\\o E_1)$, \n$b_{{}^\\o E_1}=b_{E_1}$ and $f_{{}^\\o E_1}=f_{E_1}$. It follows that \n$j_{\\WW_J}^\\WW(E_1)=j_{\\WW_{\\un\\o(J)}}^\\WW({}^\\o E_1)$. Thus $(\\un\\o(J),{}^\\o E_1)\\in\\cz_E$ and\n$\\o:(J,E_1)\\m(\\un\\o(J),{}^\\o E_1)$ is an action of $\\Om$ on $\\cz_E$. This restricts to an action of $\\Om$ on \n$\\cz_E^\\sp$. The stabilizer in $\\Om$ of $(J,E_1)\\in\\cz_E^\\sp$ for this action is denoted by $\\Om_{J,E_1}$. We set \n$$\\fc_E=\\max_{(J,E_1)\\in\\cz_E^\\sp}|\\Om_{J,E_1}|.$$\n\n\\subhead 1.4\\endsubhead\nLet $G$ be a semisimple (adjoint) algebraic group over $\\CC$ with root datum $\\car$. \nLet $\\cx$, $C\\m\\r_C$, $C\\m\\bb_C$, $C\\m\\zz_C$, $C\\m\\tzz_C$, $\\tcs_\\WW$ be as in 0.1.\n\n\\proclaim{Theorem 1.5}(a) $\\tcs_\\WW=\\bcs_\\WW$. \n\n(b) Let $C\\in\\cx$. Set $E=\\r_C\\in\\bcs_\\WW$. Then:\n\n(b1) $\\bb_C=b_E$;\n\n(b2) $\\zz_C=\\fa_E$;\n\n(b3) $\\tzz_C\/\\zz_C=\\fc_E$.\n\\endproclaim\nFor exceptional types the proof of (a),(b1)-(b3) consists in examining the existing tables. Some relevant data is\ncollected in \\S7. The proof for the classical types is given in \\S3-\\S6 after combinatorial preliminaries in\n1.9-1.11 and \\S2.\n\n\\subhead 1.6\\endsubhead\nLet $G'$ be a connected reductive group over $\\CC$  such that $G$ is the quotient of $G'$ by its centre. \n\nNote that 1.5(a) is closely connected to the definition of a unipotent support\nof a character sheaf on $G'$ provided by \\cite{\\USU, 10.7}. In fact, \n\\cite{\\USU, 10.7(iii)} provides a proof of the inclusion $\\bcs_\\WW\\sub\\tcs_\\WW$\nwithout case by case checking.\n\nFor any $g\\in G'$ let $g_u$ be the unipotent part of $g$. We now state an \nalternative conjectural definition of the unipotent support of a character\nsheaf on $G'$.\n\n\\proclaim{Conjecture 1.7} Let $A$ be a character sheaf on $G'$. There exists a unique unipotent class $C$ in $G'$\nsuch that:\n\n(i) $A|_{\\{g\\}}\\ne0$ for some $g\\in G'$ with $g_u\\in C$;\n\n(ii) if $g'\\in G'$ satisfies $A|_{\\{g'\\}}\\ne0$ then the conjugacy class of $g'_u$ in $G'$ has dimension $<\\dim(C)$.\n\\endproclaim\n\n\\subhead 1.8\\endsubhead\nTheorem 1.5 remains valid if $\\CC$ is replaced by an algebraically closed field whose characteristic is either $0$\nor a prime which is good for $G$ and which (if $G$ is of type $A_{n-1}$) does not divide $n$.\n\n\\subhead 1.9\\endsubhead\nIn the rest of this section we discuss some preliminaries to the proof of 1.5.\n\nIf $J,J'\\in\\tca$, $J\\sub J'$ and $E_1\\in\\Irr(\\WW_J)^\\da$, there is a unique $E'_1\\in\\Irr(\\WW_{J'})$ such that \n$b_{E_1}=b_{E'_1}$ and $[E'_1:\\Ind_{\\WW_J}^{\\WW_{J'}}(E_1)]_{\\WW_{J'}}\\ge1$. (Then \n$[E'_1:\\Ind_{\\WW_J}^{\\WW_{J'}}(E_1)]_{\\WW_{J'}}=1$ and $E'_1\\in\\Irr(\\WW_{J'})^\\da$.) We write \n$E'_1=j_{\\WW_J}^{\\WW_{J'}}(E_1)$. Note that \n\n(a) $j_{\\WW_J}^\\WW(E_1)=j_{\\WW_{J'}}^\\WW(j_{\\WW_J}^{\\WW_{J'}}(E_1))$;\n\n(b) if, in addition, $E_1\\in\\cs_{\\WW_J}$, then $E'_1\\in\\cs_{\\WW_{J'}}$ and $f_{E_1}\\le f_{E'_1}$.\n\\nl\n(See \\cite{\\SPE, Sec.4}.)\n\nLet $\\cp'$ be the collection of parahoric subgroups $W$ of $\\WW$ such that $W=\\WW_J$ for some $J\\sub\\tI$, \n$|J|=|\\tI|-1$. From (a),(b) we see that\n$$\\bcs_\\WW=\\{E\\in\\Irr(\\WW);E=j_W^\\WW(E_1)\\text{ for some }W\\in\\cp'\n\\text{ and some }E_1\\in\\cs_W\\},\\tag c$$\n$$\\fa_E=\\max_{(J,E_1)\\in\\cz_E;|J|=|\\tI|-1}f_{E_1}\\text{ for }E\\in\\bcs_\\WW.\\tag d$$\nIf $W=W_1\\T W_2$ where $W_1,W_2$ are finite crystallographic Coxeter groups and $E_1\\in\\Irr(W_1)$, \n$E_2\\in\\Irr(W_2)$ then $E:=E_1\\bxt E_2\\in\\Irr(W)$ belongs to $\\cs_W$ if and only if $E_1\\in\\cs_{W_1}$ and \n$E_2\\in\\cs_{W_2}$; in this case we have \n\n(e) $a_E=a_{E_1}+a_{E_2}$, $f_E=f_{E_1}f_{E_2}$. \n\n\\subhead 1.10\\endsubhead\nWe show: \n\n(a) {\\it if $J,J'\\in\\tca$ and $\\WW_J=\\WW_{J'}\\ne\\WW$ then $J=J'$.}\n\\nl\nIt is enough to show that if $J,J'\\in\\tca$ and $\\WW_J\\sub\\WW_{J'}\\ne\\WW$ then $J\\sub J'$. To see this we may\nassume that $J$ consists of a single element $j$. We have $s_j\\in\\WW_{J'}$. Assume that $j\\n J'$. If \n$J'\\cup\\{j\\}\\ne\\tI$ then $\\WW_{J'\\cup\\{j\\}}$ is a Coxeter group on the generators $\\{s_h;h\\in J'\\cup\\{j\\}\\}$. In \nparticular $s_j$ is not contained in the subgroup $\\WW_{J'}$ generated by $\\{s_h;h\\in J'\\}$, a contradiction. Thus\nwe have $J'\\cup\\{j\\}=\\tJ$. We see that $\\WW_{J'}$ contains $\\{s_h;h\\in J'\\cup\\{j\\}\\}$ which generates $\\WW$. Thus\n$\\WW_J=\\WW$ which is again a contradiction. This proves (a).\n\n\\subhead 1.11\\endsubhead\nFor a subgroup $\\tiO$ of $\\Om$ let $\\cp^{\\tiO}$ be the collection of parahoric subgroups $W$ of $\\WW$ such that \n$W=\\WW_J$ for some $J\\in\\tca$ where $J$ is $\\tiO$-stable and is maximal with this property. From the\ndefinitions we have\n$$\\fc_E=\\max|\\tiO|,$$\nwhere the maximum is taken over all subgroups\n $\\tiO\\sub\\Om$ and all $(J,E_1)\\in\\cz_E^\\sp$ such that $\\WW_J\\in\\cp^{\\tiO}$,\n$\\tiO\\sub\\Om_{J,E_1}$.\n\n\\head 2. Combinatorics\\endhead\n\\subhead 2.1\\endsubhead\nIn this section we fix $m\\in\\NN$.\n\nLet $Z_m=\\{z_*=(z_0,z_1,z_2,\\do,z_m)\\in\\NN^{m+1};z_0<z_1<\\do<z_m\\}$. Let $z_*^0=z_*^{0,m}=(0,1,2,\\do,m)\\in Z_m$. \nFor any $z_*\\in Z_m$ we have $z_*-z^0_*\\in\\NN^{m+1}$. Hence \n\n$\\r_0:Z_m@>>>\\NN$, $z_*\\m\\sum_{i\\in[0,m]}(z_i-z^0_i)$ and \n\n$\\b_0:Z_m@>>>\\NN$, $z_*\\m\\sum_{0\\le i<j\\le m}(z_i-z_i^0)$\n\\nl\nare well defined. For any $n\\in\\NN$ we set $Z_m^n=\\{z_*\\in Z_m;\\r_0(z_*)=n\\}$. \n\n\\subhead 2.2\\endsubhead\nLet $X_m$ be the set of all $x_*=(x_0,x_1,x_2,\\do,x_m)\\in\\NN^{m+1}$ such that $x_i\\le x_{i+1}$ for $i\\in[0,m-1]$,\n$x_i<x_{i+2}$ for $i\\in[0,m-2]$. For $x_*\\in X_m$ let $\\fS(x_*)$ be the set of all $i\\in[0,m]$ such that \n$x_{i-1}<x_i<x_{i+1}$ (with the convention $x_{-1}=-\\iy,x_{m+1}=\\iy$). Note that \n\n(a) $|\\fS(x_*)|\\cong m-1\\mod2$;\n\n(b) $\\fS(x_*)=\\em$ if and only if $m$ is odd and $x_i=x_{i+1}$ for $i=0,1,\\do,(m-1)\/2$. \n\n\\subhead 2.3\\endsubhead\nLet $Y_m$ be the set of all $y_*=(y_0,y_1,y_2,\\do,y_m)\\in\\NN^{m+1}$ such that $y_i\\le y_{i+1}$ for $i\\in[0,m-1]$,\n$y_i\\ll y_{i+2}$ for $i\\in[0,m-2]$. For $y_*\\in Y_m$ let $\\fI(y_*)$ be the set of all intervals $[i,j]\\sub[0,m]$\n(with $i\\le j$) such that \n\n$y_{i-1}-(i-1)<y_i-i=y_{i+1}-{i+1}=\\do=y_j-j<y_{j+1}-(j+1)$\n\\nl\n(with the convention $y_{-1}=-\\iy,y_{m+1}=\\iy$). We have \n\n(a) $\\fI(y_*)=\\em$ if and only if $m$ is odd and $y_i=y_{i+1}$ for $i=0,1,\\do,(m-1)\/2$.\n\\nl\nLet \n\n$\\fI'(y_*)=\\{\\ci\\in\\fI(y_*);\\ci=[i,j]\\text{ with $|[i,j]|=$odd}\\}$, \n\n$\\fI''(y_*)=\\{\\ci\\in\\fI(y_*);\\ci=[i,j]\\text{ with $|[i,j]|=$even}\\}$. \n\\nl\nWe have \n\n(b) $|\\fI'(y_*)|\\cong m-1\\mod2$.\n\\nl\nLet $R(y_*)$ be the set of all $k\\in[0,m]$ such that $k=i$ or $k=j$ for some (necessarily unique) \n$[i,j]\\in\\fI(y_*)$. Let $R_0(y_*)$ be the set of all $k\\in[0,m]$ such that $k=i$ for some (necessarily unique) \n$[i,j]\\in\\fI(y_*)$ with $i=j$. Clearly,\n\n(c) $|R(y_*)|+|R_0(y_*)|=2|\\fI(y_*)|.$\n\n\\subhead 2.4\\endsubhead\nLet $x_*,x'_*\\in X_m$ and let $y_*=x_*+x'_*\\in\\NN^{m+1}$. Note that $y\\in Y_m$. If $k\\in\\fS(x_*)$ then \n$x_{k-1}<x_k<x_{k+1}$, $x'_{k-1}\\le x'_k\\le x'_{k+1}$ (and at least one of the last two $\\le$ is $<$). Hence \n$y_{k-1}<y_k<y_{k+1}$ (and at least one of the last two $<$ is $\\ll$). Hence $k\\in R(y_*)$. Thus \n$\\fS(x_*)\\sub R(y_*)$. Similarly, $\\fS(x'_*)\\sub R(y_*)$. We see that $\\fS(x_*)\\cup\\fS(x'_*)\\sub R(y_*)$. If \n$k\\in\\fS(x_*)\\cap\\fS(x'_*)$ then $x_{k-1}<x_k<x_{k+1}$, $x'_{k-1}<x'_k<x'_{k+1}$ hence $y_{k-1}\\ll y_k\\ll y_{k+1}$\nso that $k\\in R_0(y_*)$. Thus, $\\fS(x_*)\\cap\\fS(x'_*)\\sub R_0(y_*)$ and\n$$|\\fS(x_*)|+|\\fS(x'_*)|=|\\fS(x_*)\\cup\\fS(x'_*)|+|\\fS(x_*)\\cap\\fS(x'_*)|\\le|R(y_*)|+|R_0(y_*)|.$$\nUsing this and 2.3(c) we see that\n$$|\\fS(x_*)|+|\\fS(x'_*)|\\le2|\\fI(y_*)|,\\tag a$$\nwith equality if and only if $\\fS(x_*)\\cup\\fS(x'_*)=R(y_*)$ and $\\fS(x_*)\\cap\\fS(x'_*)=R_0(y_*)$.\n \n\\subhead 2.5\\endsubhead\nLet $y_*\\in Y_m$. We consider a partition $[0,m]=\\cj_0\\sqc\\cj_1\\sqc\\do\\cj_t$ where for each $s\\in[0,t]$ we have\n$\\cj_s=[m_s,m'_{s+1}]$ with $m_s\\le m'_{s+1}$, $m_0=0$, $m'_{t+1}=m$ and for each $s\\in[1,t]$ we have \n$m_s=m'_s+1$. We require that for $s\\in[1,t]$ we have $y_{m'_s}\\ll y_{m_s}$ and for any $s\\in[0,t]$ we have either\n\n(i) $|\\cj_s|=2$ and $(y_{m_s},y_{m'_{s+1}})=(a_s,a_s)$, or\n\n(ii) $(y_{m_s},y_{m_s+1},\\do,y_{m'_{s+1}})=(a_s,a_s+1,a_s+2,\\do)$. \n\\nl\nfor some $a_s\\in\\NN$. Such a partition exists and is unique. Let\n\n$\\cg_1(y_*)=\\{s\\in[0,t];s\\text{ is as in (i)}\\}$, $\\cg_2(y_*)=\\{s\\in[0,t];s\\text{ is as in (ii)}\\}$. \n\\nl\nWe have \n\n$\\fI(y_*)=\\{[i,j];i=m_s,j=m'_{s+1}\\text{ for some $s\\in\\cg_2(y_*)$}\\}$;\n\n$R(y_*)=\\{i\\in[0,m];i=m_s\\text{ or $i=m'_{s+1}$ for some $s\\in\\cg_2(y_*)$}\\}$;\n\n$R_0(y_*)=\\{i\\in[0,m];i=m_s=m'_{s+1}\\text{ for some $s\\in\\cg_2(y_*)$}\\}$.\n\n\\subhead 2.6\\endsubhead\nLet $y_*\\in Y_m$. Let $S'(y_*)$ be the set consisting of all pairs \n\n$x_*=(x_0,x_1,\\do,x_m)$, $x'_*=(x'_0,x'_1,\\do,x'_m)$\n\\nl\nin $\\NN^{m+1}$ which satisfy (i)-(iv) below (notation in 2.5):\n\n(i) for any $s\\in\\cg_1(y_*)$ we have $(x_{m_s},x_{m'_{s+1}})=(u_s,u_s)$, $(x'_{m_s},x'_{m'_{s+1}})=(u'_s,u'_s)$, \n$u_s+u'_s=a_s$;\n \n(ii) for any $s\\in\\cg_2(y_*)$ we have either\n\n(ii1) $(x_{m_s},x_{m_s+1},\\do,x_{m'_{s+1}})=(u_s,u_s+1,u_s+1,u_s+2,u_s+2,u_s+3,\\do)$,\n$(x'_{m_s},x'_{m_s+1},\\do,x'_{m'_{s+1}})=(u'_s,u'_s,u'_s+1,u'_s+1,u'_s+2,u'_s+2,\\do)$, $u_s+u'_s=a_s$, or\n\n(ii2) $(x_{m_s},x_{m_s+1},\\do,x_{m'_{s+1}})=(u_s,u_s,u_s+1,u_s+1,u_s+2,u_s+2,\\do)$,\n\n$(x'_{m_s},x'_{m_s+1},\\do,x'_{m'_{s+1}})=(u'_s,u'_s+1,u'_s+1,u'_s+2,u'_s+2,u'_s+3,\\do)$, $u_s+u'_s=a_s$;\n\n(iii) for any $s\\in[1,t]$ we have $x_{m'_s}<x_{m_s}$, $x'_{m'_s}<x'_{m_s}$;\n\n(iv) if $\\fI'(y_*)=\\em$ then for any $s\\in\\cg_2(y_*)$, \n$(x_{m_s},x_{m_s+1},\\do,x_{m'_{s+1}})$, \\lb\n$(x'_{m_s},x'_{m_s+1},\\do,x'_{m'_{s+1}})$ are as in (ii1).\n\\nl\nAn element $(x_*,x'_*)$ of $S'(y_*)$ can be constructed by induction as follows. Assume that the entries \n$x_i,x'_i$ have been already chosen for $i\\in\\cj_0\\cup\\cj_1\\cup\\do\\cup\\cj_{s-1}$ for some $s\\in[0,t]$ so that \n(i)-(iii) hold as far as it makes sense. In the case where $s>0$ let $\\x=x_{m'_s},\\x'=x'_{m'_s}$; in the case \nwhere $s=0$ let $\\x=\\x'=-\\iy$. In any case we have $\\x+\\x'\\le a_s-2$ hence we can find $u_s,u'_s$ in $\\NN$ such \nthat $\\x<u_s$, $\\x'<u'_s$, $u_s+u'_s=a_s$. (The number of choices is $y_{m_s}-y_{m'_s}-1$ if $s>0$ and $y_0+1$ if\n$s=0$.) Then we define \n\n$(x_{m_s},x_{m_s+1},\\do,x_{m'_{s+1}})$, $(x'_{m_s},x'_{m_s+1},\\do,x'_{m'_{s+1}})$\n\\nl\nby (i)\nif $s\\in\\cg_1(y_*)$ and by (ii) if $s\\in\\cg_2(y_*)$. This gives two choices for each $s\\in\\cg_2(y_*)$ such that \n$|\\cj_s|>1$, unless $\\fI'(y_*)=\\em$ when there is only one choice. This completes the inductive definition of \n$x_*,x'_*$. We see that $S'(y_*)\\ne\\em$.\n\nLet $S(y_*)$ be the set of all $(x_*,x'_*)\\in X_m\\T X_m$ such that (v),(vi),(vii) below hold:\n\n(v) $x_*+x'_*=y_*$,\n\n(vi) $\\fS(x_*)\\cup\\fS(x'_*)=R(y_*)$, $\\fS(x_*)\\cap\\fS(x'_*)=R_0(y_*)$ (or equivalently \n$|\\fS(x_*)|+|\\fS(x'_*)|=2|\\fI(y_*)|$),\n\n(vii) if $\\fI'(y_*)=\\em$ (so that $m$ is odd), then $\\fS(x'_*)=\\em$.\n\\nl\nFrom the definitions we see that $S(y_*)=S'(y_*)$. Hence \n\n(a) $S(y_*)\\ne\\em$.\n\\nl\nFrom 2.4(a) we see that:\n\n(b) if $\\fI(y_*)=\\em$ and $(x_*,x'_*)\\in S(y_*)$ then $\\fS(x_*)=\\em$, $\\fS(x'_*)=\\em$.\n\\nl\nOn the other hand,\n\n(c) if $\\fI'(y_*)\\ne\\em$ and $(x_*,x'_*)\\in S(y_*)$ then $\\fS(x_*)\\ne\\em$, $\\fS(x'_*)\\ne\\em$.\n\\nl\nIndeed, let $[i,j]\\in\\fI'(y_*)$. Then we have either $i\\in\\fS(x_*),j\\in\\fS(x'_*)$ or $i\\in\\fS(x'_*),j\\in\\fS(x_*)$;\nin both cases the conclusion of (c) holds.\n\n\\subhead 2.7\\endsubhead\nIn this subsection we assume that $m$ is even, $\\ge2$. We set \n\n$\\tX_m=\\{x_*\\in X_m;x_0=0,x_1\\ge1\\}$, \n$\\tY_m=\\{y_*\\in Y_m;y_1\\ge1\\}$. \n\\nl\nIf $x_*\\in X_m$, $x'_*\\in\\tX_m$, then $x_*+x'_*\\in\\tY_m$. \n\nLet $y_*\\in\\tY_m$ be such that \n\n(a) $y_0=0,y_1=1$. \n\\nl\n(Thus $\\fI(y_*)$ contains an interval of form $[0,\\a]$ hence $\\fI(y_*)\\ne\\em$.) Let $\\tS'(y_*)$ be the set \nconsisting of all pairs $x_*=(x_0,x_1,\\do,x_m)$, $x'_*=(x'_0,x'_1,\\do,x'_m)$ in $\\NN^{m+1}$ which satisfy the \nconditions (i)-(iii) in 2.6 together with conditions (i),(ii) below (notation in 2.5):\n\n(i) for $s=0$ (necessarily in $\\cg_2(y_*)$) we have \n\n$(x_0,x_1,\\do,x_{m'_1})=(0,0,1,1,2,2,\\do)$, $(x'_0,x'_1,\\do,x'_{m'_1})=(0,1,1,2,2,3,3,\\do)$\n\\nl\n(so that $0\\in\\fS(x'_*)$);\n\n(ii) if $\\fI(y_*)=\\{[0,\\a]\\}\\cup\\fI''(y_*)$ (so that $\\fI'(y_*)=\\{[0,\\a]\\}$) then \nfor any $s\\in\\cg_2(y_*)-\\{0\\}$, $(x_{m_s},x_{m_s+1},\\do,x_{m'_{s+1}})$, \n$(x'_{m_s},x'_{m_s+1},\\do,x'_{m'_{s+1}})$ are as in 2.6(ii1).\n\\nl\nWe can construct an element in $\\tS'(y_*)$ by the same method as in 2.6. In particular, $\\tS'(y_*)\\ne\\em$. \n\nNow let $\\tS(y_*)$ be the set of all $(x_*,x'_*)\\in X_m\\T\\tX_m$ such that \n\n(iii) $x_*+x'_*=y_*$,\n\n(iv) $\\fS(x_*)\\cup\\fS(x'_*)=R(y_*)$, $\\fS(x_*)\\cap\\fS(x'_*)=R_0(y_*)$ (or equivalently \n$|\\fS(x_*)|+|\\fS(x'_*)|=2|\\fI(y_*)|$),\n\n(v) if $\\fI(y_*)=\\{[0,\\a]\\}\\cup\\fI''(y_*)$, then $\\fS(x'_*)=\\{0\\}$.\n\\nl\nFrom the definitions we see that $\\tS(y_*)=\\tS'(y_*)$. Hence \n\n(b) $\\tS(y)\\ne\\em$.\n\\nl\nNote that\n\n(c) if $\\fI(y_*)=\\{[0,\\a]\\}$ and $(x_*,x'_*)\\in\\tS(y_*)$, then $\\fS(x_*)=\\{\\a\\}$, $\\fS(x'_*)=\\{0\\}$.\n\\nl\nIndeed from 2.4(a) we see that $|\\fS(x_*)|+|\\fS(x'_*)|\\le2$. On the other hand, we have $0\\in\\fS(x'_*)$ and\n$\\a\\in\\fS(x_*)$ (see (i)) since in this case $\\a$ is even; (c) follows. Note that\n\n(d) if $\\fI'(y_*)$ contains at least one interval $\\ne[0,\\a]$ and $(x_*,x'_*)\\in\\tS(y_*)$, then $|\\fS(x'_*)|\\ge3$.\n\\nl\nIndeed, let $[i,j]\\in\\fI'(y_*)$, $[i,j]\\ne[0,\\a]$. Then we have either $i\\in\\fS(x_*),j\\in\\fS(x'_*)$ or \n$i\\in\\fS(x'_*),j\\in\\fS(x_*)$. Since $0\\in\\fS(x'_*)$ we see that $|\\fS(x'_*)|\\ge2$. Since $|\\fS(x'_*)|$ is odd we \nsee that $|\\fS(x'_*)|\\ge3$.\n\n\\subhead 2.8\\endsubhead\nLet $x_*^0\\in X_m$ be $(0,0,1,1,\\do,(n-1),(n-1),n)$ if $m=2n$ and \\lb $(0,0,1,1,\\do,n,n)$ if $m=2n+1$. For any \n$x_*\\in X_m$ we have $x_i\\ge x^0_i$ for all $i\\in[0,m]$. Hence \n\n$\\r:X_m@>>>\\NN$, $\\x_*\\m\\sum_{i\\in[0,m]}(x_i-x^0_i)$ and \n\n$\\b:X_m@>>>\\NN$, $x_*\\m\\sum_{0\\le i<j\\le m}(x_i-x_i^0)$\n\\nl\nare well defined. \n\nLet $y_*^0\\in Y_m$ be $(0,0,2,2,\\do,(m-2),(m-2),m)$ if $m$ is even and \\lb\n$(0,0,2,2,\\do,(m-1),(m-1))$ if $m$ is odd.\nFor any $y_*\\in Y_m$ we have $y_i\\ge y^0_i$ for all $i\\in[0,m]$. Hence \n\n$\\r':Y_m@>>>\\NN$, $y_*\\m\\sum_{i\\in[0,m]}(y_i-y^0_i)$ and \n\n$\\b':Y_m@>>>\\NN$, $y_*\\m\\sum_{0\\le i<j\\le m}(y_i-y_i^0)$\n\\nl\nare\nwell defined. Since $x^0_*+x^0_*=y^0_*$ we have \n\n$\\r'(x_*+x'_*)=\\r(x_*)+\\r(x'_*)$, $\\b'(x_*+x'_*)=\\b(x_*)+\\b(x'_*)$\n\\nl\nfor any $x_*,x'_*\\in X_m$. For any $n\\in\\NN$ we set $X_m^n=\\{x_*\\in X_m;\\r(x_*)=n\\}$, \n$Y_m^n=\\{y_*\\in Y_m;\\r'(y_*)=n\\}$.\n\nAssume that $m=2k$, $k\\ge1$. Let $\\tx_*^0\\in\\tX_m$ be $(0,1,1,\\do,k,k)$. For any $x_*\\in\\tX_m$ we have \n$x_i\\ge\\tx^0_i$ for all $i$. Hence \n\n$\\ti\\r:\\tX_m@>>>\\NN$, $\\x_*\\m\\sum_{i\\in[0,m]}(x_i-\\tx^0_i)$ and\n\n$\\tib:\\tX_m@>>>\\NN$, $x_*\\m\\sum_{0\\le i<j\\le m}(x_i-\\tx_i^0)$\n\\nl\nare well defined. Let \n$\\ty_*^0=(0,1,2,3,\\do,m)\\in Y_m$. For any $y_*\\in\\tY_m$ we have $y_i\\ge\\ty^0_i$ for all $i$. Hence \n\n$\\ti\\r':\\tY_m@>>>\\NN$, $y_*\\m\\sum_{i\\in[0,m]}(y_i-\\ty^0_i)$ and \n\n$\\tib':\\tY_m@>>>\\NN$, $y_*\\m\\sum_{0\\le i<j\\le m}(y_i-\\ty_i^0)$\n\\nl\nare well defined. Since $x^0_*+\\tx^0_*=\\ty^0_*$ we have \n\n$\\ti\\r'(x_*+x'_*)=\\r(x_*)+\\ti\\r(x'_*)$, $\\tib'(x_*+x'_*)=\\b(x_*)+\\tib(x'_*)$ \n\\nl\nfor any $x_*\\in X_m$, $x'_*\\in\\tX_m$. For any $n\\in\\NN$ we set \n\n$\\tX_m^n=\\{x_*\\in\\tX_m;\\ti\\r(x_*)=n\\}$, $\\tY_m^n=\\{y_*\\in\\tY_m;\\ti\\r'(y_*)=n\\}$.\n\n\\subhead 2.9\\endsubhead\nLet $\\ce_m$ be the set of all $e_*=(e_0,e_1,\\do,e_m)\\in\\NN^{m+1}$ such that $e_0\\le e_1\\le\\do\\le e_m$. For any \n$n\\in\\NN$ let $\\ce_m^n=\\{e_*\\in\\ce_m;\\sum_ie_i=n\\}$.\n\nLet $x_*\\in X_m$. We associate to $x_*$ an element $\\hx_*\\in X_m$ as follows. Let $i_0<i_1<\\do<i_s$ be the \nelements of $\\fS(x_*)$ in increasing order. Clearly, each of the sets \n$[0,i_0-1],[i_0+1,i_1-1],\\do,[i_{s-1}+1,i_s-1],[i_s+1,m]$ has even cardinal, say \n$2t_0,2(t_1-1),\\do,2(t_s-1),2t_{s+1}$ (respectively). We define $\\hx_*=(\\hx_0,\\hx_1,\\do,\\hx_m)\\in\\NN^{m+1}$ by\n$$(\\hx_0,\\hx_1,\\do,\\hx_{i_0}-1)=(0,0,1,1,\\do,t_0-1,t_0-1), h_{i_0}=t_0,$$\n$$\\align&(\\hx_{i_0+1},\\hx_{i_0+2},\\do,\\hx_{i_1-1})\\\\&\n=(t_0+1,t_0+1,t_0+2,t_0+2,\\do,t_0+t_1-1,t_0+t_1-1), h_{i_1}=t_0+t_1,\\endalign$$\n$$\\align&(\\hx_{i_1+1},\\hx_{i_1+2},\\do,\\hx_{i_2-1})=\\\\&\n(t_0+t_1+1,t_0+t_1+1,t_0+t_1+2,t_0+t_1+2,\\do,t_0+t_1+t_2-1,\nt_0+t_1+t_2-1),\\\\& h_{i_2}=t_0+t_1+t_2,\\endalign$$\n$$\\do$$\n$$\\align&(\\hx_{i_s+1},\\hx_{i_s+2},\\do,hx_m)\\\\&\n=(t_0+t_1+\\do+t_s+1,t_0+t_1+\\do+t_s+1,t_0+t_1+\\do+t_s+2,\\\\&\nt_0+t_1+\\do+t_s+2,\\do,t_0+t_1+\\do+t_{s+1},t_0+t_1+\\do+t_{s+1}).\\endalign$$\nNote that $\\hx_*$ depends only on $\\fS(x_*)$, not on $x_*$ itself. We have $\\hx_*\\in X_m$, $\\fS(\\hx_*)=\\fS(x_*)$.\nLet $e_*=x_*-\\hx_*$. We have $e_*\\in\\ce_m$. Moreover for any $i\\in[0,m-1]$ such that $\\hx_i=\\hx_{i+1}$ we have \n$e_i=e_{i+1}$.\n\n\\subhead 2.10\\endsubhead\nLet $x_*\\in X_m$, $e_*\\in\\ce_m$. Then $x_*+e_*\\in X_m$ hence $y_*:=x_*+e_*+x_*\\in Y_m$. Assume that \n$\\fS(e_*+x_*)=\\fS(x_*)$ and $(x_*,e_*+x_*)\\in S(y_*)$. Then \n\n$\\fS(x_*)\\cup\\fS(e_*+x_*)=R(y_*)$, $\\fS(x_*)\\cap\\fS(e_*+x_*)=R_0(y_*)$\n\\nl\nhence $\\fS(x_*)=R(y_*)=R_0(y_*)$. It follows that for any $\\ci\\in\\fI(y_*)$ we \nhave $|\\ci|=1$.  \n\n\\subhead 2.11\\endsubhead\nConversely, let $y_*\\in Y_m^n$ be such that for any $\\ci\\in\\fI(y_*)$ we have $|\\ci|=1$. By 2.6(a) we can find \n$(x_*,x'_*)\\in S(y_*)$. We have $x_*+x'_*=y_*$, $\\fS(x_*)\\cup\\fS(x'_*)=R(y_*)$, $\\fS(x_*)\\cap\\fS(x'_*)=R_0(y_*)$.\nFrom our assumption we have $R_0(y_*)=R(y_*)$. Hence $\\fS(x_*)\\cup\\fS(x'_*)=\\fS(x_*)\\cap\\fS(x'_*)$ so that \n$\\fS(x_*)=\\fS(x'_*)$. By 2.9 we have $\\hx_*=\\hx'_*\\in X_m$ and $e_*:=x_*-\\hx_*\\in\\ce_m,e'_*:=x'_*-\\hx'_*\\in\\ce_m$.\nMoreover, if $i\\in[0,m-1]$ and $\\hx_i=\\hx_{i+1}$ then $e_i=e_{i+1}$ and $e'_i=e'_{i+1}$ hence $\\te_i=\\te_{i+1}$ \nwhere $\\te_*=e_*+e'_*\\in\\ce_m$. It follows that $\\fS(\\te_*+\\hx_*)=\\fS(\\hx_*)=\\fS(x_*)=R(y_*)=R_0(y_*)$. Since \n$|\\fS(x_*)|+|\\fS(x'_*)|=2|\\fI(y_*)|$ we have $|\\fS(\\hx_*)|+|\\fS(\\te_*+\\hx_*)|=2|\\fI(y_*)|$. Also, if\n$\\fI'(y_*)=\\em$ then by 2.6(vii) we have $\\fS(x'_*)=\\em$. Hence $\\fS(\\hx_*)=\\fS(\\te_*+\\hx_*)=\\em$. In any case we\nsee that $y_*=\\hx_*+\\te_*+\\hx_*$, $(\\hx_*,\\te_*+\\hx_*)\\in S(y_*)$,\n$\\fS(\\te_*+\\hx_*)=\\fS(\\hx_*)$.  \n\n\\head 3. Type $A_{n-1}$\\endhead\n\\subhead 3.1\\endsubhead\nFor $n\\in\\NN$ let $S_n$ be the group of all permutations of $\\{1,2,\\do,n\\}$. We have $S_0=S_1=\\{1\\}$; for $n\\ge2$\nwe regard $S_n$ as a Coxeter group whose generators are the transpositions $(i,i+1)$ for $i\\in[1,n-1]$. We have \n$\\cs_{S_n}=\\Irr(S_n)$. If $k$ is large (relative to $n$) we have a natural bijection $\\Irr(S_n)\\lra Z_k^n$, \n$[z_*]\\lra z_*$, see \\cite{\\OR, 4.4}. For example, $[(0,1,\\do,k-n,k-n+2,\\do,k,k+1)]$ is the sign representation\nof $S_n$. For any $z_*\\in Z_k^n$ we have $\\b_0(z_*)=b_{[z_*]}$, see \\cite{\\OR, (4.4.2)}. \n\nAssume now that $n=n'+n''$ with $n',n''$ in $\\NN$. The set of permutations of $\\{1,2,\\do,n\\}$ which leave stable \neach of the subsets $\\{1,2,\\do,n'\\}$, $\\{n'+1,n'+2,\\do,n\\}$ is a standard parabolic subgroup of $S_n$ which may be\nidentified with $S_{n'}\\T S_{n''}$.\n\nFor $z'_*\\in Z_k^{n'}$, $z''_*\\in Z_k^{n'}$ we have $z'_*+z''_*-z^0_*\\in Z_k^n$ and from the definitions we have:\n\n(a) $[[z'_*+z''_*-z^0_*]:\\Ind_{S_{n'}\\T S_{n''}}^{S_n}([z'_*]\\bxt[z''_*])]_{S_n}=1$.\n\\nl\nNote also that $\\b_0(z'_*)+\\b_0(z''_*)=\\b_0(z'_*+z''_*-z^0_*)$ hence\n$b_{[z'_*]}+b_{[z''_*]}=b_{[z'_*+z''_*-z^0_*]}$, so that\n\n(b) $[z'_*+z''_*-z^0_*]=j_{S_{n'}\\T S_{n''}}^{S_n}([z'_*]\\bxt[z''_*])$.\n\n\\subhead 3.2\\endsubhead\nIn this subsection we assume that $G$ is of type $A_{n-1}$ ($n\\ge2)$. In this case 1.5(a),(b1),(b2) are immediate.\nWe prove 1.5(b3).\n\nFor $C\\in\\cx$ let $E=\\r_C$. We have $E=[z_*]$ for a unique $z_*\\in Z_k^n$. We have $\\zz_C=1$ and\n$\\tzz_C=\\text{g.c.d.}\\{n,z_j-z_j^0(j\\in[0,k])\\}$ where $\\text{g.c.d.}$ denotes the greatest common divisor. We \nidentify $\\{1,2,\\do,n\\}=\\ZZ\/n$ in the obvious way. We also identify $\\WW=S_n$ as Coxeter groups so that the\nreflections $s_i(i\\in\\tI)$ are the transpositions $(i,i+1)$ with $i\\in\\ZZ\/n$ (with $i+1$ computed in $\\ZZ\/n$.) Now\n$\\Om$ is a cyclic group of order $n$ with generator $\\o:i\\m i+1$ for all $i\\in\\ZZ\/n$. For any $d|n$ (divisor \n$d\\ge1$ of $n$) let $\\Om_d$ be the subgroup of $\\Om$ generated by $\\o^{n\/d}$. For any coset $P$ of $\\Om_d$ in \n$\\Om$ \nlet $S_n^P$ be the set of all permutations $w$ of $\\ZZ\/n$ such that for any $r\\in P$ the subset \n$\\{r+1,r+2,\\do,r+(n\/d)\\}$ is $w$-stable. We may identify $S_n^P$ with a product of $d$ copies of $S_{n\/d}$. Note \nthat $\\cp^{\\Om_d}$ (see 1.11) \nconsists of the subgroups $S_n^P$ as above; each of these subgroups is stable under the\nconjugation action of $\\Om_d$ on $\\WW$. An irreducible representation $\\bxt_{h=1}^d[\\tz_*^{(h)}]$ (with \n$\\tz_*^{(h)}\\in Z_k^{n\/d}$) of $S_n^P$ (identified with $S_{n\/d}^d$) is $\\Om_d$-stable if and only if \n$\\tz_*^{(h)}=\\tz_*$ is independent of $h$; in this case we have\n$$j_{S_n^P}^{S_n}(\\bxt_{h=1}^d[\\tz_*^{(h)}])=[\\sum_{h=1}^d\\tz^{(h)}_*-(d-1)z^0_*]=[d\\tz_*-(d-1)z^0_*]$$\nas we see by applying $(d-1)$ times 3.1(b). Using this and 1.11 we see that \n$$\\fc_{[z_*]}=\\max d$$\nwhere $\\max$ is taken over all divisors $d\\ge1$ of $n$ such that $z_*-z^0_*=d(\\tz_*-z^0_*)$ for some \n$\\tz_*\\in Z_k^{n\/d}$. Equivalently, we have\n$$\\fc_{[z_*]}=\\text{g.c.d.}\\{n,z_j-z^0_j (j\\in[0,k])\\}.$$\nSince this is equal to $\\tzz_C\/\\zz_C$ we see that 1.5(b3) is proved in our case.\n\n\\head 4. Type $B_n$\\endhead\n\\subhead 4.1\\endsubhead\nFor $n\\in\\NN$ let $W_n$ be the group of permutations of the set \\lb\n$\\{1,2,\\do,n,n',\\do,2',1'\\}$ which commute with \nthe involution $i\\m i',i'\\m i(i\\in[1,n])$. We have $W_0=\\{1\\}$; for $n\\ge1$ we regard $W_n$ as a Coxeter group of\ntype $B_n=C_n$ whose generators are the transposition $(n,n')$ and the products of two transpositions \n$(i,i+1)((i+1)',i')$ for $i\\in[1,n-1]$. By \\cite{\\CL, \\S2} we have $\\Irr(W_n)=\\Irr(W_n)^\\da$.\n\n\\subhead 4.2\\endsubhead\nIn the remainder of this section we fix an even integer $m=2k$ which is large relative to $n$.\n\nLet $U_k^n=\\{(z_*;z'_*)\\in Z_k\\T Z_{k-1};\\r_0(z_*)+\\r_0(z'_*)=n\\}$. As in \\cite{\\OR, 4.5} we have a bijection \n\n(a) $\\Irr(W_n)\\lra U_k^n$, $[z_*;z'_*]\\lra(z_*;z'_*)$.\n\\nl\n(In {\\it loc.cit.} the notation $\\left(\\sm z_*\\\\z'_*\\esm\\right)$ was used instead of $(z_*;z'_*)$.) By \n\\cite{\\CL, \\S2} we have \n\n(b) $b_{[z_*;z'_*]}=2\\b_0(z_*)+2\\b_0(z'_*)+\\r_0(z'_*)$.\n\\nl\nThere is a unique bijection $\\z_n:\\cs_{W_n}@>\\si>>X_m^n$ under which $x_*\\in X_m^n$ corresponds to$\\{[z_*,z'_*]\\}$\nwhere $z_*=(x_0,x_2,x_4,\\do,x_m)$, $z'_*=(x_1,x_3,x_5,\\do,x_{m-1})$. This bijection has the following property: if\n$E\\in\\cs_{W_n},x_*=\\z_n(E)$ then $b_E=\\b(x_*)$, $f_E=2^{(|\\fS(x_*)|-1)\/2}$.\n\n\\subhead 4.3\\endsubhead\nLet $u_*\\in Z_m$. Define $\\ddu_*\\in Z_k$, $\\du_*\\in Z_{k-1}$ by $\\ddu_i=u_{2i}-i$ for $i\\in[0,k]$, \n$\\du_i=u_{2i+1}-i-1$ for $i\\in[0,k-1]$. \n\n\\subhead 4.4\\endsubhead\nLet $(p,q)\\in\\NN^2$ be such that $p+q=n$. The group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in $W_n$ \nthat leave stable each of the subsets\n$$\\{1,2,\\do,p\\}, \\{p',\\do,2',1'\\}, \\{p+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(p+1)'\\}$$\nis a standard parabolic subgroup of $W_n$ which may be identified with $S_p\\T W_q$ in an obvious way.\n\nLet $(\\tz_*;\\tz'_*)\\in U_k^q$, $u_*\\in Z_m^p$. Let \n$v_*=\\tz_*+\\ddu_*-z^{0,k}_*$, $v'_*=\\tz'_*+\\du_*-z^{0,k-1}_*$. Then $(v_*;v'_*)\\in U_k^n$, $[u_*]\\in\\Irr(S_p)$, \n$[\\tz_*;\\tz'_*]\\in\\Irr(W_q)$, $[v_*;v'_*]\\in\\Irr(W_n)$. We show:\n\n(a) $[v_*;v'_*]=j_{S_p\\T W_q}^{W_n}([u_*]\\bxt[\\tz_*;\\tz'_*])$. \n\\nl\nWe can assume that $p\\ge1$ and that the result holds for $p$ replaced by $\\tp<p$. In the case where $[u_*]$ is the\nsign representation of $S_p$, (a) can be proved along the lines of \n\\cite{\\UC, 2.7}. If $[u_*]$ is not the sign \nrepresentation of $S_p$, we can find $p',p''$ in $\\NN_{>0}$ such that $p'+p''=p$ and $u'\\in Z_{2k}^{p'}$, \n$u'\\in Z_{2k}^{p''}$ such that $u_*=u'_*+u''_*-z^{0,m}_*$. By 3.1(b), we have \n$[u_*]=j_{S_{p'}\\T S_{p''}}^{S_p}([u'_*]\\bxt[u''_*])$. Hence \n$$[u_*]\\bxt[\\tz_*;\\tz'_*]=j_{S_{p'}\\T S_{p''}\\T W_q}^{S_p\\T W_q}([u'_*]\\bxt[u''_*]\\bxt[\\tz_*;\\tz'_*])$$\nand \n$$\\align j_{S_p\\T W_q}^{W_n}([u_*]\\bxt[\\tz_*;\\tz'_*])&=\nj_{S_p\\T W_q}^{W_n}j_{S_{p'}\\T S_{p''}\\T W_q}^{S_p\\T W_q}([u'_*]\\bxt[u''_*]\\bxt[\\tz_*;\\tz'_*])\\\\&=\nj_{S_{p'}\\T S_{p''}\\T W_q}^{W_n}([u'_*]\\bxt[u''_*]\\bxt[\\tz_*;\\tz'_*])\\\\&=\nj_{S_{p'}\\T W_{p''+q}}^{W_n}j_{S_{p'}\\T S_{p''}\\T W_q}^{S_{p'}\\T W_{p''+q}}\n([u'_*]\\bxt[u''_*]\\bxt[\\tz_*;\\tz'_*])\\\\&=\nj_{S_{p'}\\T W_{p''+q}}^{W_n}([u'_*]\\bxt[\\tz_*+\\ddu''_*-z_*^{0,k};\\tz'_*+\\du''_*-z_*^{0,k-1}])\\\\&\n=[\\tz_*+\\ddu''_*+\\ddu'_*-2z_*^{0,k};\\tz'_*+\\du''_*+\\du'_*-2z_*^{0,k-1}]\\\\&\n=[\\tz_*+\\ddu_*-z_*^{0,k};\\tz'_*+\\du_*-z_*^{0,k-1}].\\endalign$$\n(We have used the induction hypothesis for $p$ replaced by $p'$ or $p''$.) This proves (a).\n\n\\subhead 4.5\\endsubhead\nIn the remainder of this section we assume that $G$ has type $B_n$ ($n\\ge2$). We identify $\\WW=W_n$ as Coxeter \ngroups in the standard way. The reflections $s_j(j\\in\\tI)$ are the transpositions $(n,n')$, $(1,1')$ and the \nproducts of two transpositions $(i,i+1)(i',(i+1)')$ for $i\\in[1,n-1]$. The group $\\Om$ has order $2$ with generator\ngiven by the involution $i\\m(n+1-i)',i'\\m(n+1-i)$ for $i\\in[1,n]$.\n\nLet $(r,p,q)\\in\\NN^3$ be such that $r+p+q=n$. The group of all permutations of \\lb\n$\\{1,2,\\do,n,n',\\do,2',1'\\}$ in \n$W_n$ that leave stable each of the subsets\n$$\\align&\\{1,2,\\do,r\\}\\cup\\{r',\\do,2',1'\\}, \\{r+1,r+2,\\do,r+p\\}, \\\\&\n\\{(r+p)',\\do,(r+2)',(r+1)'\\}, \\\\&\n\\{r+p+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(r+p+1)'\\}\\endalign$$\nis a parahoric subgroup of $\\WW$ which may be identified with $W_r\\T S_p\\T W_q$ in an obvious way.\n\nLet $(z_*;z'_*)\\in U_k^r$, $(\\tz_*;\\tz'_*)\\in U_k^q$, $u_*\\in Z_{2k}^p$, Define $\\ddu_*\\in Z_k$,$\\du_*\\in Z_{k-1}$\nas in 4.3. Let $w_*=z_*+\\tz_*+\\ddu_*-2z^{0,k}_*$, $w'_*=z'_*+\\tz'_*+\\du_*-2z^{0,k-1}_*$. Then \n$(w_*,w'_*)\\in U_k^n$, $[z_*;z'_*]\\in\\Irr(W_r)$, $[u_*]\\in\\Irr(S_p)$, $[\\tz_*;\\tz'_*]\\in\\Irr(W_q)$, \n$[w_*,w'_*]\\in\\Irr(W_n)$. We show:\n\n(a) $[w_*;w'_*]=j_{W_r\\T S_p\\T W_q}^{W_n}([z_*;z'_*]\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*])$. In particular,\n\n$[[w_*;w'_*]:\\Ind_{W_r\\T S_p\\T W_q}^{W_n}([z_*;z'_*]\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*])]_{W_n}=1$.\n\\nl\nAssume first that $p=0$. We have:\n$$[[z_*+\\tz_*-z^{0,k}_*;z'_*+\\tz'_*-z^{0,k-1}_*]:\\Ind_{W_r\\T W_q}^{W_n}([z_*;z'_*]\\bxt[\\tz_*;\\tz'_*])]_{W_n}=1.\n\\tag b$$\nUsing the definitions this can be deduced from the analogous statement for $S_n$, see 3.1(a). Moreover we have \n$b_{[z_*+\\tz_*-z^{0,k}_*;z'_*+\\tz'_*-z^{0,k-1}_*]}=b_{[z_*;z'_*]}+b_{[\\tz_*;\\tz'_*]}$. It follows that\n$$[z_*+\\tz_*-z^{0,k}_*;z'_*+\\tz'_*-z^{0,k-1}_*]=j_{W_r\\T W_q}^{W_n}([z_*;z'_*]\\bxt[\\tz_*;\\tz'_*]).\\tag c$$\nThus (a) holds in this special case. \n\nIn the general case we use 4.4(a) with $n$ replaced by $p+q$ and (c) applied to $n,r,0,p+q$ instead of $n,r,p,q$.\nWe obtain\n$$\\align&j_{W_r\\T S_p\\T W_q}^{W_n}([z_*;z'_*]\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*])\\\\&=\nj_{W_r\\T W_{p+q}}^{W_n}(j_{W_r\\T S_p\\T W_q}^{W_r\\T W_{p+q}}([z_*;z'_*]\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*]))\\\\&=\nj_{W_r\\T W_{p+q}}^{W_n}([z_*;z'_*]\\bxt[\\tz_*+\\ddu_*-z^{0,k}_*;\\tz'_*+\\du_*-z^{0,k-1}_*])=[w_*;w'_*].\\endalign$$\nThis proves (a).\n\n\\subhead 4.6\\endsubhead\nBy \\cite{\\ICC, \\S13}, there is a unique bijection $\\t:\\tcs_\\WW@>>>Y_m^n$ such that for any $y_*\\in Y_m^n$, the \nfibre $\\t\\i(y_*)$ is $[z_*,z'_*]$ where $z_*=(y_0,y_2-1,y_4-2,\\do,y_m-m\/2)$, \n$z'_*=(y_1,y_3-1,y_5-2,\\do,y_{m-1}-(m-2)\/2)$. This bijection has the following property: if $C\\in\\cx$ and \n$y_*=\\t(\\r_C)$, then $\\bb_C=\\b'(y_*)$, $\\zz_C=2^{|\\fI(y_*)|-1}$. From \\cite{\\ICC, \\S14} we see that:\n\n$\\tzz_C\/\\zz_C=2$ if $|\\ci|=1$ for any $\\ci\\in\\fI(y_*)$,\n\n$\\tzz_C\/\\zz_C=1$ if $|\\ci|>1$ for some $\\ci\\in\\fI(y_*)$.\n\n\\subhead 4.7\\endsubhead\nIn the setup of 4.5 we assume that $[z_*;z'_*]\\in\\cs_{W_r}$, $[\\tz_*;\\tz'_*]\\in\\cs_{W_q}$. Define $x_*\\in X_m^r$,\n$\\tx_*\\in X_m^q$ by $\\z_r([z_*;z'_*])=x_*$, $\\z_q([\\tz_*;\\tz'_*])=\\tx_*$. Let $e_*=u_*-z^{0,m}_*\\in\\ce_m$. We \nshow: \n\n(a) $[w_*,w'_*]\\in\\tcs_\\WW$ and $\\t([w_*,w'_*])=x_*+e_*+\\tx_*$.\n\\nl\nWe have $w_i=x_{2i}+\\tx_{2i}+u_{2i}-i-2i$ for $i\\in[0,k]$, $w'_i=x_{2i+1}+\\tx+_{2i+1}+u_{2i+1}-i-1-2i$ for \n$i\\in[0,k-1]$. Define $y_*\\in\\NN^{m+1}$ by $w_i=y_{2i}-i$ for $i\\in[0,k]$, $w'_i=y_{2i+1}-i$ for $i\\in[0,k-1]$.\nThen $y_*=x_*+\\tx_*+e_*$. Since $x_*\\in X_m,\\tx_*\\in X_m,e_*\\in\\ce_m$ we have $y_*\\in Y_m$. More precisely,\n$y_*\\in Y_m^n$. Using 4.6 we deduce that $[w_*,w'_*]\\in\\tcs_\\WW$ and (a) follows.\n\nFrom (a) and 4.5(a) we see that for $(r,p,q)$ as in 4.5, the assignment \n\n$(E_1,E_2,\\tE_1)\\m j_{W_r\\T S_p\\T W_q}^{W_n}(E_1\\bxt E_2\\bxt\\tE_1)$\n\\nl\nis a map $j:\\cs_{W_r}\\T\\cs_{S_p}\\T\\cs_{W_q}@>>>\\tcs_\\WW$ and we have a commutative diagram\n$$\\CD\n\\cs_{W_r}\\T\\cs_{S_p}\\T\\cs_{W_q}@>j>>\\tcs_\\WW\\\\\n@V\\z_r\\T\\x_p\\T\\z_qVV @V\\t VV\\\\\nX_m^r\\T\\ce_m^p\\T X_m^q@>h>>Y_m^n \\endCD$$                          \nwhere $h$ is given by $(x_*,e_*,\\tx_*)\\m x_*+e_*+\\tx_*$ and $\\x_p:\\cs_{S_p}@>>>\\ce_m^p$ is the bijection \n$[e_*+z_*^{0,m}]\\lra e_*$.\n\n\\subhead 4.8\\endsubhead\nNote that $\\cp'$ (see 1.9) is exactly the collection of parahoric subgroups $W_r\\T S_0\\T W_q$ of $W_n$ with \n$(r,p,q)$ as in 4.5 and $p=0$. By 4.7, $j_{W_r\\T S_0\\T W_q}^{W_n}$ carries $\\cs_{W_r}\\T\\cs_{S_0}\\T\\cs_{W_q}$ into\n$\\tcs_\\WW$. Hence $\\bcs_\\WW\\sub\\tcs_\\WW$.\n\nConversely, let $E\\in\\tcs_\\WW$. With $\\t$ as in 4.6, let $y_*=\\t(E)\\in Y_m^n$. By 2.6(a) we can find \n$(x_*,\\tx_*)\\in S(y_*)$. Define $r,q$ in $\\NN$ by $x_*\\in X_m^r,\\tx_*\\in X_m^q$. We must have $r+q=n$. Let \n$e_*=(0,0,\\do,0)\\in\\ce_m^0$. In the commutative diagram in 4.7 (with $p=0$) we have $h(x_*,e_*,\\tx_*)=y_*$, \n$(x_*,e_*,\\tx_*)=(\\z_r(E_1),\\x_p(\\QQ),\\z_q(\\tE_1))$ where $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W_q}$ (recall that \n$\\z_r,\\z_q$ are bijections) and $\\t(j(E_1,\\QQ,\\tE_1))=\\t(E)$. Since $\\t$ is bijective we deduce that \n$E=j(E_1,\\QQ,\\tE_1)$. Thus, $E\\in\\bcs_\\WW$. Thus, $\\tcs_\\WW\\sub\\bcs_\\WW$. We see that $\\tcs_\\WW=\\bcs_\\WW$. This \nproves 1.5(a) in our case.\n\n\\subhead 4.9\\endsubhead\nIn the remainder of this section we fix $C\\in\\cx$ and we set $E=\\r_C\\in\\tcs_\\WW$, $y_*=\\t(E)\\in Y_m^n$ ($\\t$ as in\n4.6).\n\nLet $(r,q)\\in\\NN^2$, $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W_q}$ be such that $r+q=n$, \n\n$E=j_{W_r\\T S_0\\T W_q}^{W_n}(E_1\\bxt\\QQ\\bxt\\tE_1)$. \n\\nl\n(These exist since $E\\in\\bcs_\\WW$.) We set\n$x_*=\\z_r(E_1)\\in X_m^r$, $\\tx_*=\\z_q(\\tE_1)\\in X_m^q$. From the commutative diagram in 4.7 we see that \n$x_*+\\tx_*=y_*$. By 4.6 we have $\\bb_C=\\b'(y_*)$. Since $\\b'(x_*+\\tx_*)=\\b(x_*)+\\b(\\tx_*)$, we have \n$\\bb_C=\\b(x_*)+\\b(\\tx_*)$. Since $\\b(x_*)=b_{E_1}$, $\\b(\\tx_*)=b_{\\tE_1}$, we have $\\bb_C=b_{E_1}+b_{\\tE_1}$ hence\n$\\bb_C=b_{E_1\\bxt\\QQ\\bxt\\tE_1}$. Since $E=j_{W_r\\T S_0\\T W_q}^{W_n}(E_1\\bxt\\QQ\\bxt\\tE_1)$ we have \n$b_{E_1\\bxt\\QQ\\bxt\\tE_1}=b_E$ hence $\\bb_C=b_E$, proving 1.5(b1) in our case. \n\nNext we note that $f_{E_1}=2^{(|\\fS(x_*)|-1)\/2}$, $f_{\\tE_1}=2^{(|\\fS(\\tx_*)|-1)\/2}$, $\\zz_C=2^{|\\fI(y_*)|-1}$, \n$|\\fS(x_*)|+|\\fS(\\tx_*)|\\le2|\\fI(y_*)|$. Hence\n$$f_{E_1\\bxt\\QQ\\bxt\\tE_1}=2^{(|\\fS(x_*)|+|\\fS(\\tx_*)|-2)\/2}\\le 2^{|\\fI(y_*)|-1}=\\zz_C.$$\nTaking maximum over all $r,q,E_1,\\tE_1$  as above we obtain $\\fa_E\\le\\zz_C$.\n\nUsing again 2.6(a) we can find $(x_*,\\tx_*)\\in S(y_*)$. Define $r,q$ in $\\NN$ by $x_*\\in X_m^r$, $\\tx_*\\in X_m^q$.\nWe must have $r+q=n$. Define $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W_q}$ by $x_*=\\z_r(E_1),\\tx_*=\\z_q(\\tE_1)$. As \nearlier in the proof we have $E=j_{W_r\\T\\QQ\\T W_q}^{W_n}(E_1\\bxt\\QQ\\bxt\\tE_1)$. We have \n$$f_{E_1\\bxt\\QQ\\bxt\\tE_1}=2^{(|\\fS(x_*)|+|\\fS(\\tx_*)|-2)\/2}=2^{|\\fI(y_*)|-1}=\\zz_C.$$ \nIt follows that $\\fa_E=\\zz_C$, proving 1.5(b2) in our case.\n\n\\subhead 4.10\\endsubhead\nAssume now that $\\tzz_C\/\\zz_C=2$. By 4.6, for any $\\ci\\in\\fI(y_*)$ we have $|\\ci|=1$. By 2.11 we can find \n$(r,p,q)$ as in 4.5 with $q=r$ and $x_*\\in X_m^r$, $e_*\\in\\ce_m^p$ such that $y_*=x_*+e_*+x_*$, \n$(x_*,e_*+x_*)\\in S(y_*)$, $\\fS(e_*+x_*)=\\fS(x_*)$. Define $E_1\\in\\cs_{W_r}$, $E_2\\in\\cs_{S_p}$ by\n$x_*=\\z_r(E_1)$, $e_*=\\x_p(E_2)$. Using the commutative diagram in 4.7 we see that \n$E=j_{W_r\\T S_p\\T W_r}^{W_n}(E_1\\bxt E_2\\bxt E_1)$. Moreover, \n$$f_{E_1\\bxt E_2\\bxt E_1}=2^{(|\\fS(x_*)|+|\\fS(x_*)|-2)\/2}=2^{(|\\fS(x_*)|+|\\fS(e_*+x_*)|-2)\/2}=2^{|\\fI(y_*)|-1}\n=\\zz_C.$$ \nWe have $W_r\\T S_p\\T W_r=\\WW_J$ for a unique $J$ which is $\\Om$-stable. Moreover, $E_1\\bxt E_2\\bxt E_1$ is \n$\\Om$-stable. We see that $\\fc_E=2$.\n\n\\subhead 4.11\\endsubhead\nConversely, assume that $\\fc_E=2$. Using 1.11 we see that there exist $(r,p,q)$ as in 4.5 with $q=r$ and \n$E_1\\in\\cs_{W_r}$, $E_2\\in\\cs_{S_p}$ such that $E=j_{W_r\\T S_p\\T W_r}^{W_n}(E_1\\bxt E_2\\bxt E_1)$, \n$f_{E_1\\bxt E_2\\bxt E_1}=\\zz_C$. We set $x_*=\\z_r(E_1)\\in X_m^r$, $e_*=\\x_p(E_2)$. We have $y_*=x_*+e_*+x_*$ and\n$$2^{(|\\fS(x_*)|+|\\fS(x_*)|-2)\/2}=2^{|\\fI(y_*)|-1};$$\nhence $|\\fS(x_*)|+|\\fS(x_*)|=|\\fI(y_*)|$. Let $E'_1=j_{S_p\\T W_r}^{W_{p+r}}(E_2\\bxt E_1)\\in\\cs_{W_{p+r}}$. Then \n$E=j_{W_r\\T W_{p+r}}^{W_n}(E_1\\bxt E'_1)$. Using 1.5(b2) and the definition we have \n$f_{E_1\\bxt E'_1}\\le\\fa_E=\\zz_C$. By 1.9(b) we have $f_{E_2\\bxt E_1}\\le f_{E'_1}$. Hence \n$\\zz_C=f_{E_1\\bxt E_2\\bxt E_1}\\le f_{E_1\\bxt E'_1}\\le\\zz_C$; this forces $f_{E_2\\bxt E_1}=f_{E'_1}$. The last \nequality can be rewritten as\n$$2^{(|\\fS(x_*)|-1)\/2}=2^{(|\\fS(e_*+x_*)|-1)\/2}$$\nsince $e_*+x_*=\\z_{p+r}(E'_1)$ (a consequence of 4.4(a)). Hence $|\\fS(e_*+x_*)|=|\\fS(x_*)|$ and \n$|\\fS(x_*)|+|\\fS(e_*+x_*)|=2|\\fI(y_*)|$. Thus, $(x_*,e_*+x_*)\\in S(y_*)$. Using 2.10 we see that for any \n$\\ci\\in\\fI(y_*)$ we have $|\\ci|=1$. By 4.6 we have $\\tzz_C\/\\zz_C=2$. \n\n\\subhead 4.12\\endsubhead\nFrom 4.10, 4.11, we see that $\\tzz_C\/\\zz_C=2$ if and only if $\\fc_E=2$. Since $\\fc_E\\in[1,2]$ and \n$\\tzz_C\/\\zz_C\\in[1,2]$ we see that $\\tzz_C\/\\zz_C=\\fc_E$; this proves 1.5(b3) in our case.\n\n\\head 5. Type $C_n$\\endhead\n\\subhead 5.1\\endsubhead\nFor $n\\in\\NN$ let $W'_n$ be the set of all elements in $W_n$ which are even permutations of\n$\\{1,2,\\do,n,n',\\do,2',1'\\}$. We have $W'_0=W'_1=\\{1\\}$. For $n\\ge2$ we regard $W'_n$ as a Coxeter group of type \n$D_n$ whose generators are the products of two transpositions $(i,i+1)((i+1)',i')$ for $i\\in[1,n-1]$ and\n$(n-1,n')(n,(n-1)')$.\n\n\\subhead 5.2\\endsubhead\nIn this subsection we fix an integer $k$ which is large relative to $n$.\n\nLet $V_k^n$ be the set of unordered pairs $(z_*,z'_*)$ in $Z_{k-1}\\T Z_{k-1}$ such that $\\r_0(z^*)+\\r_0(z'_*)=n$.\nIf $n\\ge2$ we have as in \\cite{\\OR, 4.5} a map $\\io:\\Irr(W'_n)@>>>V_k^n$. (In {\\it loc.cit.} the notation \n$\\left(\\sm z_*\\\\z'_*\\esm\\right)$ was used instead of $(z_*,z'_*)$.) Now $\\io$ is also defined when \n$n\\in\\{0,1\\}$; it is the unique map between two sets of cardinal $1$. \n\nLet ${}^\\da V_k^n$ be the set of {\\it ordered} pairs $(z_*;z'_*)$ in $Z_{k-1}\\T Z_{k-1}$ such that \n$\\r_0(z^*)+\\r_0(z'_*)=n$ and either $\\r_0(z_*)>\\r_0(z'_*)$ or $z_*=z'_*$. We regard ${}^\\da V_k^n$ as a subset of\n$V_k^n$ by forgetting the order of a pair. We define a partition ${}^\\da V_k^n={}'V_k^n\\sqc{}''V_k^n$ by\n\n${}''V_k^n=\\{(z_*;z'_*)\\in{}^\\da V_k^n;z_*=z'_*\\}$ if $n\\ge2$, ${}''V_k^n=\\em$ if $n\\le1$,\n\n${}'V_k^n=\\{(z_*;z'_*)\\in{}^\\da V_k^n;z_*\\ne z'_*\\}$ if $n\\ge1$, ${}'V_k^n={}^\\da V_k^n$ if \n$n=0$.\n\\nl\nBy \\cite{\\CL, \\S2} we have $\\Irr(W'_n)^\\da=\\io\\i({}^\\da V_k^n)$. For $(z_*;z'_*)\\in^\\da V_k^n$ and $\\k\\in\\{0,1\\}$ \nwe define $[z_*,z'_*]^\\k\\in\\Irr(W'_n)^\\da$ by the following requirements: if $(z_*;z'_*)\\in{}'V_k^n$, then \n$\\io\\i(z_*;z'_*)$ has a single element $[z_*;z'_*]^0=[z_*;z'_*]^1$; if $(z_*;z'_*)\\in{}''V_k^n$, then \n$\\io\\i(z_*;z'_*)$ consists of two elements $[z_*;z'_*]^0,[z_*;z'_*]^1$.\n\nBy \\cite{\\CL, \\S2}, if $(z_*;z'_*)\\in{}^\\da V_k^n$ then $b_{[z_*;z'_*]^\\k}=2\\b_0(z_*)+2\\b_0(z'_*)+\\r_0(z'_*)$. \n\nThere is a unique map $\\z'_n:\\cs_{W'_n}@>>>X_{2k-1}^n$ such that for any $x_*\\in X_{2k-1}^n$, $\\z'_n{}\\i(x_*)$ is\n$\\{[z_*;z'_*]^0=[z_*;z'_*]^1\\}$ (if $\\fS(x_*)\\ne\\em$ or if $n=0$) and is $\\{[z_*;z'_*]^0,[z_*;z'_*]^1\\}$ (if\n$\\fS(x_*)=\\em$ and $n\\ge2$) where \n\n$z_*=(x_1,x_3,x_5,\\do,x_{2k-1})$, $z'_*=(x_0,x_2,x_4,\\do,x_{2k-2})$. \n\\nl\nThis map has the following property: if $E\\in\\cs_{W'_n},x_*=\\z'_n(E)$ then $b_E=\\b(x_*)$, \n$f_E=2^{\\max((|\\fS(x_*)|-2)\/2,0)}$.\n\nThere is a unique map $\\tiz_n:\\cs_{W'_n}@>>>\\tX_{2k}^n$ such that for any $x_*\\in\\tX_{2k}^n$, $\\tiz_n\\i(x_*)$ is\n$\\{[z_*;z'_*]^0=[z_*,z'_*]^1\\}$ (if $\\fS(x_*)\\ne\\{0\\}$ or if $n=0$) and is $\\{[z_*;z'_*]^0,[z_*,z'_*]^1\\}$ (if \n$\\fS(x_*)=\\{0\\}$ and $n\\ge2$) where $z_*=(x_2-1,x_4-1,\\do,x_{2k}-1)$, $z'_*=(x_1-1,x_3-1,x_5-1,\\do,x_{2k-1}-1)$.\n\nThis map has the following property: if $E\\in\\cs_{W'_n},x_*=\\tiz_n(E)$, then $b_E=\\b(x_*)$, \n$f_E=2^{\\max((|\\fS(x_*)|-3)\/2,0)}$.\n\n\\subhead 5.3\\endsubhead\nIn the remainder of this section we assume that $G$ is of type $C_n$ ($n\\ge3$) and we identify $\\WW=W_n$ as \nCoxeter groups in the standard way; we also fix an even integer $m=2k$ which is large relative to $n$. The \nreflections $s_j(j\\in\\tI)$ are the transposition $(1,1')$ and the products of two transpositions \n$(i,i+1)(i',(i+1)')$ for $i\\in[1,n-1]$ and $(n-1,n')(n,(n-1)')$. The group $\\Om$ has order $2$ with generator given\nby the transposition $(n,n')$.\n\nLet $(r,q)\\in\\NN^2$ be such that $r+q=n$. The group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in $W_n$ \nthat leave stable the subset $\\{1,2,\\do,r\\}\\cup\\{r',\\do,2',1'\\}$ and which restrict to an even permutation of \n$\\{r+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(r+1)'\\}$, is a parahoric subgroup of $\\WW$ which may be identified with \n$W_r\\T W'_q$ in an obvious way. Let $(z_*;z'_*)\\in U_k^r$, \n$(\\tz_*;\\tz'_*)\\in{}^\\da V_k^q$. Let \n\n$\\tz_*^!=(0,\\tz_0+1,\\tz_1+1,\\do,\\tz_{k-1}+1)\\in Z_k$. \n\\nl\nLet $w_*=z_*+\\tz_*^!-z^{0,k}_*$, $w'_*=z'_*+\\tz'_*-z^{0,k-1}_*$. Then $[z_*;z'_*]\\in\\Irr(W_r)$, \n$[\\tz_*;\\tz'_*]^\\k\\in\\Irr(W'_q)^\\da$ $(k=0,1$), $[w_*;w'_*]\\in\\Irr(W_n)$ are well defined and we have\n\n(a) $[[w_*;w'_*]:\\Ind_{W_r\\T W'_q}^{W_n}([z_*;z'_*]\\bxt[\\tz_*;\\tz'_*]^\\k)]_{W_n}=1$.\n\\nl\n(This can be deduced from the second sentence in 4.5(a) with $p=0$.) Moreover, we have \n$b_{[w_*;w'_*]}=b_{[z_*;z'_*]}+b_{[\\tz_*;\\tz'_*]^\\k}$. It follows that\n\n(b) $[w_*;w'_*]=j_{W_r\\T W'_q}^{W_n}([z_*;z'_*]\\bxt[\\tz_*;\\tz'_*]^\\k)$.\n\n\\subhead 5.4\\endsubhead\nBy \\cite{\\ICC, \\S12}, there is a unique bijection $\\ti\\t:\\tcs_\\WW@>>>\\tY_m^n$ such that for any $y_*\\in\\tY_m^n$, \nthe fibre $\\ti\\t\\i(y_*)$ is $\\{[z_*,z'_*]\\}$ where $z_*=(y_0,y_2-1,y_4-2,\\do,y_m-m\/2)$, \n$z'_*=(y_1-1,y_3-2,y_5-3,\\do,y_{m-1}-m\/2)$. This bijection has the following property: if $C\\in\\cx$ and \n$y_*=\\ti\\t(\\r_C)$ then $\\bb_C=\\tib'(y_*)$, $\\zz_C=2^{|\\fI(y_*)|-1-\\ti\\d_{y_*})}$ where $\\ti\\d_{y_*}=1$ if there \nexists $\\ci\\in\\fI'(y_*)$ such that $0\\n\\ci$ and $\\ti\\d_{y_*}=0$ if there is no $\\ci\\in\\fI'(y_*)$ such that \n$0\\n\\ci$. Moreover, $\\tzz_C=2^{|\\fI(y_*)|-1)}$. Hence $\\tzz_C\/\\zz_C=2^{\\ti\\d_{y_*}}$.\n\n\\subhead 5.5\\endsubhead\nIn the setup of 5.3 we assume that $[z_*;z'_*]\\in\\cs_{W_r}$, $[\\tz_*;\\tz'_*]^k\\in\\cs_{W'_q}$. We set \n$x_*=\\z_r([z_*;z'_*])\\in X_m^r$, $\\tx_*=\\tiz_q([\\tz_*;\\tz'_*]^\\k)\\in\\tX_m^q$. We show:\n\n(a) $[w_*,w'_*]\\in\\tcs_\\WW$ and $\\ti\\t([w_*,w'_*])=x_*+\\tx_*$.\n\\nl\nWe have $w_i=x_{2i}+\\tx_{2i}-i$, $w'_i=x_{2i+1}+\\tx_{2i+1}-1-i$. Define $y_*\\in\\NN^{m+1}$ by $y_{2i}=w_i+i$ for \n$i\\in[0,k]$, $y_{2i+1}=w'_i+i+1$ for $i\\in[0,k-1]$. We have $y_*=x_*+\\tx_*$. Since $x_*\\in X_m,\\tx_*\\in X_m,$ we \nhave $y_*\\in\\tY_m$. More precisely, $y_*\\in\\tY_m^n$. Using 5.4 we deduce that $[w_*,w'_*]\\in\\tcs_\\WW$ and (a) \nfollows.\n\nFrom (a) and 5.3(b) we see that for $(r,q)$ as in 5.3, the assignment \n$(E_1,\\tE_1)\\m j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$ is a map $j:\\cs_{W_r}\\T\\cs_{W'_q}@>>>\\tcs_\\WW$ and we have a \ncommutative diagram\n$$\\CD\n\\cs_{W_r}\\T\\cs_{W'_q}@>j>>\\tcs_\\WW\\\\\n@V\\z_r\\T\\tiz_qVV @V\\ti\\t VV\\\\\nX_m^r\\T\\tX_m^q@>h>>\\tY_m^n\\endCD$$                          \nwhere $h$ is given by $(x_*,\\tx_*)\\m x_*+\\tx_*$.\n\n\\subhead 5.6\\endsubhead\nNote that $\\cp'$ is exactly the collection of subgroups $W_r\\T W'_q$ of $W_n$ with $(r,q)$ as in 5.3 and $q\\ne1$.\n(On the other hand $W_{n-1}\\T W'_1$ is a maximal parabolic subgroup of the Coxeter group $W_n$.) By 5.5, \n$j_{W_r\\T W'_q}^{W_n}$ carries $\\cs_{W_r}\\T\\cs_{W'_q}$ into $\\tcs_\\WW$. Hence $\\bcs_\\WW\\sub\\tcs_\\WW$.\n\nConversely, let $E\\in\\tcs_\\WW$. With $\\ti\\t$ as in 5.4, let $y_*=\\ti\\t(E)\\in\\tY_m^n$. By 2.7(b) we can find \n$(x_*,\\tx_*)\\in\\tS(y_*)$. (The assumption 2.7(a) is automatically satisfied since $m$ is large relative to $n$.) \nDefine $r,q$ in $\\NN$ by $x_*\\in X_m^r,\\tx_*\\in\\tX_m^q$. We must have $r+q=n$. In the commutative diagram in 5.5 \nwe have $h(x_*,\\tx_*)=y_*$, $(x_*,\\tx_*)=(\\z_r(E_1),\\tiz_q(\\tE_1))$ where $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W'_q}$\n(recall that $\\z_r$, $\\tiz_q$ are surjective) and $\\ti\\t(j(E_1,\\tE_1))=\\ti\\t(E)$. Since $\\ti\\t$ is bijective we \ndeduce that $E=j(E_1,\\tE_1)$. Thus, $E\\in\\bcs_\\WW$ and $\\tcs_\\WW\\sub\\bcs_\\WW$. We see that $\\tcs_\\WW=\\bcs_\\WW$. \nThis proves 1.5(a) in our case.\n\n\\subhead 5.7\\endsubhead\nIn the remainder of this section we fix $C\\in\\cx$ and we set $E=\\r_C\\in\\tcs_\\WW$, $y_*=\\ti\\t(E)\\in\\tY_m^n$ (with \n$\\ti\\t$ as in 5.4). \n\nLet $(r,q)\\in\\NN^2$, $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W'_q}$ be such that $r+q=n$, \n$E=j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$. (These exist since $E\\in\\bcs_\\WW$.) We set $x_*=\\z_r(E_1)\\in X_m^r$,\n$\\tx_*=\\tiz_q(\\tE_1)\\in\\tX_m^q$. From the commutative diagram in 5.5 we see that $x_*+\\tx_*=y_*$. By 5.4 we have \n$\\bb_C=\\tib'(y_*)$. Since $\\tib'(x_*+\\tx_*)=\\b(x_*)+\\tib(\\tx_*)$ we have $\\bb_C=\\b(x_*)+\\tib(\\tx_*)$. Since \n$\\b(x_*)=b_{E_1}$, $\\tib(\\tx_*)=b_{\\tE_1}$, we have $\\bb_C=b_{E_1}+b_{\\tE_1}$ hence $\\bb_C=b_{E_1\\bxt\\tE_1}$. \nSince $E=j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$ we have $b_{E_1\\bxt\\tE_1}=b_E$ hence $\\bb_C=b_E$, proving 1.5(b1) in\nour case. \n\nIf $|\\fS(\\tx_*)|\\ge3$ then \n$$\\align&f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-1)\/2}2^{(|\\fS(\\tx_*)|-3)\/2}\\\\&=\n2^{(|\\fS(x_*)|+|\\fS(\\tx_*)|-4)\/2}\\le 2^{|\\fI(y_*)|-2}\\le\\zz_C.\\endalign$$\nIf $|\\fS(\\tx_*)|=1$ and $|\\fS(x_*)|\\le2|\\fI(y_*)|-3$ then \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-1)\/2}\\le 2^{|\\fI(y_*)|-2}\\le\\zz_C.$$\nIf $|\\fS(\\tx_*)|=1$ (hence $\\fS(\\tx_*)=\\{0\\}$) and $|\\fS(x_*)|=2|\\fI(y_*)|-1$ then $\\ti\\d_{y_*}=0$ so that \n$\\zz_C=2^{|\\fI(y_*)|-1}$ and \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-1)\/2}=2^{|\\fI(y_*)|-1}=\\zz_C.$$\nThus in any case we have $f_{E_1\\bxt\\tE_1}\\le\\zz_C$. Taking maximum over all $r,q,E_1,\\tE_1$ as above we obtain \n$\\fa_E\\le\\zz_C$.\n\n\\subhead 5.8\\endsubhead\nAssume now that $\\ti\\d_{y_*}=1$. Then $|\\fI(y_*)|\\ge2$. By 2.7(b) we can find $(x_*,\\tx_*)\\in\\tS(y_*)$. By 2.7(d)\nwe have $|\\fS(\\tx_*)|\\ge3$. Define $(r,q)\\in\\NN^2$ by $x_*\\in X_m^r,\\tx_*\\in\\tX_m^q$. We must have $r+q=n$. We can\nfind $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z_r(E_1),\\tx_*=\\tiz_q(\\tE_1)$. As earlier in the \nproof, we have $E=j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$ and \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-1)\/2+(|\\fS(\\tx_*)|-3)\/2}=2^{|\\fI(y_*)|-2}=\\zz_C.$$\n\n\\subhead 5.9\\endsubhead\nNext we assume that $\\ti\\d_{y_*}=0$. By 2.7(b) we can find $(x_*,\\tx_*)\\in\\tS(y_*)$. By 2.7(v) we have\n$\\fS(\\tx_*)=\\{0\\}$. Then $|\\fS(x_*)|=2|\\fI(y_*)|-1$. Define $(r,q)\\in\\NN^2$ by $x_*\\in X_m^r,\\tx_*\\in\\tX_m^q$. We\nmust have $r+q=n$. We can find $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z_r(E_1)$,\n$\\tx_*=\\tiz_q(\\tE_1)$. We have $E=j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$ and \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-1)\/2}=2^{|\\fI(y_*)|-1}=\\zz_C.$$\nUsing this and 5.8 we see that in any case, $\\fa_E=\\zz_C$, proving 1.5(b2) in our case.\n\n\\subhead 5.10\\endsubhead\nAssume first that $\\d_{y_*}=1$. Let $r,q,x_*,\\tx_*,E_1,\\tE_1$ be as in 5.8. Then $|\\fS(\\tx_*)|\\ge3$ hence $q\\ge1$\n(so that the unique $J$ such that $\\WW_J=W_r\\T W'_q$ is $\\Om$-stable) and $\\tE_1$ is $\\Om$-stable. It follows that \n$\\fc_E=2$.\n\nConversely, assume that $\\fc_E=2$. Using 1.11 we see that there exist $(r,q)\\in\\NN^2$ be such that $r+q=n$ with \n$q\\ge1$ and $E_1\\in\\cs_{W_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $\\tE_1$ is $\\Om$-stable, \n$E=j_{W_r\\T W'_q}^{W_n}(E_1\\bxt\\tE_1)$, $f_{E_1\\bxt\\tE_1}=\\zz_C$. We set $x_*=\\z_r(E_1)\\in X_m^r$, \n$\\tx_*=\\tiz_q(\\tE_1)\\in\\tX_m^q$. We have $y_*=x_*+\\tx_*$. Since $\\tE_1$ is $\\Om$-stable, we have $|\\fS(\\tx_*)|\\ge3$.\nHence \n$$2^{|\\fI(y_*)|-2}\\le\\zz_C=f_{E_1\\bxt\\tE_1}=2^{(|\\fS(x_*)|-1)\/2+(|\\fS(\\tx_*)|-3)\/2}\\le2^{|\\fI(y_*)|-2}.$$\nIt follows that $2^{|\\fI(y_*)|-2}=\\zz_C$ so that $\\ti\\d_{y_*}=1$. \n\nWe see that $\\tzz_C\/\\zz_C=2$ if and only if $\\fc_E=2$. Since $\\fc_E\\in[1,2]$ and $\\tzz_C\/\\zz_C\\in[1,2]$ we see \nthat $\\tzz_C\/\\zz_C=\\fc_E$; this proves 1.5(b3) in our case.\n \n\\head 6. Type $D_n$\\endhead\n\\subhead 6.1\\endsubhead\nIn this section we assume that $G$ is of type $D_n$ ($n\\ge4$). We identify $\\WW=W'_n$ as Coxeter groups in the \nusual way. The reflections $s_j(j\\in\\tI)$ are the products of two transpositions $(i,i+1)(i',(i+1)')$ for \n$i\\in[1,n-1]$ and $(n-1,n')(n,(n-1)')$, $(1,2')(2,1')$. Define $\\o_1\\in W'_n$ by $i\\m(n+1-i)',i'\\m n+1-i$ for \n$i\\in[1,n-1]$, $n\\m1$, $n'\\m1'$ (if $n$ is even) and by $i\\m(n+1-i)',i'\\m n+1-i$ for $i\\in[1,n]$ (if $n$ is \neven). Define $\\o_2\\in W'_n$ by $i\\m i$ for $i\\in[2,n-1]$, $1\\m1',1'\\m1,n\\m n',n'\\m n$. We have $\\o_1,\\o_2\\in\\Om$.\nIf $n$ is odd, $\\Om$ is cyclic of order $4$ with generator $\\o_1$ such that $\\o_1^2=\\o_2$. If $n$ is even, $\\Om$ is\nnoncyclic of order $4$ with generators $\\o_1,\\o_2$ of order $2$.\n\n\\subhead 6.2\\endsubhead\nIn the remainder of this section we fix an odd integer $m=2k-1$ which is large relative to $n$.\n\nLet $(p,q)\\in\\NN^2$ be such that $p+q=n$, $q\\ge1$. The group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ \nin $W_n$ that leave stable each of the subsets $\\{1,2,\\do,p\\}$, $\\{p',\\do,2',1'\\}$ and induce an even permutation\non the subset $\\{p+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(p+1)'\\}$ is a standard parabolic subgroup of $W'_n$ which may\nbe identified with $S_p\\T W'_q$ in an obvious way.\n\nLet $(\\tz_*;\\tz'_*)\\in{}'V_k^q$, $u_*\\in Z_{2k-1}^p$. Define $\\ddu_*\\in Z_{k-1}$, $\\du_*\\in Z_{k-1}$ by \n$\\ddu_i=u_{2i}-i$, $\\du_i=u_{2i+1}-i-1$ for $i\\in[0,k-1]$. Let $v_*=\\tz_*+\\du_*-z^{0,k-1}_*$, \n$v'_*=\\tz'_*+\\ddu_*-z^{0,k-1}_*$. Then $(v_*;v'_*)\\in{}'V_k^n$, $[u_*]\\in\\Irr(S_p)$, \n$[\\tz_*;\\tz'_*]\\in\\Irr(W'_q)$, $[v_*;v'_*]\\in\\Irr(W'_n)$. We have:\n\n(a) $[v_*;v'_*]^0=j_{S_p\\T W'_q}^{W'_n}([u_*]\\bxt[\\tz_*;\\tz'_*]^0)$. \n\\nl\nThe proof is similar to that of 4.4(a).\n\n\\subhead 6.3\\endsubhead\nLet $(r,p,q)\\in\\NN^3$ be such that $r+p+q=n$. The group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in \n$W'_n$ that leave stable each of the subsets \n$$\\{r+1,r+2,\\do,r+p\\}, \\{(r+p)',\\do,(r+2)',(r+1)'\\}$$\nand induce an even permutation on each of the subsets \n$$\\{1,2,\\do,r\\}\\cup\\{r',\\do,2',1'\\}, \\{r+p+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(r+p+1)'\\}$$\nis a parahoric subgroup of $\\WW$ which may be identified with $W'_r\\T S^{(0)}_p\\T W'_q$ in an obvious way.\n($S^{(0)}_p$ is a copy of $S_p$.)\n\nWhen $r=0,p\\ge2$, the group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in $W'_n$ that leave stable each \nof the subsets \n$$\\{1',2,\\do,p\\},\\{p',\\do,2',1\\},\\{p+1,\\do,n-1,n\\}\\cup\\{n',(n-1)',\\do,(p+1)'\\}$$\nis a parahoric subgroup of $\\WW$ which may be identified with $W'_r\\T S_p^{(1)}\\T W'_q$. ($S^{(1)}_p$ is a copy of\n$S_p$.)\n\nWhen $p\\ge2,q=0$, the group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in $W'_n$ that leave stable each \nof the subsets \n$$\\{r+1,r+2,\\do,n-1,n'\\},\\{n,(n-1)'\\do,(r+2)',(r+1)'\\},\\{1,2,\\do,r\\}\\cup\\{r',\\do,2',1'\\}$$ \nis a parahoric subgroup of $\\WW$ which may be identified with $W'_r\\T S_p^{(2)}\\T W'_q$. ($S^{(2)}_p$ is a copy of\n$S_p$.)\n\nWhen $r=q=0$, the group of all permutations of $\\{1,2,\\do,n,n',\\do,2',1'\\}$ in $W'_n$ that leave stable each of \nthe subsets $\\{1',2,3,\\do,n-1,n'\\}$, $\\{n,(n-1)'\\do,3',2',1\\}$ is a parahoric subgroup of $\\WW$ which may be \nidentified with $W'_r\\T S_p^{(3)}\\T W'_q$. ($S^{(3)}_p$ is a copy of $S_p$.)\n\nThus the parahoric subgroup $W'_r\\T S_p^{(\\l)}\\T W'_q$ is defined in the following cases:\n\n(a) $\\l=0$; $p\\ge2,r=0,\\l=1$; $p\\ge2,q=0,\\l=2$; $r=q=0,\\l=3$.\n\\nl\nWhen $p=0$ we write also $W'_r\\T W'_q$ instead of $W'_r\\T S_p^{(0)}\\T W'_q$.\n\nLet $(z_*;z'_*)\\in{}^\\da V_k^r$, $(\\tz_*;\\tz'_*)\\in{}^\\da V_k^q$, $u_*\\in Z_{2k-1}^p$, Define $\\ddu_*\\in Z_{k-1}$,\n$\\du_*\\in Z_{k-1}$ by $\\ddu_i=u_{2i}-i$, $\\du_i=u_{2i+1}-i-1$ for $i\\in[0,k-1]$. Let \n$w_*=z_*+\\tz_*+\\du_*-2z^{0,k-1}_*$, $w'_*=z'_*+\\tz'_*+\\ddu_*-2z^{0,k-1}_*$. Then $(w_*,w'_*)\\in{}^\\da V_k^n$.\n\nFor $\\k,\\ti\\k,\\k'\\in\\{0,1\\}$ we have $[z_*;z'_*]^\\k\\in\\Irr(W'_r)^\\da$, $[u_*]\\in\\Irr(S_p)$, \n$[\\tz_*;\\tz'_*]^{\\ti\\k}\\in\\Irr(W'_q)^\\da$, $[w_*,w'_*]^{\\k'}\\in\\Irr(W'_n)^\\da$. For $\\l$ as in (a) we have:\n\n(b) $[w_*;w'_*]^{\\k'}=j_{W'_r\\T S_p^{(\\l)}\\T W'_q}^{W'_n}([z_*;z'_*]^\\k\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*]^{\\ti\\k})$\n\\nl\nwith the following restriction on $\\k'$: if $z_*=z'_*,\\tz_*=\\tz'_*$, \n$\\du_*=\\ddu_*$, then $w_*=w'_*$ and $\\k'$ in \n(b) is uniquely determined by $\\k,\\ti\\k,\\l$; moreover, both $\\k'=0$ and $\\k'=1$ are obtained from some \n$(\\k,\\ti\\k,\\l)$.\n\nNow (b) can be proved in a way similar to 4.5(a); alternatively, from the second statement of 4.5(a) one can \ndeduce that\n$$[[w_*;w'_*]^{\\k'}:\n\\Ind_{W'_r\\T S_p^{(\\l)}\\T W'_q}^{W'_n}([z_*;z'_*]^\\k\\bxt[u_*]\\bxt[\\tz_*;\\tz'_*]^{\\ti\\k})]_{W'_n}\\ge1;$$\nwe can also check directly that $b_{[w_*;w'_*]^{\\k'}}=b_{[z_*;z'_*]^\\k}+b_{[u_*]}+b_{[\\tz_*;\\tz'_*]^{\\ti\\k}}$ \nand (b) follows.\n\n\\subhead 6.4\\endsubhead\nBy \\cite{\\ICC, \\S13}, we have $\\tcs_\\WW\\sub\\Irr(W'_n)^\\da$ and there is a unique map $\\t:\\tcs_\\WW@>>>Y_m^n$ such \nthat for $y_*\\in Y_m^n$, $\\t\\i(y_*)$ consists of $[z_*;z'_*]^0=[z_*;z'_*]^1$ (if $\\fI(y_*)\\ne\\em$) and consists of\n$[z_*;z'_*]^0,[z_*;z'_*]^1$ (if $\\fI(y_*)=\\em$) where \n\n$z_*=(y_1,y_3-1,y_5-2,\\do,y_m-(m-1)\/2)$, \n\n$z'_*=(y_0,y_2-1,y_4-2,\\do,y_{m-1}-(m-1)\/2)$.\n\\nl\nThis map has the following property: if $C\\in\\cx$ and \n$y_*=\\t(\\r_C)$, then $\\bb_C=\\b'(y_*)$, $\\zz_C=2^{\\max(|\\fI(y_*)|-1-\\d_{y_*},0)}$ where $\\d_{y_*}=1$ if\n$\\fI'(y_*)\\ne\\em$ and $\\d_{y_*}=0$ if $\\fI'(y_*)=\\em$. Moreover, $\\tzz_C\/\\zz_C$ is:\n\n$4$ if $\\d_{y_*}=1$ and $|\\ci|=1$ for any $\\ci\\in\\fI(y_*)$,\n\n$2$ if $\\d_{y_*}=1$ and $|\\ci|>1$ for some $\\ci\\in\\fI(y_*)$,\n\n$2$ if $\\fI(y_*)=\\em$,\n\n$1$ if $\\d_{y_*}=0$ and $|\\ci|>1$ for some $\\ci\\in\\fI(y_*)$.\n\\nl\nMore precisely, let $\\uG@>>>G$ be a double covering which is a special orthogonal group and let $\\uzz_C$ be the \nnumber of connected components of the centralizer in $\\uG$ of a unipotent element of $\\uG$ which maps to an \nelement of $C$. From \\cite{\\ICC, \\S14} we see that:\n\n$\\tzz_C\/\\uzz_C=2$ if $|\\ci|=1$ for any $\\ci\\in\\fI(y_*)$,\n\n$\\tzz_C\/\\zz_C=1$ if $|\\ci|>1$ for some $\\ci\\in\\fI(y_*)$.\n\\nl\nOn the other hand, from $\\uzz_C=2^{\\max(|\\fI(y_*)|-1,0)}$, $\\zz_C=2^{\\max(|\\fI(y_*)|-1-\\d_{y_*},0)}$, we see that\n$\\uzz_C\/\\zz_C=2^{\\d_{y_*}}$.\n\n\\subhead 6.5\\endsubhead\nIn the setup of 6.3 we assume that $[z_*;z'_*]^\\k\\in\\cs_{W'_r}$, $[\\tz_*;\\tz'_*]^{\\ti\\k}\\in\\cs_{W'_q}$ and $\\k'$ \nis as in 6.3(b). Define $x_*\\in X_m^r,\\tx_*\\in X_m^q$ by $\\z'_r([z_*;z'_*]^\\k)=x_*$, \n$\\z'_q([\\tz_*;\\tz'_*]^{\\ti\\k})=\\tx_*$. Let $e_*=u_*-z^{0,m}_*\\in\\ce_m$. We show: \n\n(a) $[w_*,w'_*]^{\\k'}\\in\\tcs_\\WW$ and $\\t([w_*,w'_*]^{\\k'})=x_*+e_*+\\tx_*$.\n\\nl\nWe have $w_i=x_{2i+1}+\\tx_{2i+1}+u_{2i+1}-i-2i-1$, $w'_i=x_{2i}+\\tx_{2i}+u_{2i}-i-2i$ for $i\\in[0,k-1]$. Define \n$y_*\\in\\NN^{m+1}$ by $w_i=y_{2i+1}-i$, $w'_i=y_{2i}-i$ for $i\\in[0,k-1]$. Then $y_*=x_*+\\tx_*+e_*$. Since \n$x_*\\in X_m,\\tx_*\\in X_m,e_*\\in\\ce_m$ we have $y_*\\in Y_m$. More precisely, $y_*\\in Y_m^n$. Using 6.4 we deduce\nthat $[w_*,w'_*]^{\\k'}\\in\\tcs_\\WW$ and (a) follows.\n\nFrom (a) and 6.3(b) we see that for $\\l$ as in 6.3(a), the assignment \n$(E_1,E_2,\\tE_1)\\m j_{W_r\\T S_p^{(\\l)}\\T W_q}^{W_n}(E_1\\bxt E_2\\bxt\\tE_1)$ is a map\n$j:\\cs_{W'_r}\\T\\cs_{S_p}\\T\\cs_{W'_q}@>>>\\tcs_\\WW$ and we have a commutative diagram\n$$\\CD\n\\cs_{W'_r}\\T\\cs_{S_p}\\T\\cs_{W'_q}@>j>>\\tcs_\\WW\\\\\n@V\\z'_r\\T\\x_p\\T\\z'_qVV @V\\t VV\\\\\nX_m^r\\T\\ce_m^p\\T X_m^q@>h>>Y_m^n \\endCD$$                          \nwhere $h$ is given by $(x_*,e_*,\\tx_*)\\m x_*+e_*+\\tx_*$ and $\\x_p:\\cs_{S_p}@>>>\\ce_m^p$ is the bijection \n$[e_*+z_*^{0,m}]\\lra e_*$.\n\n\\subhead 6.6\\endsubhead\nNote that $\\cp'$ is exactly the collection of parahoric subgroups $W'_r\\T W'_q$ of $W'_n$ with \n$(r,q)\\in\\NN^2$ such that $r+q=n$, $r\\ne1$, $q\\ne1$. (On the other hand $W'_{n-1}\\T W'_1$, $W'_1\\T W'_{n-1}$ are maximal \nparabolic subgroup of the Coxeter group $W'_n$.) By 6.5, $j_{W'_r\\T W'_q}^{W'_n}$ carries $\\cs_{W'_r}\\T\\cs'_{W_q}$\ninto $\\tcs_\\WW$. Hence $\\bcs_\\WW\\sub\\tcs_\\WW$.\n\nConversely, let $E\\in\\tcs_\\WW$, $y_*=\\t(E)\\in Y_m^n$ ($\\t$ as in 6.4). By 2.6(a) we can find \n$(x_*,\\tx_*)\\in S(y_*)$. Define $r,q$ in $\\NN$ by $x_*\\in X_m^r$, $\\tx_*\\in X_m^q$. We must have $r+q=n$. In the \ncommutative diagram in 6.5 we have $h(x_*,\\tx_*)=y_*$, $x_*=\\z'_r(E_1),\\tx_*=\\z'_q(\\tE_1)$ where \n$E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ (recall that $\\z'_r$, $\\z'_q$ are surjective) and\n$\\t(j(E_1,\\tE_1)=\\t(E)$. Thus $j(E_1,\\tE_1),E$ are in the same fibre of $\\io:{}^\\da V_k^n@>>>\\Irr(W'_n)^\\da$. \nReplacing $E_1$ or $\\tE_1$ by an element in the same fibre of $\\io:{}^\\da V_k^r@>>>\\Irr(W'_r)^\\da$ or \n$\\io:{}^\\da V_k^q@>>>\\Irr(W'_q)^\\da$ we see that we can assume that $j(E_1,\\tE_1)=E$. Thus, $E\\in\\bcs_\\WW$. Thus, $\\tcs_\\WW\\sub\\bcs_\\WW$. We see that $\\tcs_\\WW=\\bcs_\\WW$. This proves 1.5(a) in our case.\n\n\\subhead 6.7\\endsubhead\nIn the remainder of this section we fix $C\\in\\cx$ and we set $E=\\r_C\\in\\tcs_\\WW$, $y_*=\\t(E)\\in Y_m^n$ (with\n$\\t$ as in 6.4).\n\nLet $(r,q)\\in\\NN^2$, $E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ be such that $r+q=n$, \n$E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$. (These exist since $E\\in\\bcs_\\WW$.) Define $x_*\\in X_m^r,\\tx_*\\in X_m^q$\nby $x_*=\\z'_r(E_1),\\tx_*=\\z'_q(\\tE_1)$. From the commutative diagram in 6.5 we see that $x_*+\\tx_*=y_*$. By 6.4,\nwe have $\\bb_C=\\b'(y_*)$. Since $\\b'(x_*+\\tx_*)=\\b(x_*)+\\b(\\tx_*)$ we have $\\bb_C=\\b(x_*)+\\b(\\tx_*)$. Since\n$\\b(x_*)=b_{E_1}$, $\\b(\\tx_*)=b_{\\tE_1}$, we have $\\bb_C=b_{E_1}+b_{\\tE_1}$ hence $\\bb_C=b_{E_1\\bxt\\tE_1}$. Since \n$E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ we have $b_{E_1\\bxt\\tE_1}=b_E$ hence $\\bb_C=b_E$, proving 1.5(b1) in our\ncase.\n\nIf $|\\fS(x_*)|\\ge2$, $|\\fS(\\tx_*)|\\ge2$, then\n$$\\align&f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-2)\/2}2^{(|\\fS(\\tx_*)|-2)\/2)}\\\\&=\n2^{(|\\fS(x_*)|+|\\fS(\\tx_*)|-4)\/2}\\le 2^{|\\fI(y_*)|-2}\\le\\zz_C.\\endalign$$ \nIf $|\\fS(x_*)|=0$, $2\\le|\\fS(\\tx_*)|\\le2|\\fI(y_*)|-2$ then\n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(\\tx_*)|-2)\/2)}\\le 2^{|\\fI(y_*)|-2}\\le\\zz_C.$$ \nSimilarly, if $2\\le|\\fS(x_*)|\\le2|\\fI(y_*)|-2$, $|\\fS(\\tx_*)|=0$, then $f_{E_1\\bxt\\tE_1}\\le\\zz_C$. If \n$|\\fS(x_*)|=0$, $2\\le|\\fS(\\tx_*)|=2|\\fI(y_*)|$ then $\\fI'(y_*)=\\em$ and \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(\\tx_*)|-2)\/2)}=2^{|\\fI(y_*)|-1}\\le\\zz_C.$$ \nSimilarly, if $2\\le|\\fS(x_*)|=2|\\fI(y_*)|$, $|\\fS(\\tx_*)|=0$ then $f_{E_1\\bxt\\tE_1}\\le\\zz_C$. If $|\\fS(x_*)|=0$, \n$|\\fS(\\tx_*)|=0$, then \n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=1=\\zz_C.$$ \nThus in any case we have $f_{E_1\\bxt\\tE_1}\\le\\zz_C$. Taking maximum over all $r,q,E_1,\\tE_1$ as above we obtain \n$\\fa_E\\le\\zz_C$.\n\n\\subhead 6.8\\endsubhead\nAssume now that $\\d_{y_*}=1$. Then $|\\fI(y_*)|\\ge2$. By 2.6(a) we can find $(x_*,\\tx_*)\\in S(y_*)$. By 2.6(c) we\nhave $|\\fS(x_*)|\\ge2$, $|\\fS(\\tx_*)|\\ge2$. Define $(r,q)\\in\\NN^2$ by $x_*\\in X_m^r,\\tx_*\\in X_m^q$. We must have \n$r+q=n$. We can find $E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z'_r(E_1),\\tx_*=\\z'_q(\\tE_1)$. As \nearlier in the proof we can assume that $E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ and we have\n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-2)\/2+(|\\fS(\\tx_*)|-2)\/2}=2^{|\\fI(y_*)|-2}=\\zz_C.$$\n\n\\subhead 6.9\\endsubhead\nNext we assume that $\\fI(y_*)\\ne\\em$ and $\\d_{y_*}=0$. By 2.6(a) we can find $(x_*,\\tx_*)\\in S(y_*)$. By 2.6(vii)\nwe have $\\fS(\\tx_*)=\\em$. Then $|\\fS(x_*)|=2|\\fI(y_*)|$. Define $(r,q)\\in\\NN^2$ by $x_*\\in X_m^r,\\tx_*\\in X_m^q$.\nWe must have $r+q=n$. We can find $E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z'_r(E_1)$,\n$\\tx_*=\\z'_q(\\tE_1)$. We can assume that $E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ and we have\n$$f_{E_1\\bxt\\tE_1}=f_{\\tE_1}=2^{(|\\fS(\\tx_*)|-2)\/2}=2^{|\\fI(y_*)|-1}=\\zz_C.$$\nNow we assume that $\\fI(y_*)=\\em$. By 2.6(a) we can find $(x_*,\\tx_*)\\in S(y_*)$. By 2.6(b) we have\n$\\fS(x_*)=\\em$, $\\fS(\\tx_*)=\\em$. Define $(r,q)\\in\\NN^2$ by $x_*\\in X_m^r,\\tx_*\\in X_m^q$. We must have $r+q=n$. \nWe can find $E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z'_r(E_1),\\tx_*=\\z'_q(\\tE_1)$. We can assume\nthat $E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ and we have $f_{E_1\\bxt\\tE_1}=1=\\zz_C$.\n\nWe see that in any case, $\\fa_E=\\zz_C$, proving 1.5(b2) in our case.\n\n\\subhead 6.10\\endsubhead\nFor $g\\in\\Om$ let $\\la g\\ra$ be the subgroup of $\\Om$ generated by $g$.\n\nWhen $n$ is even the subgroups of $\\Om$ are $\\{1\\},\\la\\o_1\\ra,\\la\\o_2\\ra,\\la\\o_1\\o_2\\ra,\\Om$; when $n$ is odd the \nsubgroups of $\\Om$ are $\\{1\\},\\la\\o_2\\ra,\\Om$. \n\n(a) The collection of subgroups $W'_r\\T S_p^{(0)}\\T W'_q$ (with $r=q\\ge1$) contains all subgroups in $\\cp^\\Om$.\n\n(b) The collection of subgroups $W'_r\\T W'_q$ contains all subgroups in $\\cp^{\\la\\o_2\\ra}$.\n\n(c) For $n$ even, the collection in (a) together with the subgroups $W'_0\\T S_p^{(\\l)}\\T W'_0$ (with $\\l=0$ or \n$3$) contains all subgroups in $\\cp^{\\la\\o_1\\ra}$.\n\n(d) For $n$ even, the collection in (a) together with the subgroups $W'_0\\T S_p^{(\\l)}\\T W'_0$ (with $\\l=1$ or \n$2$) contains all subgroups in $\\cp^{\\la\\o_1\\o_2\\ra}$.\n\n\\subhead 6.11\\endsubhead\nAssume that $\\tzz_C\/\\zz_C=4$. Then $\\d_{y_*}=1$ and $|\\ci|=1$ for any $\\ci\\in\\fI(y_*)$. By 2.11 we can find $r,p$,\n$x_*\\in X_m^r$, $e_*\\in\\ce_m^p$ (with $r+p+r=n$) such that $y_*=x_*+e_*+x_*$, $(x_*,e_*+x_*)\\in S(y_*)$, \n$\\fS(x_*)=\\fS(e_*+x_*)\\ne\\em$. Note that $r\\ge1$. Define $E_1\\in\\cs_{W'_r}$ by $\\z'_r(E_1)=x_*$, $E_2\\in\\cs_{S_p}$\nby $\\x_p(E_2)=e_*$. We have $E=j_{W'_r\\T S_p^{(0)}\\T W'_r}^{W'_n}(E_1\\bxt E_2\\bxt E_1)$ and\n$$\\align&f_{E_1\\bxt E_2\\bxt E_1}=2^{(|\\fS(x_*)|-2)\/2}2^{(|\\fS(x_*)|-2)\/2}=\n2^{(|\\fS(x_*)|-2)\/2}2^{(|\\fS(e_*+x_*)|-2)\/2}\\\\&=2^{|\\fI(y_*)|-2}=\\zz_C.\\endalign$$\nWe have $W'_r\\T S^{0)}_p\\T W'_r\\in\\cp^\\Om$. Moreover, $E_1\\bxt E_2\\bxt E_1$ is $\\Om$-stable. We see that $\\fc_E=4$. \n\n\\subhead 6.12\\endsubhead\nConversely, assume that $\\fc_E=4$. By 1.11 and 6.10(a), there exist $(r,p,q)$ as in 6.3 with $q=r\\ge1$ and \n$E_1\\in\\cs_{W'_r}$, $E_2\\in\\cs_{S_p}$ such that $E=j_{W'_r\\T S_p^{(0)}\\T W'_r}^{W'_n}(E_1\\bxt E_2\\bxt E_1)$, \n$f_{E_1\\bxt E_2\\bxt E_1}=\\zz_C$ and such that $E_1$ extends to a $W_r$-module. We set $x_*=\\z'_r(E_1)\\in X_m^r$, \n$e_*=\\x_p(E_2)$. We have $y_*=x_*+e_*+x_*$. Since $E_1$ extends to a $W_r$-module we have $\\fS(x_*)\\ne\\em$, hence\n$\\fI(y_*)\\ne\\em$. Thus, $\\zz_C=2^{|\\fI(y_*)|-1-\\d_{y_*}}$, \n$2^{(|\\fS(x_*)|-2+|\\fS(x_*)|-2)\/2}=2^{|\\fI(y_*)|-1-\\d_{y_*}}$ and $|\\fS(x_*)|+|\\fS(x_*)|=2|\\fI(y_*)|+1-\\d_{y_*}$. \nSince $|\\fS(x_*)|+|\\fS(x_*)|\\le 2|\\fI(y_*)|$, we have $1-\\d_{y_*}\\le0$ hence $\\d_{y_*}=1$ and \n$|\\fS(x_*)|+|\\fS(x_*)|=2|\\fI(y_*)|$. \n\nLet $E'_1=j_{S_p\\T W'_r}^{W'_{p+r}}(E_2\\bxt E_1)\\in\\cs_{W'_{p+r}}$. Then \n$E=j_{W'_r\\T W'_{p+r}}^{W'_n}(E_1\\bxt E'_1)$. By 1.5(b2) we have $f_{E_1\\bxt E'_1}\\le\\zz_C$. By 1.9(b) we have \n$f_{E_2\\bxt E_1}\\le f_{E'_1}$. Hence $\\zz_C=f_{E_1\\bxt E_2\\bxt E_1}\\le f_{E_1\\bxt E'_1}\\le\\zz_C$; this forces \n$f_{E_2\\bxt E_1}=f_{E'_1}$. The last equality can be rewritten as\n$$2^{(|\\fS(x_*)|-2)\/2}=2^{(|\\fS(e_*+x_*)|-2)\/2}$$\nsince $e_*+x_*=\\z'_{p+r}(E'_1)$ (a consequence of 6.2(a)). Hence $|\\fS(e_*+x_*)|=|\\fS(x_*)|$. We have also \n$(x_*,e_*+x_*)\\in S(y_*)$. Using 2.10, we see that for any $\\ci\\in\\fI(y_*)$ we have $|\\ci|=1$. Thus, \n$\\tzz_C\/\\zz_C=4$.\n\nUsing this together with 6.11, we see that $\\fc_E=4$ if and only if $\\tzz_C\/\\zz_C=4$.\n\n\\subhead 6.13\\endsubhead\nAssume that $\\fI(y_*)=\\em$. Then $n$ is even. Define $e_*\\in\\NN^{m+1}$ by $y_*=x^0_*+e_*+x^0_*$. We have \n$e_*\\in\\ce_m^n$. Define $E_1\\in\\cs_{W'_0}$ by $\\z'_0(E_1)=x_*^0$, $E_2\\in\\cs_{S_n}$ by $\\x_n(E_2)=e_*$. For some \n$\\l\\in[0,3]$ we have $E=j_{W'_0\\T S_n^{(\\l)}\\T W'_0}^{W'_n}(E_1\\bxt E_2\\bxt E_1)$, see 6.3. We have \n$f_{E_1\\bxt E_2\\bxt E_1}=1=\\zz_C$. Note that $W'_0\\T S_n^{(\\l)}\\T W'_0\\in\\cp^{\\Om_1}$ where $\\Om_1$ is\n$\\la\\o_1\\ra$ or $\\la\\o_1\\o_2\\ra$; moreover $E_1\\bxt E_2\\bxt E_1$ is $\\Om_1$-stable. We see that $\\fc_E\\ge2$. By \n6.12 we cannot have $\\fc_E=4$. Hence $\\fc_E=2$.\n\n\\subhead 6.14\\endsubhead\nAssume that $\\d_{y_*}=1$ and $|\\ci|>1$ for some $\\ci\\in\\fI(y_*)$. We have $|\\fI(y_*)|\\ge2$. By 2.6(a) we can find\n$(x_*,\\tx_*)\\in S(y_*)$. By 2.6(c) we have $|\\fS(x_*)|\\ge2$, $|\\fS(\\tx_*)|\\ge2$. Define $(r,q)\\in\\NN^2$ by \n$x_*\\in X_m^r,\\tx_*\\in X_m^q$. We must have $r+q=n$ and $r\\ge1$, $q\\ge1$. We can find uniquely $E_1\\in\\cs_{W'_r}$,\n$\\tE_1\\in\\cs_{W'_q}$ such that $x_*=\\z'_r(E_1),\\tx_*=\\z'_q(\\tE_1)$. We have \n$E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ and\n$$f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-2)\/2+(|\\fS(\\tx_*)|-2)\/2}=2^{|\\fI(y_*)|-2}=\\zz_C.$$\nWe have $W'_r\\T W'_q\\in\\cp^{\\la\\o_2\\ra}$ and $E_1\\bxt\\tE_1$ is $\\la\\o_2\\ra$-stable. We see that $\\fc_E\\ge2$. By \n6.12 we cannot have $\\fc_E=4$. Hence $\\fc_E=2$.\n\n\\subhead 6.15\\endsubhead\nAssume that $\\fc_E=2$. By 1.11 and 6.10, either (i) or (ii) below holds.\n\n(i) there exist $(r,p,q)$ as in 6.3 with $q=r$, $\\l\\in[0,3]$ (with $\\l=0$ unless $r=0$) and $E_1\\in\\cs_{W'_r}$, \n$E_2\\in\\cs_{S_p}$ such that $E=j_{W'_r\\T S_p^{(\\l)}\\T W'_r}^{W'_n}(E_1\\bxt E_2\\bxt E_1)$, \n$f_{E_1\\bxt E_2\\bxt E_1}=\\zz_C$;\n\n(ii) there exist $(r,q)$ with $r+q=n$ and $E_1\\in\\cs_{W'_r}$, $\\tE_1\\in\\cs_{W'_q}$ such that $E_1$ extends to a \n$W_r$-module, $\\tE_1$ extends to a $W_q$-module, $E=j_{W'_r\\T W'_q}^{W'_n}(E_1\\bxt\\tE_1)$ and\n$f_{E_1\\bxt\\tE_1}=\\zz_C$.\n\\nl\nAssume first that (i) holds. We set $x_*=\\z'_r(E_1)\\in X_m^r$. If $r\\ge1$ and $E_1$ extends to a $W_r$-module then\n$E_1\\bxt E_2\\bxt E_1$ is $\\Om$-stable (note that $W'_r\\T S_p^{(\\l)}\\T W'_r\\in\\cp^\\Om$) so that $\\fc_E=4$ \ncontradicting $\\fc_E=2$. Thus, either $r\\ge1$ and $E_1$ does not extend to a $W_r$-module or $r=0$. It follows \nthat $\\fS(x_*)=\\em$ and $f_{E_1}=1$ so that $\\zz_C=1$. Hence either $|\\fI(y_*)|=0$ or $|\\fI(y_*)|=2,\\d_{y_*}=1$. \nIn the first case we have $\\tzz_C\/\\zz_C=2$. In the second case, using $\\d_{y_*}=1$ we see that $\\tzz_C\/\\zz_C\\ge2$;\nif we had $\\tzz_C\/\\zz_C=4$ we would have $\\fc_E=4$, a contradiction. Thus in both cases we have $\\tzz_C\/\\zz_C=2$. \n\nNext assume that (ii) holds. We set $x_*=\\z'_r(E_1)\\in X_m^r$, $\\tx_*=\\z'_q(\\tE_1)\\in\\tX_m^q$. We have \n$y_*=x_*+\\tx_*$. Since $E_1$ extends to a $W_r$-module and $\\tE_1$ extends to a $W_q$-module we have \n$|\\fS(x_*)|\\ge2$, $|\\fS(\\tx_*)|\\ge2$. Hence\n$$2^{|\\fI(y_*)|-2}\\le\\zz_C=f_{E_1\\bxt\\tE_1}=f_{E_1}f_{\\tE_1}=2^{(|\\fS(x_*)|-2)\/2+(|\\fS(\\tx_*)|-2)\/2}\\le \n2^{|\\fI(y_*)|-2}.$$\nIt follows that $2^{|\\fI(y_*)|-2}=\\zz_C$ so that $\\d_{y_*}=1$. This implies that $\\tzz_C\/\\zz_C\\ge2$; if we had \n$\\tzz_C\/\\zz_C=4$ we would have $\\fc_E=4$, a contradiction. Thus we have $\\tzz_C\/\\zz_C=2$. \n\nUsing this together with 6.13, 6.14, we see that $\\fc_E=2$ if and only if $\\tzz_C\/\\zz_C=2$.\n\n\\subhead 6.16\\endsubhead\nBy 6.12, we have $\\fc_E=4$ if and only if $\\tzz_C\/\\zz_C=4$. By 6.15, we have $\\fc_E=2$ if and only if \n$\\tzz_C\/\\zz_C=2$. Since $\\fc_E\\in\\{1,2,4\\}$ and $\\tzz_C\/\\zz_C\\in\\{1,2,4\\}$ we see that $\\fc_E=\\tzz_C\/\\zz_C$; this \nproves 1.5(b3) in our case.\n\n\\head 7. Exceptional types\\endhead\n\\subhead 7.1\\endsubhead\nIn this section we assume that $G$ is an exceptional group. For each type we give a table with rows indexed by the\nunipotent conjugacy classes in $G$ in which the row corresponding to $C\\in\\cx$ has four entries:\n$$\\r_C\\qquad\\bb_C\\qquad a\\T a'\\quad(J,E_1)$$\nwhere $a=\\zz_C$, $a'=\\tzz_C\/\\zz_C$ and $(J,E_1)$ is an example of an element of $\\cz_E$ ($E=\\r_C$) such that \n$f_{E_1}=\\zz_C$ and $|\\Om_{J,E_1}|=\\tzz_C\/\\zz_C$. (When $\\Om=\\{1\\}$ we have $a'=1$ and we write $a$ instead of \n$a\\T a'$). \nWe specify an irreducible representation $E_1$ of a Weyl group either by using the notation of \\cite{\\OR, Ch.4} \n(for type $E_6,E_7,E_8$) or by specifying its degree. The representation is then determined by its $b_{E_1}$ which\nequals $\\bb_C$ in the table or (in the case of $G_2$, $F_4$) by other information in the same row of the table. On\nthe other hand, $\\e$ always denotes the sign representation. In a pair $(J,E_1)$, $J$ is any subset of $\\tI$ such \nthat $\\WW_J$ has the specified type; in addition, for type $F_4$, we denote by $A_2$ (resp. $A'_2$) a subset $J$ \nof $\\tI$ such that $\\WW_J$ is of type $A_2$ and is contained (resp. not contained) in a parahoric subgroup of type\n$B_4$).\n\nThe group $\\Om$ is $\\{1\\}$ for types $G_2,F_4$ and is a cyclic group of order\n$9-n$ for type $E_n (n=6,7,8)$.\n\nType $G_2$     \n$$\\allowdisplaybreaks\\alignat4\n\\r_C&&\\qquad\\bb_C&&\\qquad a\\T a'&&\\quad(J,E_1)\\\\\n1&&\\qquad0 &&\\qquad   1 && \\qquad       (\\em,1)           \\\\\n2&&\\qquad1 && \\qquad  6 && \\qquad       (G_2,2)          \\\\\n2&&\\qquad2 &&\\qquad  1  &&\\qquad        (A_1A_1,\\e)       \\\\\n1&&\\qquad3 &&\\qquad  1  &&\\qquad        (A_2,\\e) \\\\        \n1&&\\qquad6 &&\\qquad  1  &&\\qquad         (G_2,\\e)            \n\\endalignat$$           \n\nType $F_4$\n$$\\allowdisplaybreaks\\alignat4\n\\r_C&&\\qquad\\bb_C&&\\qquad a\\T a'&&\\quad(J,E_1)\\\\\n1&&\\qquad0 &&\\qquad    1&&\\qquad   (\\em,1)   \\\\\n4&&\\qquad1 &&\\qquad    2&&\\qquad       (F_4,4) \\\\     \n9&&\\qquad2 &&\\qquad    2&&\\qquad       (F_4,9) \\\\      \n8&&\\qquad3 &&\\qquad    1&&\\qquad      (A_2,\\e)  \\\\      \n8&&\\qquad3 &&\\qquad    1 &&\\qquad    (A'_2,\\e) \\\\         \n12&&\\qquad4 &&\\qquad   24 &&\\qquad    (F_4,12)\\\\       \n16&&\\qquad5 &&\\qquad  2 &&\\qquad   (C_3A_1,3\\bxt\\e)\\\\  \n9&&\\qquad6 &&\\qquad    2&&\\qquad    (B_4,6)\\\\       \n6&&\\qquad6 &&\\qquad    1&&\\qquad       (A_2A'_2,\\e)\\\\    \n4&&\\qquad7 &&\\qquad    1&&\\qquad       (A_3A_1,\\e)\\\\    \n8&&\\qquad9 &&\\qquad    1&&\\qquad       (C_3,\\e)\\\\        \n8&&\\qquad9 &&\\qquad    2&&\\qquad        (B_4,4)\\\\  \n9&&\\qquad10&&\\qquad    1&&\\qquad     (C_3A_1,\\e) \\\\     \n4&&\\qquad13&&\\qquad    2&&\\qquad      (F_4,4)\\\\         \n2&&\\qquad16&&\\qquad    1&&\\qquad       (B_4,\\e)\\\\       \n1&&\\qquad24&&\\qquad    1&&\\qquad       (F_4,\\e)\n\\endalignat$$           \n              \nType $E_6$\n$$\\allowdisplaybreaks\\alignat4\n\\r_C&&\\qquad\\bb_C&&\\qquad a\\T a'&&\\quad(J,E_1)\\\\\n1_p&&\\qquad0 &&\\qquad   1\\T3  &&\\qquad        (\\em,1)           \\\\\n6_p&&\\qquad1 &&\\qquad   1\\T3  &&\\qquad      (D_4,4) \\\\          \n20_p&&\\qquad2&&\\qquad   1\\T1&&\\qquad       (E_6,20_p)\\\\       \n30_p&&\\qquad3&&\\qquad   2\\T3 &&\\qquad     (D_4,8)\\\\    \n15_q&&\\qquad4 &&\\qquad  1\\T3 &&\\qquad     (A_1A_1A_1A_1,\\e)\\\\ \n64_p&&\\qquad4 &&\\qquad  1\\T1&&\\qquad         (E_6,64_p)       \\\\\n60_p&&\\qquad5&&\\qquad   1\\T1&&\\qquad         (E_6,60_p)    \\\\\n24_p&&\\qquad6&&\\qquad   1\\T1&&\\qquad          (E_6,24_p)       \\\\\n81_p&&\\qquad6 &&\\qquad  1\\T1&&\\qquad         (E_6,81_p)         \\\\\n80_s&&\\qquad7&&\\qquad   6\\T1&&\\qquad       (E_6,80_s)           \\\\\n60_s&&\\qquad8&&\\qquad   1\\T1&&\\qquad          (A_3A_1A_1,\\e)    \\\\\n10_s&&\\qquad9&&\\qquad   1\\T3&&\\qquad       (A_2A_2A_2,\\e)    \\\\\n81'_p&&\\qquad10&&\\qquad  1\\T1&&\\qquad          (E_6,81'_p)          \\\\\n60'_p&&\\qquad11&&\\qquad  1\\T1&&\\qquad         (E_6,60'_p)      \\\\\n24'_p&&\\qquad12&&\\qquad  1\\T3&&\\qquad       (D_4,\\e)          \\\\\n64'_p&&\\qquad13&&\\qquad  1\\T1&&\\qquad         (E_6,64'_p)      \\\\\n30'_p&&\\qquad15&&\\qquad  2\\T1&&\\qquad       (E_6,30'_p)      \\\\\n15'_q&&\\qquad16&&\\qquad  1\\T1&&\\qquad          (A_5A_1,\\e)\\\\     \n20'_p&&\\qquad20&&\\qquad  1\\T1&&\\qquad     (E_6,20'_p)\\\\      \n6'_p&&\\qquad25 &&\\qquad 1\\T1&&\\qquad       (E_6,6'_p)            \\\\\n1'_p&&\\qquad36  &&\\qquad1\\T1&&\\qquad        (E_6,\\e)           \n\\endalignat$$\n\nType $E_7$\n$$\\allowdisplaybreaks\\alignat4\n\\r_C&&\\qquad\\bb_C&&\\qquad a\\T a'&&\\quad(J,E_1)\\\\\n1_a&&\\qquad0 &&\\qquad   1\\T 2 &&\\qquad     (\\em,1)     \\\\\n7'_a&&\\qquad1&&\\qquad    1\\T 2 &&\\qquad    (E_6,6_p)            \\\\\n27_a&&\\qquad2 &&\\qquad  1\\T 2 &&\\qquad     (E_6,20_p)        \\\\\n56'_a&&\\qquad3 &&\\qquad  2\\T2 &&\\qquad      (E_6,30_p)     \\\\\n21'_b&&\\qquad3 &&\\qquad  1\\T1  &&\\qquad      (E_7,21'_b)    \\\\\n120_a&&\\qquad4 &&\\qquad 2\\T1 &&\\qquad    (E_7,120_a)  \\\\\n35_b&&\\qquad4  &&\\qquad 1\\T2 &&\\qquad    (A_7,14) \\\\   \n189'_b&&\\qquad5 &&\\qquad2\\T2 &&\\qquad     (A_1D_4A_1,\\e\\bxt8\\bxt\\e) \\\\\n105_b&&\\qquad6 &&\\qquad 1\\T1 &&\\qquad        (E_7,105_b)\\\\ \n210_a&&\\qquad6 &&\\qquad  1\\T2 &&\\qquad      (A_7,35)\\\\          \n168_a&&\\qquad6 &&\\qquad  1\\T2 &&\\qquad    (A_7,56)\\\\            \n315'_a&&\\qquad7&&\\qquad   6\\T2&&\\qquad     (E_6,80_s)\\\\        \n189'_c&&\\qquad7&&\\qquad   1\\T1&&\\qquad        (E_7,189'_c)\\\\    \n405_a&&\\qquad8&&\\qquad   2\\T1&&\\qquad      (E_7,405_a)\\\\  \n280_b&&\\qquad8&&\\qquad   1\\T2&&\\qquad      (A_7,56)\\\\           \n70'_a&&\\qquad9&&\\qquad   1\\T2&&\\qquad      (A_2A_2A_2,\\e)\\\\   \n216'_a&&\\qquad9&&\\qquad   1\\T1&&\\qquad        (D_6A_1,30\\bxt\\e)      \\\\\n378'_a&&\\qquad9&&\\qquad  1\\T2 &&\\qquad    (A_7,70)\\\\          \n420_a&&\\qquad10&&\\qquad  2\\T1 &&\\qquad     (E_7,420_a)\\\\    \n210_b&&\\qquad10&&\\qquad  1\\T1&&\\qquad         (E_7,210_b)\\\\          \n512'_a&&\\qquad11&&\\qquad  2\\T1&&\\qquad     (E_7,512'_a)\\\\                \n105_c&&\\qquad12&&\\qquad  1\\T2&&\\qquad      (D_4,\\e)             \\\\\n84_a&&\\qquad12 &&\\qquad 1\\T2 &&\\qquad       (A_7,14)\\\\                        \n420'_a&&\\qquad13 &&\\qquad  2\\T1&&\\qquad       (D_6,24)      \\\\         \n210_b&&\\qquad13  &&\\qquad1\\T2 &&\\qquad       (A_3A_3A_1,\\e)            \\\\\n378'_a&&\\qquad14 &&\\qquad 2\\T1&&\\qquad  (D_6A_1,24\\bxt\\e)  \\\\       \n105'_c&&\\qquad15 &&\\qquad 1\\T1  &&\\qquad   (A_5A_2,\\e\\bxt1)\\\\             \n405'_a&&\\qquad15 &&\\qquad 2\\T2 &&\\qquad  (E_6,30'_p)\\\\           \n216_a&&\\qquad16  &&\\qquad1\\T2 &&\\qquad    (A_7,20)\\\\      \n315_a&&\\qquad16  &&\\qquad6\\T1 &&\\qquad        (E_7,315_a)\\\\                  \n280'_b&&\\qquad17  &&\\qquad1\\T1 &&\\qquad         (D_6A_1,15\\bxt\\e) \\\\\n70_a&&\\qquad18  &&\\qquad1\\T1 &&\\qquad              (A_5A_2,\\e)             \\\\\n189_c&&\\qquad20 &&\\qquad1\\T2 &&\\qquad        (E_6,20'_p)\\\\               \n210'_a&&\\qquad21 &&\\qquad 1\\T1&&\\qquad        (E_7,210'_a)               \\\\ \n168'_a&&\\qquad(21 &&\\qquad    1\\T1&&\\qquad        (E_7,168'_a)            \\\\\n105'_b&&\\qquad21 &&\\qquad1\\T2 &&\\qquad         (A_7,7)         \\\\\n189_b&&\\qquad22   &&\\qquad1\\T1&&\\qquad         (E_7,189_b)    \\\\\n120'_a&&\\qquad25  &&\\qquad   2\\T1&&\\qquad     (E_7,120'_a)                  \\\\\n15_a&&\\qquad28  &&\\qquad   1\\T2 &&\\qquad        (A_7,\\e)                   \\\\\n56_a&&\\qquad30  &&\\qquad   2\\T1 &&\\qquad       (E_7,56_a)                      \\\\\n35'_b&&\\qquad31 &&\\qquad   1\\T1 &&\\qquad        (D_6A_1,\\e)             \\\\\n21_b&&\\qquad36 &&\\qquad    1\\T2&&\\qquad       (E_6,\\e)                \\\\\n27'_a&&\\qquad37 &&\\qquad   1\\T1 &&\\qquad        (E_7,27'_a)                    \\\\\n7_a&&\\qquad46  &&\\qquad   1\\T1&&\\qquad           (E_7,7_a)               \\\\\n1'_a&&\\qquad63 &&\\qquad    1\\T1  &&\\qquad        (E_7,\\e)                 \n\\endalignat$$\n\nType $E_8$\n$$\\allowdisplaybreaks\\alignat4\n\\r_C&&\\qquad\\bb_C&&\\qquad a\\T a'&&\\quad(J,E_1)\\\\\n1_x&&\\qquad0 &&\\qquad   1 &&\\qquad                (\\em,1)           \\\\\n8_z&&\\qquad1 &&\\qquad     1&&\\qquad              (E_8,8_z)\\\\          \n35_x&&\\qquad2&&\\qquad     1&&\\qquad              (E_8,35_x)\\\\       \n112_z&&\\qquad3&&\\qquad   2 &&\\qquad             (E_8,112_z) \\\\\n84_x&&\\qquad4&&\\qquad 1  &&\\qquad             (E_7A_1,21'_b\\bxt\\e)    \\\\\n210_x&&\\qquad4&&\\qquad   2&&\\qquad              (E_8,210_x)\\\\          \n560_z&&\\qquad5&&\\qquad   2 &&\\qquad             (E_7A_1,120_a\\bxt\\e)\\\\\n567_x&&\\qquad6&&\\qquad   1 &&\\qquad             (E_8,567_x)\\\\           \n700_x&&\\qquad6&&\\qquad   2 &&\\qquad             (E_8,700_x)\\\\         \n400_x&&\\qquad7 &&\\qquad1   &&\\qquad           (A_2A_1A_1A_1A_1,\\e)\\\\ \n1400_z&&\\qquad7 &&\\qquad 6 &&\\qquad              (E_6,80_s)            \\\\\n1400_x&&\\qquad8 &&\\qquad 6 &&\\qquad              (E_8,1400_x)  \\\\  \n1344_x&&\\qquad8 &&\\qquad 1 &&\\qquad               (E_7A_1,189'_c\\bxt\\e)  \\\\\n448_z&&\\qquad9 &&\\qquad 1  &&\\qquad                (A_2A_2A_2,\\e)      \\\\\n3240_z&&\\qquad9 &&\\qquad 2  &&\\qquad                (E_7A_1,405_a\\bxt\\e)   \\\\\n2240_x&&\\qquad10 &&\\qquad6  &&\\qquad       (E_6A_2,80_s\\bxt\\e)\\\\    \n2268_x&&\\qquad10 &&\\qquad   2 &&\\qquad (E_8,2268_x)         \\\\\n4096_x&&\\qquad11 &&\\qquad 2 &&\\qquad (E_7,512'_a)\\\\       \n1400_z&&\\qquad11 &&\\qquad 1 &&\\qquad  (E_7A_1,210_b\\bxt\\e)\\\\   \n525_x&&\\qquad12 &&\\qquad  1 &&\\qquad       (D_4,\\e)\\\\              \n4200_x&&\\qquad12&&\\qquad    2&&\\qquad    (E_8,4200_x) \\\\\n 972_x&&\\qquad12&&\\qquad  1 &&\\qquad      (A_3A_3,\\e) \\\\         \n2800_z&&\\qquad13&&\\qquad    2&&\\qquad      (E_8,2800_z)\\\\               \n4536_z&&\\qquad13&&\\qquad   2 &&\\qquad    (D_8,560)\\\\    \n6075_x&&\\qquad14&&\\qquad    2 &&\\qquad    (D_8,280) \\\\\n2835_x&&\\qquad14&&\\qquad   1  &&\\qquad     (A_4A_2A_1,\\e)\\\\         \n4200_z&&\\qquad15&&\\qquad   1  &&\\qquad   (A_5,\\e)\\\\             \n5600_z&&\\qquad15 &&\\qquad 2  &&\\qquad       (E_6,30'_p)\\\\           \n4480_y&&\\qquad16 &&\\qquad120 &&\\qquad     (E_8,4480_y)               \\\\\n3200_x&&\\qquad16 &&\\qquad  1 &&\\qquad    (A_5A_1,\\e)         \\\\\n7168_w&&\\qquad17 &&\\qquad  6&&\\qquad  (E_7A_1,315_a\\bxt\\e)\\\\\n4200_y&&\\qquad18 &&\\qquad   2 &&\\qquad       (D_8,252)              \\\\\n3150_y&&\\qquad18 &&\\qquad    2 &&\\qquad     (E_6A_2,30'_p\\bxt\\e)       \\\\\n2016_w&&\\qquad19 &&\\qquad    1 &&\\qquad    (A_5A_2A_1,\\e)       \\\\\n1344_w&&\\qquad19 &&\\qquad  1  &&\\qquad       (D_5A_3,5\\bxt\\e)         \\\\\n2100_y&&\\qquad20 &&\\qquad     1 &&\\qquad     (D_5,\\e)              \\\\\n420_y&&\\qquad20  &&\\qquad     1 &&\\qquad      (A_4A_4,\\e)           \\\\\n5600'_z&&\\qquad21 &&\\qquad2 &&\\qquad         (E_8,5600'_z)             \\\\\n4200'_z&&\\qquad21 &&\\qquad   2&&\\qquad  (D_8,224)         \\\\\n3200'_x&&\\qquad22 &&\\qquad   1 &&\\qquad      (E_7A_1,168_a\\bxt\\e) \\\\   \n6075'_x&&\\qquad22 &&\\qquad1 &&\\qquad        (E_8,6075'_x)            \\\\\n2835'_x&&\\qquad22  &&\\qquad   1 &&\\qquad     (A_6A_1,\\e)          \\\\\n4536'_z&&\\qquad23  &&\\qquad   1 &&\\qquad     (D_5A_2,\\e)         \\\\\n4200'_x&&\\qquad24  &&\\qquad 2 &&\\qquad     (E_8,4200'_x)         \\\\\n2800'_z&&\\qquad25 &&\\qquad  2  &&\\qquad      (E_7,120'_a)              \\\\\n4096'_x&&\\qquad26 &&\\qquad    2 &&\\qquad     (E_8,4096'_x)          \\\\\n840'_x&&\\qquad26  &&\\qquad  1  &&\\qquad      (D_5A_3,\\e)          \\\\\n700'_x&&\\qquad28  &&\\qquad 1  &&\\qquad       (A_7,\\e)                   \\\\\n2240'_x&&\\qquad28 &&\\qquad  2 &&\\qquad     (E_8,2240'_x)                \\\\\n1400'_z&&\\qquad29 &&\\qquad  1 &&\\qquad     (A_7A_1,\\e)                  \\\\\n2268'_x&&\\qquad30 &&\\qquad     2&&\\qquad         (E_7,56_a)                \\\\\n3240'_z&&\\qquad31 &&\\qquad    2&&\\qquad     (E_7A_1,56_a\\bxt\\e)      \\\\\n1400'_x&&\\qquad32 &&\\qquad    6&&\\qquad    (E_8,1400'_x)            \\\\\n1050'_x&&\\qquad34 &&\\qquad     1 &&\\qquad      (D_8,28)                \\\\\n 525'_x&&\\qquad36 &&\\qquad   1  &&\\qquad        (E_6,\\e))                \\\\\n175'_x&&\\qquad36  &&\\qquad      1&&\\qquad       (A_8,\\e)                 \\\\\n1400'_z&&\\qquad37 &&\\qquad  6 &&\\qquad           (E_8,1400'_z)                \\\\\n1344'_x&&\\qquad38 &&\\qquad  1 &&\\qquad   (E_7A_1,27'_a\\bxt\\e)              \\\\\n448'_z&&\\qquad39  &&\\qquad   1 &&\\qquad          (E_6A_2,\\e)\\\\               \n700'_x&&\\qquad42  &&\\qquad    2 &&\\qquad         (E_8,700'_x)\\\\                       \n400'_z&&\\qquad43  &&\\qquad    1 &&\\qquad   (D_8,8)               \\\\\n567'_x&&\\qquad46  &&\\qquad  1  &&\\qquad           (E_7,7_a)                       \\\\\n560'_z&&\\qquad47  &&\\qquad   1 &&\\qquad      (E_7A_1,7_a\\bxt\\e)          \\\\\n210'_x&&\\qquad52  &&\\qquad     2&&\\qquad         (E_8,210'_x)                   \\\\\n50'_x&&\\qquad56   &&\\qquad     1&&\\qquad      (D_8,\\e)                        \\\\\n112'_z&&\\qquad63  &&\\qquad    2&&\\qquad   (E_8,112'_z)               \\\\\n84'_x&&\\qquad64  &&\\qquad    1 &&\\qquad       (E_7A_1,\\e)                    \\\\\n 35'_x&&\\qquad74  &&\\qquad    1 &&\\qquad        (E_8,35'_x)                  \\\\\n8'_z&&\\qquad91   &&\\qquad    1  &&\\qquad      (E_8,8'_z)\\\\                          \n 1'_x&&\\qquad120 &&\\qquad    1  &&\\qquad       (E_8,\\e)                  \n\\endalignat$$\n\n\\head Index \\endhead\n0.1: $\\cx,\\r_C,\\bb_C,\\zz_C,\\tzz_C,\\tcs_\\WW$\n\n1.1: $\\Irr(W),f_E,a_E,b_E,\\cs_W,\\Irr(W)^\\da$\n\n1.2: $\\car,\\tI,\\tca,\\WW,\\WW_J,s_i,\\Om$\n\n1.3: $j_{\\WW_J}^\\WW(E_1),\\bcs_\\WW,\\cz_E,\\fa_E,\\cz_E^\\sp,\\Om_{J,E_1},\\fc_E$\n\n1.4: $G$\n\n1.9: $\\cp'$\n\n1.11: $\\cp^{\\tiO}$\n\n2.9: $\\ce_m$\n\n4.1: $W_n$\n\n4.2: $U_k^n,\\z_n$\n\n4.6: $\\t$\n\n5.1: $W'_n$\n\n5.2: $V_k^n,{}^\\da V_k^n,\\z'_n,\\tiz_n$\n\n5.4: $\\ti\\t$\n\n6.4: $\\t$\n\\widestnumber\\key{AL}\n\\Refs\n\\ref\\key\\AL\\by D.Alvis\\paper Induce\/restrict matrices for exceptional Weyl groups\\jour math.RT\/0506377\\endref\n\\ref\\key\\ALL\\by D.Alvis and G.Lusztig\\paper On Springer's correspondence for simple groups of type $E_n$\n($n=6,7,8$)\\jour Math. Proc. Camb. Phil. Soc.\\vol92\\yr1982\\pages65-78\\endref\n\\ref\\key\\CL\\by G.Lusztig\\paper Irreducible representations of finite classical groups\\jour Inv.Math.\\vol43\\yr1977\n\\pages125-175\\endref\n\\ref\\key\\SPE\\by G.Lusztig\\paper A class of irreducible representations of a Weyl group\\jour Proc. Kon. Nederl.\nAkad. (A)\\vol82\\yr1979\\pages323-335\\endref\n\\ref\\key\\UC\\by G.Lusztig\\paper Unipotent characters of the symplectic and odd\northogonal groups over a finite field\\jour Invent.math.\\vol64\\yr1981\\pages263-296\\endref\n\\ref\\key\\OR\\by G.Lusztig\\book Characters of reductive groups over a finite field\\bookinfo Ann.Math.Studies\\vol107\n\\publ Princeton U.Press\\yr1984\\endref \n\\ref\\key\\ICC\\by G.Lusztig\\paper Intersection cohomology complexes on a reductive group\\jour Invent.Math.\\vol75\n\\yr1984\\pages205-272\\endref \n\\ref\\key\\USU\\by G.Lusztig\\paper A unipotent support for irreducible representations\\jour Adv.in Math.\\vol 94\n\\yr1992\\pages139-179\\endref\n\\ref\\key\\SH\\by T.Shoji\\paper On the Springer representations of Weyl groups of classical algebraic groups\\jour \nComm.in Alg.\\vol7\\yr1979\\pages1713-1745,2027-2033\\endref\n\\ref\\key\\SHO\\by T.Shoji\\paper On the Springer representations of Chevalley groups of type $F_4$\\jour Comm.in Alg.\n\\vol8\\yr1980\\pages409-440\\endref\n\\ref\\key\\SP\\by T.A.Springer\\paper Trigonometric sums, Green functions of finite groups and representations of Weyl\ngroups\\jour Invent.Math.\\vol36\\yr1976\\pages173-207\\endref\n\\endRefs\n\\enddocument","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Introduction}\nSpins in semiconductor quantum dots are among the most promising candidates for the realization of a scalable quantum bit (qubit) \\cite{Loss1998,Kloeffel2013}. For such spin qubits, the qubit energy is the Zeeman energy $\\Delta = g \\mu_{\\rm B} B$, where $\\mu_{\\rm B}$ is the Bohr magneton, $B$ is the magnetic field and $g$ is the g-factor. Hence, a detailed understanding of the g-factor is an important ingredient for the addressability of spin qubits. In multi-qubit devices, local g-factor differences between the individual qubits allow to address them selectively, and can also be utilized for the realization of quantum logic gates \\cite{Veldhorst2015, Jones2018}.\n\nIn addition to the addressability, the g-factor can also impact the coherence of spin qubits.\nWhile the spin of an electron in GaAs is strongly influenced by the host nuclear spins through the hyperfine interaction \\cite{Merkulov2002, Khaetskii2002, Petta2005, Koppens2008, Cywinski2009, Bluhm2011}, the resulting magnetic noise is slow \\cite{Delbecq2016}, allowing for effective countermeasures: The short dephasing time of an unprotected spin, of order 10 ns \\cite{Petta2005, Koppens2008}, has been extended to order $\\mu$s by postprocessing \\cite{Nakajima2019} or active compensation \\cite{Shulman2014, Nakajima2020} and to order ms by dynamical decoupling \\cite{Malinowski2016}. At such long time scales, additional decoherence sources need to be considered, and fluctuations in the g-factor are one of them.\nThese fluctuations originate from charge noise as $g$ is sensitive to the local electric field \\cite{Veldhorst2014,Ferdous2018}. In addition, this type of decoherence will be a major issue in group-IV semiconductors with little or no nuclear spins, such as silicon \\cite{Tanttu2018} and Si\/SiGe heterostructures \\cite{Yoneda2018}.\n\nIn semiconductors, g-factor corrections arise from the spin-orbit interaction (SOI) \\cite{Winkler2003,Stano2018gfactor}, and spatial variations of $g$ might occur due to local electric fields which modulate this interaction \\cite{Tanttu2018,Ferdous2018,Stano2018gfactor}. Recently, measurements in a SiMOS spin qubit \\cite{Tanttu2018} showed that these corrections are small for electrons in silicon due to comparably weak SOI. For holes, experiments in silicon MOSFETs \\cite{Voisin2016, Crippa2018}, in a GaAs heterostructure \\cite{Wang2016}, and in a silicon-germanium core-shell nanowire \\cite{Brauns2016} show that these corrections are more pronounced due to stronger SOI \\cite{Kloeffel2013}.\n\nHere, we present an experiment where we can separate the isotropic and anisotropic g-factor corrections in two GaAs single-electron spin qubits with slightly different wafer properties. In a recent model by \\textit{Stano et al.} \\cite{Stano2018gfactor}, the Rashba SOI, together with a bulk structure SOI term at finite magnetic field, lead to isotropic corrections, while the Dresselhaus SOI is giving rise to an anisotropic correction. The experiment is in good agreement with these predictions and thus provides clear evidence for a profound, detailed model of the g-factor corrections, serving as the key characteristic of a GaAs quantum dot spin qubit. Previous experiments in this material did not identify the g-factor anisotropy \\cite{Michal2018, Zumbuhl2004}.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.95\\columnwidth]{fig1.pdf} \\vspace{-3mm}\n\t\\caption{(a) Electron micrograph of a cofabricated device with dot position (solid ellipse) and sensor dot (dashed ellipse). The sensor conductance $G_{\\text{sensor}}$ reads the real-time charge state of the dot.\n\t(b) Two-step pulses (I) ionize and (II) load applied on the dot gate $C_P$, used to measure the tunneling rate $\\Gamma$ into the empty dot for detuning $\\Delta E$ from the reservoir chemical potential $\\mu$. The sensor plunger $C_{SP}$ is compensated to maintain read-out sensitivity. A magnetic field splits the dot states $\\left|\\uparrow\\right\\rangle$, $\\left|\\downarrow\\right\\rangle$ as well as the conduction band (blue and green) by the Zeeman energy $\\Delta$.\n\t(c) Sensor conductance $G_{\\text{sensor}}$ for two cycles (dashed pulses). Low (high) $G_{\\text{sensor}}$ indicates an empty (occupied) dot, respectively. The ionization rate during (I) is faster than the sensor bandwidth. The electron loading times $t_L$, appearing as clear steps (red traces), are histogrammed to extract the tunnel rate $\\Gamma$ via exponential fit, shown in (d) for two examples, with typical error bars $\\pm 10$~Hz. }\n\t\\label{fig:gfactor:Fig1}\\vspace{-4mm}\n\\end{figure}\n\n\nThe experiment was performed on two separate quantum dots, each in the single-electron regime, with adjacent quantum dot charge sensor, all defined with depletion gates, layout shown in Fig.~\\ref{fig:gfactor:Fig1}a, on two slightly different GaAs 2D electron gases (2DEGs), see Sec.~1 in Ref.~\\cite{som} for details. The crystal axes as labeled were carefully tracked from the wafer flats. Here, the quantum dot is tunnel coupled only to the left reservoir. The sensor conductance reads the charge state \\cite{Field1993,Barthel2010}, here with a bandwidth of $\\sim$30\\,kHz. The device is mounted on a piezo rotator stage (Attocube ANRv51), allowing magnetic fields up to 14\\,T to be applied in an arbitrary in-plane direction. The misalignment is less than 2$^{\\circ}$ and thus here negligible \\cite{Camenzind2017}. Measurements are carried out in a dilution refrigerator at an electron temperature of 200 mK.\n\nTo calculate the Zeeman energy $\\Delta$, it is necessary to convert changes of the voltage applied to the plunger gate $C_P$ to changes in energy of the quantum dot levels (see Fig.~\\ref{fig:gfactor:Fig1}b). We calibrate the lever arm of $C_P$ by probing the Fermi-Dirac distribution of the reservoir at an increased temperature ($\\sim$600\\,mK) where we assume that the electronic temperature is the same as the temperature of the mixing chamber \\cite{Maradan2014}. We checked that the lever arm shows no significant dependence on the strength or direction of the external magnetic field. Nevertheless, the lever arm ends up giving the dominant contribution to the accuracy of the extracted Zeeman splitting and g-factors, as will be discussed later. The right barrier directly next to $C_P$ is opaque, and the left reservoir barrier is relatively far from $C_P$, such that voltage changes on this plunger-gate do not affect the tunneling barrier by much.\n\nBecause the g-factor corrections depend on the shape of the quantum dot, we performed a recently developed spectroscopy of the quantum dot orbitals with in-plane magnetic fields \\cite{Camenzind2018}. For device 1, we found orbital energies that suggest a slightly ellipsoidal, disc shaped quantum dot which is elongated along the x-axis of the device as depicted in Fig.~\\ref{fig:gfactor:Fig1}a. For device 2, we found a more asymmetric dot shape than in the first device. We note that for this difference in dot shapes, the theory predicts an immeasurable small difference in the g-factor (see Sec.~II in Ref.~\\cite{som}).\n\nWe obtain $g$ by measuring the tunnel rate $\\Gamma$ into the spin states of an empty quantum dot, taking advantage of the increase in $\\Gamma$ when both spin states are energetically available. From these rates we extract $\\Delta$, and from the dependence of $\\Delta$ on the magnetic field strength we fit $g$. We measure $\\Gamma$ by applying a two-step pulse to plunger gate $C_P$ (see Fig.~\\ref{fig:gfactor:Fig1}a), repeatedly ionizing and loading the quantum dot as shown by the energy diagrams in Fig.~\\ref{fig:gfactor:Fig1}b: to ionize, the energy level of the charged quantum dot is pulsed above the chemical potential $\\mu$ of the reservoir such that an electron will tunnel into the empty states of the reservoir and thermalize.\nWe chose this ionization pulse such that the ionization efficiency is close to unity.\nTo load, we pulse the empty quantum dot to an energy detuning $\\Delta E$ below $\\mu$. At this energy, filled states are available in the reservoir and an electron can elastically tunnel through the barrier into the quantum dot. The time constant of this probabilistic tunnel process is given by $\\Gamma$.\n\nWe obtain $\\Gamma$ by monitoring the charge sensor conductance $G_{\\text{sensor}}$ and extract the times of these loading events $t_L$ as shown in Fig~\\ref{fig:gfactor:Fig1}c: the tunneling of an electron leads to a change of the charge state from empty to loaded, which results in an observable switch to an higher $G_{\\text{sensor}}$. We cycle through this pulse scheme between 2k and 20k times and extract $\\Gamma$ by fitting an exponential function to a histogram of $t_L$ (see Fig~\\ref{fig:gfactor:Fig1}d). When changing the pulse amplitudes, we obtain $\\Gamma$ as a function of the detuning $\\Delta E$.\n\nTwo important comments about the experiment: first, to stay in the sweet spot of the sensor during the pulse sequence, we have to compensate the crosstalk between the pulses applied to $C_P$ and the sensor quantum dot by applying pulses of opposite polarity to the sensor plunger gate $C_{SP}$ (see Fig.~\\ref{fig:gfactor:Fig1}a) \\cite{Biesinger2015}.\nSecond, we divide the total number of pulse cycles into segments in order to mitigate drift-related effects: In every segment, 100 pulses are applied at each selected detuning $\\Delta E$ before an automated feed-back loop is used to compensate for time-dependent drifts of the quantum dot levels by retrieving the position of $\\Delta E = 0$ \\cite{Amasha2008a}. We exclude hysteresis effects by selecting the sequence of detunings $\\Delta E$ to which we pulse randomly for each round.\n\nIn Fig.~\\ref{fig:gfactor:Fig2}a we show data of $\\Gamma (\\Delta E)$ for increasing magnetic fields up to 12~T. Due to orbital effects of the in-plane magnetic field \\cite{Camenzind2018}, the tunnel barriers have to be readjusted for each field configuration in order to keep the tunnel rates at a couple of hundred Hz (see Sec.~4 in Ref.~\\cite{som}). As a consequence, the magnitudes of $\\Gamma(\\Delta E)$ for the different traces are not comparable and were therefore normalized in Fig.~\\ref{fig:gfactor:Fig2}a. As the dot ground state is pulled below the reservoir and $\\Delta E$ starts to increase from zero, electrons may start to tunnel onto the dot, leading to the rising flank as seen in Fig. \\ref{fig:gfactor:Fig2}a for $\\Delta E\\gtrsim 0$. The observed broadening is given by the reservoir temperature. As the dot level is pulled further below the reservoir, eventually also the excited spin state becomes available, thus increasing the tunnel rate above the ground state rate, as indicated by the yellow arrow. The separation of the two steps is thus identified as the Zeeman splitting $\\Delta$, and grows with magnetic field, as seen on Fig. \\ref{fig:gfactor:Fig2}a. The observed exponential suppression of $\\Gamma$ with increasing $\\Delta E$ is attributed to an effective increase of the tunnel barrier potential experienced by the electrons when the gate voltage is increased \\cite{MacLean2007,Amasha2008b,Stano2010,Simmons2011}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width=1\\columnwidth]{fig2.pdf} \n\t\\caption{(a) Examples of the normalized tunnel rate $\\Gamma$ into the empty quantum dot for different detunings $\\Delta E$ and magnetic field strengths. Each trace exhibits two resonances, identified as the two spin states due to their behavior in magnetic field (yellow arrows). The fits shown here are according to a phenomenological model described in Sec.~3 of Ref.~\\cite{som}.\n In the trace taken at 4\\,T, the dashed line shows $\\Gamma_g (\\Delta E)$, the contribution of the spin ground state to the total tunnel rate, and $\\Delta$ indicates the Zeeman splitting. (b) Zeeman splittings $\\Delta$ in device 1, measured for different magnetic field strengths $B$ and directions as indicated by the labels. The error bars reflect the statistical uncertainty from the fits. The slope is the absolute value of the g-factor $\\vert g\\vert =\\Delta\/(\\mu_{\\rm{B}} \\vert B\\vert)$ and differs from the GaAs bulk g-factor due to spin-orbit interaction induced corrections. A distinct g-factor anisotropy is observed in the data. The inset shows the direction of the applied magnetic fields with respect to the crystal axes.}\n\t\\label{fig:gfactor:Fig2} \\vspace{-4mm}\n\\end{figure}\n\n\n\nNext, we look at the magnetic field strength and direction dependence of the extracted $\\Delta$ to investigate the behavior of the g-factor. We take such data for magnetic fields applied in a range of directions between the crystallographic axes $[\\bar{1}\\bar{1}0]$ ($X$) and $[1\\bar{1}0]$ ($Y$) (see Fig. \\ref{fig:gfactor:Fig1}a). The measured Zeeman splittings $\\Delta$ for device 1 are plotted in Fig.~\\ref{fig:gfactor:Fig2}b (see Sec.~5 in Ref.~\\cite{som} for device 2). We find a linear dependence for all directions, which indicates that the g-factor is independent of the strength of the magnetic field. Accordingly, we use a linear fit (without offset) on these data sets to obtain $\\vert g \\vert$, the absolute value of $g$. The statistical uncertainty obtained from the fits is in the range of one percent relative error. Also, it was not possible to obtain a reliable $\\Delta$ at some specific $B$ values and directions due to vanishing excited spin tunneling \\cite{Amasha2008a, Yamagishi2014} and\/or due to measurement artifacts such as reservoir resonances.\n\nStrikingly, our data indicates that $g$ depends on the magnetic-field direction. For device 1, the g-factor is maximal for a field along $X$ with $\\vert g \\vert \\approx 0.406$, and minimal along $Y$ where $\\vert g \\vert \\approx 0.344$. This difference is well above the statistical error bar, and similar in device 2 (see Sec.~5 in Ref.~\\cite{som}). This is in good qualitative agreement with the theory in Ref.~\\cite{Stano2018gfactor}. In that model, there are numerous terms giving corrections to the bulk g-factor. These can be separated into an isotropic and an anisotropic part, such that\n\\begin{equation}\n\tg = g_{\\text{bulk}} + \\delta g_{i} + \\delta g_{a}\\cos\\left({2\\phi + \\pi\/2}\\right),\n\t\\label{eq:gCorrections}\n\\end{equation}\nwhere $g_{\\text{bulk}} =-0.44$ is the GaAs bulk g-factor and $\\phi$ defines the in-plane angle with respect to the main crystal axis $[100]$ (see inset in Fig.~\\ref{fig:gfactor:Fig2}b). Here, terms with higher-order angle dependence are small and are neglected. We extract $\\delta g_i$ and $\\delta g_a$ experimentally, and the quantification of these two parameters for our quantum dot is the main result of this article. For most of the relevant terms, the magnitudes of the g-factor corrections depend primarily on $\\lambda_z$, the effective width of the electron wave function along the growth direction \\cite{Stano2018gfactor}. Here, $\\lambda_z$ is given by the triangular confinement potential formed by the GaAs\/AlGaAs heterostructure. We fit it from excited orbital state data and find $\\lambda_z\\approx6.5$\\,nm similar for both devices \\cite{Camenzind2018,Stano2017}, see Sec.~2 in Ref.~\\cite{som}.\n\nWe compare the experimental finding with the theoretical prediction for the magnetic field along $Y$ and the specific quantum dot confinement of device 1. We obtain $\\delta g$, the g-factor correction from $g_{\\rm{bulk}}$, from the measurement at each individual magnetic field by calculating $\\delta g = |g_{\\rm{bulk}}| - |\\Delta\/(\\mu_{\\rm{B}} B)|$. As seen in Fig.~\\ref{fig:gfactor:Fig3}a, the data of the two devices are in agreement with each other within the error bars (apart from one outlier) and show a slight trend to decrease at large fields. Also, with most data points slightly below the green theory curve, it seems fairly clear that the theory overall predicts a somewhat larger correction than measured in experiment. While only one specific direction is plotted here, we find this discrepancy generally for the isotropic correction. The model predicts an average $\\vert\\bar{g}\\vert = \\vert g_{\\rm{bulk}} + \\delta g_{i} \\vert \\approx 0.33$ for an electron confined in such a quantum dot. The data presented in Fig.~\\ref{fig:gfactor:Fig2}b suggests an isotropic correction to  $\\vert \\bar{g} \\vert \\approx 0.373$ for device 1, and $\\vert \\bar{g} \\vert \\approx 0.396$ for device 2. Thus, the theory calculates a stronger isotropic correction than seen in the experiment -- to be discussed later.\n\nThere are several predicted terms, as shown on Fig.~\\ref{fig:gfactor:Fig3}a, which contribute to the isotropic correction $\\delta g_{i}$. These terms originate either from the band structure or the heterostructure confinement, see Ref.~\\cite{Stano2018gfactor}. The field direction only matters for the anisotropic Dresselhaus correction $\\delta g_D$. From the theoretical calculations, we conclude that the isotropic correction is dominated by $\\delta  g_R$, a correction due to intrinsic Rashba SOI, and by $\\delta g_{43}$, a correction due to the generic SOI term $H_{43}$ \\cite{Braun1985, Stano2018gfactor}. The well known Rashba SOI term originates from the structural inversion asymmetry in the GaAs\/AlGaAs heterostructure, while $H_{43}$ is a bulk band structure term generated by a magnetic field. The next strongest isotropic term is the penetration correction $\\delta g_p$ which arises from the overlap of the wave function with the AlGaAs bulk where $g_{\\rm{AlGaAs}} = +0.4$ \\cite{Hanson2003}. This term is negligible in our case but becomes substantial for smaller 2DEG widths ($\\lambda_z \\lesssim 4$\\,nm).\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\t\\includegraphics[width=1\\columnwidth]{fig3.pdf} \\vspace{-3mm}\n\t\\caption{\n(a)  Cumulative g-factor corrections $\\delta g$ to $g_{\\rm{bulk}}$ as labeled. The isotropic terms are due to penetration into the AlGaAs, $\\delta g_p$, due to the $H_{43}$ term, $\\delta g_{43}$, and the Rashba correction, $\\delta g_{R}$. The Dresselhaus correction, $\\delta g_D$, is anisotropic and given for a field along $Y$ ($\\phi = 315^{\\circ}$), the same direction for which the data is shown (black device 1, yellow device 2). Here, the g-factor corrections at the respective magnetic fields are directly obtained from the individual measured Zeeman splitting. The green curve shows the total theoretical g-factor correction for this field direction. In agreement with the model, the experiment barely shows any dependence on the magnetic field strength.\n(b) The anisotropic corrections to the g-factor are dominated by $\\delta g_D$ while $\\delta g_{R}$ and $\\delta g_{43}$ give insignificant anisotropic corrections.}\n\t\\label{fig:gfactor:Fig3} \\vspace{-4mm}\n\\end{figure}\n\nThe anisotropic correction to the g-factor originates from the Dresselhaus SOI which is a consequence of bulk inversion asymmetry in the zinc blende crystal structure of GaAs. As seen in Fig.~\\ref{fig:gfactor:Fig2}b, the largest correction to $g_{\\rm{bulk}}$ is observed along $Y$. This is a strong indication that the Dresselhaus constant $\\gamma_c $ is negative \\cite{Stano2018gfactor,DettwilerPRX, Salis2001}, since a positive $\\gamma_c$ would have the largest deviation from $g_{\\rm{bulk}}$ in the $X$ direction. Our data suggests that $\\delta g_a=0.030\\pm0.002$ for device 1 and $0.025\\pm0.003$ for device 2, which is close to the predicted $\\delta g_a=0.024$. Further, for the relative correction to the g-factor, we find $\\delta g_a \/ \\vert \\bar{g}\\vert \\approx 8.1 \\pm0.5\\%$ for device 1 and $\\approx 6.3 \\pm0.8\\%$ for device 2, which is in good agreement with the model where this ratio is $\\approx 7\\%$.\n\nWe now discuss the possible origins of the discrepancy between theory and experiment in the isotropic correction. The first suspect is the lever arm: the accuracy of the mixing chamber temperature used here is not better than $5-10\\%$, leading to a systematic uncertainty. Because we found that the lever arm is independent of both strength and direction of the field, the scaling of the g-factors along all directions would be equal. Thus, the accuracy of the g-factor measurement is limited by this systematic error to about $10\\%$. For example, a smaller lever arm would result in a reduced $\\vert \\bar{g}\\vert$, and hence a larger $\\delta g_i$, closer to the model. Still, given the $\\lesssim 10\\%$ accuracy, this is not sufficient to reconcile the theory with the data from both devices. Despite the limited accuracy, the precision of the measurement originating from the statistical uncertainty is much better, around $1\\%$, allowing us to compare e.g. g-factors along different directions with high resolution.\n\nAnother source of deviations could be that the constants used for the \\textbf{k$\\cdot$p} calculations in the model were off. From the data available, however, it is not possible to conclude which term leads to the overestimation of $\\delta g_i$ when compared to the experiment.\n\nNext, strain effects could be a source of the discrepancy: in the theory of Ref. \\cite{Stano2018gfactor}, strain-induced SOI is not taken into account. Simplifications in the model of the heterointerface can also lead to a deviation from the observed g-factor: the model assumes an infinite linear slope of the triangular confinement potential and a step-like increase of the aluminum concentration at the AlGaAs\/GaAs interface. In reality, the profile is different in both aspects: the linear slope levels off away from interface and there is a finite transition region from AlGaAs to GaAs. Perhaps most importantly here, the details of the interface on the atomic level can effectively induce additional spin-orbit interactions \\cite{Rossler2002, Golub2004,Tanttu2018,Ferdous2018}.\n\nFinally, we mention the possibility that one needs to go beyond \\textbf{k$\\cdot$p} theory to  fully account for our observations. For example, Ref. \\cite{Bester2005} reports on self-assembled InGaAs\/GaAs quantum dots which are so small and strongly strained that the structure inhomogeneities impose strong deviations from properties based on bulk crystal models. However, this scenario is rather improbable for our large and weakly strained (lattice matched) gated GaAs\/AlGaAs dots.\n\n\\ \\\\\n\nIn summary, we find a clear g-factor anisotropy as well as an isotropic correction in two lateral, gate-defined quantum dots made on different GaAs\/AlGaAs heterostructures. In one device, this ranges from $\\vert g \\vert = 0.344$ to $0.406$ depending on the direction of the applied magnetic field. We compare our findings to a recently proposed theory by \\textit{Stano et al.} \\cite{Stano2018gfactor} in which the g-factor corrections to the GaAs bulk value are divided into a leading isotropic and a weaker anisotropic part. While the measured isotropic corrections are weaker than predicted, our data for the anisotropic corrections are in good agreement with the theory. Here, the isotropic corrections arise from Rashba and $H_{43}$-type of spin-orbit interaction and the anisotropic correction originates from the Dresselhaus SOI. In silicon spin qubits, the anisotropy gives a change of the g-factor of order of one percent, dominated by surface roughness \\cite{Tanttu2018,Ferdous2018}. In contrast, here, the measured anisotropy is more substantial due to the GaAs crystal lattice, i.e. the Dresselhaus SOI.\n\nOur findings substantiate the relevant g-factor corrections in  GaAs spin qubits.\nHere, the identification of the dominant g-factor correction terms could help to better understand the decoherence processes that originate from coupling to  charge noise, and in principle it could also be exploited for all-electrical spin manipulation.\nFurthermore, this work represents a major step in probing band structure parameters using quantum dots. For example, extracting the \\textbf{k$\\cdot$p} parameters from bulk measurements is often complicated by effects of the electron-electron interaction. Here, since the dot is singly occupied, such effects do not enter.\nThis work shows that a quantum dot could be used as a well-controlled probe in experiments aiming at the microscopic parameters of the semiconductor. Similar experiments could be performed for samples with significantly different heterostructure confinements. From the dependence of the g-factor corrections on the width and symmetry of the heterostructure, the \\textbf{k$\\cdot$p} parameters could be obtained with a new level of confidence \\cite{Stano2018gfactor}.\n\nThe data supporting this study are available in a Zenodo repository \\cite{gfactorZenodo}.\n\n\\subsection{Acknowledgments}\nWe thank M. Steinacher and S. Martin for technical support. This work was supported by the Swiss Nanoscience Institute (SNI), NCCR QSIT and SPIN, Swiss NSF, ERC starting grant (DMZ), and the European Microkelvin Platform (EMP). PS acknowledges support from CREST JST (JPMJCR1675).\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, the diffractive processes are attracting much attention as a\nway of amplifying the physics programme at proton colliders,\nincluding new channels searching for New Physics. The investigation of these reactions at high energies gives\nimportant information on the structure of hadrons and their\ninteraction mechanisms. Hard diffractive processes, such as the\ndiffractive production of massive electroweak bosons and dijets, allow the study of the interplay\nof small- and large-distance dynamics within QCD. The existence of\na hard scale provides the normalization of the Born term diagram. For boson hadroproduction, single-diffractive dissociation  can occur characterized by the existence of\none large rapidity gap, which can be represented by\n the Pomeron exchange. At high energies, there are important\ncontributions from unitarization effects and the suppression of the single-Pomeron Born\ncross section due to the multi-Pomeron\ncontributions depends, in general, on the particular\nhard process. At the Tevatron energy, $\\sqrt s = 1.8$~TeV, the\nsuppression is in the range 0.05--0.2~\\cite{GLM,KMRsoft,BH,KKMR}, whereas for LHC energy, $\\sqrt{s}=14$ TeV, the suppression  appears to be of order 0.08--0.1 ~\\cite{GLM,KMRsoft,KKMR}. Therefore, the correct treatment of\nthe multiple scattering effects is crucial for the reliability of the\ntheoretical predictions of the cross sections for these\ndiffractive processes.\n\nIn the present study, our motivation to perform a new analysis on diffractive boson production is twofold: produce updated theoretical estimations compatible with the current Tevatron data on single diffractive $W$ and $Z$ hadroproduction \\cite{CDF,D0} and to perform reliable predictions to the future measurements at the LHC. In order to do so, we use Regge factorization (single Pomeron exchange) and the corresponding corrections for multiple-Pomeron scatterings. Factorization for diffractive hard scattering is equivalent to the hard-scattering aspects of the Ingelman and Schlein model\n\\cite{IS}, where diffractive scattering is attributed to the\nexchange of a Pomeron, i.e. a colorless object with vacuum quantum\nnumbers. The Pomeron is treated like a real particle,\nand one considers that a diffractive electron-proton collision is\ndue to an electron-Pomeron collision and that a diffractive\nproton-proton collision is due to a proton-Pomeron collision.\nTherefore, the diffractive hard cross sections are\nobtained as a product of a hard-scattering coefficient, a known Pomeron-proton coupling, and parton densities in\nthe Pomeron. The parton densities in the Pomeron have been sistematicaly extracted from\ndiffractive DIS measurements. In particular, the quark singlet and gluon content of the Pomeron is obtained from the diffractive struture function $F_{2}^{D(3)}(x_{\\tt I\\! P},\\beta,Q^2)$.  Recently, a new analysis of these diffractive parton distributions has been presented \\cite{H1diff} by the H1 Collaboration in DESY-HERA. On the other hand, it is well known that the single Pomeron approach produces results that overestimate the experimental values by a large factor \\cite{Alvero,mara}. Thus, in the present analysis the corresponding multiple Pomeron exchange corrections will be taken into account.\n\nThis paper is organized as follows. In the next section, we present the main formulae to compute the inclusive and diffractive cross sections for $W$ and $Z$ hadroproduction. We show the details concerning the parameterization for the diffractive partons distribution in the Pomeron, extracted recently in DESY-HERA. In addition, we present the theoretical estimations for the gap survival propability factor that will be used in the comparison of our results with experimental measurements from Tevatron and extrapolations to the LHC energy.  In the last section we present our numerical results and perform predictions to future measurements in CERN LHC experiment. The compatibility with data is analysed, possible additional corrections are investigated and the comparison with other approaches are  considered.\n\n\n\n\\section{Diffractive Hadroproduction of Massive Gauge Bosons}\n\nLet us start by introducing the main expressions to compute the inclusive and diffractive cross sections.\nFor the {\\it hard} diffractive\nprocesses we will consider the Ingelman-Schlein (IS) picture \\cite{IS}, where\nthe Pomeron structure (quark and gluon content) is probed. The starting point is the generic cross section for a process in which partons of two hadrons, $A$ and $B$, interact to produce a massive electroweak boson, $\n A + B \\rightarrow (W^{\\pm}\/Z^0) + X$, \n\\begin{eqnarray}\n\\nonumber\n\\frac{d \\sigma}{dx_a\\,dx_b} = \\sum_{a,b} f_{a\/A}(x_a,\n\\mu^2)\\, f_{b\/B}(x_b, \\mu^2)\\, \\frac{d\\hat{\\sigma}(ab\\rightarrow [W\/Z]\\,X)}{d\\hat{t}}\\,,\n\\label{gen}\n\\end{eqnarray}\nwhere $x_i f_{i\/h}(x_i, \\mu^2)$ is the parton distribution function of a parton of flavour $i=a,b$ in the hadron $h=A,B$.  The quantity $d\\hat{\\sigma}\/d\\hat{t}$ gives the elementary hard cross section of the corresponding subprocess and $\\mu^2=M_{W\/Z}^2$ is the hard scale in which the pdf's are evolved in the QCD evolution. Equation above express the usual leading-order QCD procedure to obtain the non-diffractive cross section. Next-to-leading-order\ncontributions are not essential for the present purposes.\n\nIn order to obtain the corresponding expression for\ndiffractive processes, one assumes that one of the hadrons, say\nhadron $A$, emits a Pomeron whose partons interact with partons of the hadron $B$.\nThus the parton distribution  $x_a f_{a\/A}(x_a, \\mu^2)$ in\nEq.~(\\ref{gen}) is replaced by the convolution between a \ndistribution of partons in the Pomeron, $\\beta f_{a\/{\\tt I\\! P}}(\\beta,\n\\mu^2)$, and the ``emission rate\" of Pomerons by the hadron, $f_{{\\tt I\\! P}\/h}(x_{{\\tt I\\! P}},t)$. The last quantity, $f_{{\\tt I\\! P}\/h}(x_{{\\tt I\\! P}},t)$, is the Pomeron flux factor and its explicit formulation is described in\nterms of Regge theory. Therefore, we can rewrite the parton distribution as\n\\begin{eqnarray}\n\\nonumber\n\\label{convol}\nx_a f_{a\/A}(x_a, \\,\\mu^2) & =& \\int dx_{{\\tt I\\! P}} \\int d\\beta \\int dt\\,\nf_{{\\tt I\\! P}\/A}(x_{{\\tt I\\! P}},\\,t) \\\\\n&\\times & \\beta \\, f_{a\/{\\tt I\\! P}}(\\beta, \\,\\mu^2)\\,\n\\delta \\left(\\beta-\\frac{x_a}{x_{{\\tt I\\! P}}}\\right),\n\\end{eqnarray}\nand, now defining $\\bar{f} (x_{{\\tt I\\! P}}) \\equiv \\int_{-\\infty}^0 dt\\\nf_{{\\tt I\\! P\/A}}(x_{{\\tt I\\! P}},t)$, one obtains\n\\begin{eqnarray}\n\\label{convoP}\nx_a f_{a\/A}(x_a, \\,\\mu^2)\\ =\\ \\int dx_{{\\tt I\\! P}} \\\n\\bar{f}(x_{{\\tt I\\! P}})\\, {\\frac{x_a}{x_{{\\tt I\\! P}}}}\\, f_{a\/{\\tt I\\! P}}\n({\\frac{x_a}{x_{{\\tt I\\! P}}}}, \\mu^2).\n\\end{eqnarray}\n\nConcerning the $W^{\\pm}$ diffractive production, one considers the reaction\n$p + {\\bar p}(p) \\rightarrow p + \\ W (\\rightarrow e\\ \\nu ) + \\ X$, assuming that a Pomeron emitted by a proton in\nthe positive $z$ direction interacts with a $\\bar p$ (or a $p$) producing $W^{\\pm}$\nthat subsequently decays into $e^{\\pm}\\ \\nu$. The detection of this reaction is\ntriggered by the lepton ($e^+$ or\n$e^-$) that appears boosted towards negative $\\eta$ (rapidity) in coincidence\nwith a rapidity gap in the right hemisphere.\n\nBy using the same concept of the convoluted structure function, the\ndiffractive (single diffraction, SD) cross section for the inclusive lepton production for\nthis process becomes\n\\begin{widetext}\n\\begin{eqnarray}\n\\frac{d\\sigma^{\\mathrm{SD}}_{\\mathrm{lepton}}}{d\\eta_e}= \\sum_{a,b}\n\\int \\frac{dx_{\\tt I\\! P}}{x_{\\tt I\\! P}}\\, \\bar{f}(x_{\\tt I\\! P})\n\\int dE_T \\ f_{a\/{\\tt I\\! P}}(x_a, \\,\\mu^2)\\,f_{b\/\\bar{p}(p)}(x_b, \\,\\mu^2)\\\n\\left[\\frac{ V_{ab}^2\\ G_F^2}{6\\ s\\ \\Gamma_W}\\right]\\ \\frac{\\hat{t}^2}\n{\\sqrt{A^2-1}}\n\\label{dsw}\n\\end{eqnarray}\n\\end{widetext}\nwhere\n\\begin{equation}\nx_a = \\frac{M_W\\ e^{\\eta_e}}{(\\sqrt{s}\\ x_{{\\tt I\\! P}})}\\ \\left[A \\pm\n\\sqrt{(A^2-1)}\\right],\n\\label{xaw}\n\\end{equation}\n\\begin{equation}\nx_b = \\frac{M_W\\ e^{-\\eta_e}}{\\sqrt{s}}\\ \\left[A \\mp \\sqrt{(A^2-1)}\\right],\n\\label{xbw}\n\\end{equation}\nand\n\\begin{equation}\n\\hat{t}=-E_T\\ M_W\\ \\left[A+\\sqrt{(A^2-1)}\\right]\n\\label{tw}\n\\end{equation}\nwith $A={M_W}\/{2 E_T}$, $E_T$ being the lepton transverse energy, $G_F$ is the Fermi constant and the hard scale $\\mu^2=M_W^2$. The quantity  $V_{ab}$ is equal\n to the Cabibbo-Kobayashi-Maskawa matrix element if $e_a+e_{b}=\\pm 1$ and zero otherwise, where $a,b$ denote quark flavors and $e_q$ the fractional charge of quark $q$. The upper signs in Eqs.~(\\ref{xaw}) and (\\ref{xbw})\nrefer to $W^+$ production (that is, $e^+$ detection). The corresponding\ncross section for $W^-$ is obtained by using the lower signs and ${\\hat t}\n\\leftrightarrow {\\hat u}$.\n\nIn a similar way, the cross section for the diffractive hadroproduction of neutral weak vector boson $Z$\nis given by\n\\begin{widetext}\n\\begin{eqnarray}\n  \\sigma ^{\\mathrm{SD}}_{Z}\\,(\\sqrt{s}) = \\sum _{a,b} \\int \\frac{dx_{\\tt I\\! P}}{x_{\\tt I\\! P}}\n    \\int \\frac{dx_b}{x_b} \\int \\frac{dx_a}{x_a}\n    \\, \\bar{f}(x_{\\tt I\\! P})\n    \\, f_{a\/{\\tt I\\! P}}(x_{a},\\,\\mu^2)\n    \\, f_{b\/\\bar{p}(p)}(x_{b}, \\,\\mu^2)\\left[\\frac{2\\pi C_{ab}^ZG_FM_Z^2}{3\\,\\sqrt{2}\\, s}\\right]\\,\\frac{d\\hat{\\sigma}(ab\\rightarrow ZX)}\n{d\\hat{t}}\\,,\n\\label{dsz}\n\\end{eqnarray}\n\\end{widetext}\nwhere $C^Z_{q{\\bar q}}=1\/2-2|e_q|\\sin^{2}\\theta _{W}\n    +4|e_q|^{2}\\sin^{4}\\theta _{W}$, with $\\theta _{W}$ being the Weinberg or weak-mixing angle. The definitions for $x_{a,b}$ are similar as for the $W$ case and now $\\mu^2=M_Z^2$. The values of the electroweak parameters\nthat appear in the various formulae were taken from the particle data\nhandbook \\cite{PDG}, and we use only four flavors ($u$, $d$, $s$, $c$)\nin the weak mixing matrix, with the Cabibbo angle $\\theta _{C} = 0.2269$.\n\n\n\n\\subsection{The Pomeron Flux Factor}\n\nAn important element in the calculation of hard diffractive cross sections is the Pomeron flux factor, introduced in\nEq.~(\\ref{convol}). We take the experimental analysis of the diffractive structure function \\cite{H1diff}, where the $x_{\\tt I\\! P}$ dependence is parameterised using a flux factor\nmotivated by Regge theory \\cite{Collins},\n\\begin{eqnarray}\nf_{\\tt I\\! P\/p}(x_{\\tt I\\! P}, t) = A_{\\tt I\\! P} \\cdot\n\\frac{e^{B_{\\tt I\\! P} t}}{x_{\\tt I\\! P}^{2\\alpha_{\\tt I\\! P} (t)-1}} \\ ,\n\\label{eq:fluxfac}\n\\end{eqnarray}\nwhere  the Pomeron trajectory is assumed to be linear,\n$\\alpha_{\\tt I\\! P} (t)= \\alpha_{\\tt I\\! P} (0) + \\alpha_{\\tt I\\! P}^\\prime t$, and the parameters\n$B_{\\tt I\\! P}$ and $\\alpha_{\\tt I\\! P}^\\prime$ and their uncertainties are obtained from\nfits to H1 FPS data \\cite{H1FPS}. The normalisation parameter $A_{\\tt I\\! P}$ is chosen such that\n$x_{\\tt I\\! P} \\cdot \\int_{t_{\\rm cut}}^{t_{\\rm min}} f_{\\tt I\\! P\/p} \\ {\\rm d} t\n= 1$ at $x_{\\tt I\\! P} = 0.003$, where\n$|t_{\\rm min}| \\simeq m_p^2 \\, x_{\\tt I\\! P}^2 \\, \/ \\, (1 - x_{\\tt I\\! P})$ is the minimum\nkinematically accessible value of $|t|$, $m_p$ is the proton mass and\n$|t_{\\rm cut}|= 1.0 \\rm\\ GeV^{2}$ is the limit of the measurement.\n\nThe flux factor above corresponds to the standard Pomeron flux from Regge phenomenology, based on the Donnachie-Landshoff model \\cite{DLflux}. On the other hand, there is an alternative Pomeron flux, proposed first by Goulianos \\cite{Goulianos}, which considers it as a probability density. Thus, the integral over the diffractive phase space could not exceed the unit and the standard flux should be normalized. For instance, see Refs. \\cite{mara} for previous phenomenology using the normalized flux in boson hadroproduction.\n\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=0.47]{wtev1800.eps}\n\\caption{(Color online) The rapidity distribution of electron and positron generated in inclusive and diffractive $W$ hadroproduction at $\\sqrt{s}=1.8$ TeV (see text).}\n\\label{fig:1}\n\\end{figure}\n\n\n\n\\subsection{The Pomeron Structure Function}\n\nIn the estimates for the diffractive cross sections, we will consider the diffractive pdf's recently obtained by the H1 Collaboration at DESY-HERA \\cite{H1diff}. The Pomeron structure function has been modeled in terms of a\nlight flavour singlet distribution $\\Sigma(z)$, consisting of $u$, $d$ and $s$\nquarks and anti-quarks\nwith $u=d=s=\\bar{u}=\\bar{d}=\\bar{s}$,\nand a gluon distribution $g(z)$.  Here, $z$ is the longitudinal momentum\nfraction of the parton entering the hard sub-process\nwith respect to the diffractive\nexchange, such that $z=\\beta$\nfor the lowest order quark-parton model process,\nwhereas $0<\\beta<z$ for higher order processes.\nThe quark singlet and gluon distributions are parameterised\nat $Q_0^2$ with the general form,\n\\begin{equation}\nz f_i (z,\\,Q_0^2) = A_i \\, z^{B_i} \\, (1 - z)^{C_i} \\,\\exp\\left[{- \\frac{0.01}{(1-z)}}\\right]\\,\n\\label{param:general}\n\\end{equation}\nwhere the last exponential factor ensures that the diffractive pdf's vanish\nat $z = 1$. The charm and beauty quarks are treated as massive, appearing via boson gluon fusion-type processes up to order $\\alpha_s^2$. To determine experimentally the diffractive pdf's, the following cuts have been considered: $\\beta < 0.8$, $M_X > 2$ GeV and $Q^2 < 8.5 $ GeV$^2$, mostly in order to avoid regions influenced\nby higher twist contributions or large theoretical uncertainties \\cite{H1diff}.\n\nFor the quark singlet distribution,\nthe data require the inclusion of all three parameters\n$A_q$, $B_q$ and $C_q$ in equation~\\ref{param:general}.\nBy comparison, the gluon density is weakly\nconstrained by the data, which are found to be\ninsensitive to the $B_g$ parameter. The gluon density is thus\nparameterised at $Q_0^2$ using only the $A_g$ and $C_g$ parameters.\nWith this parameterisation, one has the value  $Q_0^2 = 1.75 \\ {\\rm GeV^2}$ and it is referred to as the `H1 2006 DPDF Fit A'. It is verified that the fit procedure is not sensitive to the gluon pdf and a new adjust was done with $C_g=0$. Thus, the gluon density is then a simple constant at the starting scale for evolution, which was chosen to be\n$Q_0^2 = 2.5 \\ {\\rm GeV^2}$ and it is referred to as the\n`H1 2006 DPDF Fit B'. The quark singlet distribution is well constrained,\nwith an uncertainty of typically $5 -10\\%$ and good agreement between\nthe results of both fits \\cite{H1diff}.\n\n\n\\subsection{The Gap Survival Factor}\nIn the following analysis we will consider the suppression of the hard diffractive cross section by multi-Pomeron scattering effects.\n This is taken into account through a gap survival probability factor. There has been large interest in the probability of rapidity gaps in high energy interactions to survive as they may be populated by secondary particles generated by rescattering processes. This effect can be described in terms of screening or absorptive corrections, which can be estimated using the quantity \\cite{Bj}:\n \\begin{eqnarray}\n<\\!|S|^2\\!>=\\frac{\\int|{\\cal{A}}\\,(s,b)|^2\\,e^{-\\Omega (s,b)}\\,d^2b}{\\int|{\\cal{A}}\\,(s,b)|^2\\,d^2b}\\,,\n \\end{eqnarray}\nwhere $\\cal{A}$ is the amplitude, in the impact parameter space, of the particular process of interest at center-of-mass energy $\\sqrt{s}$. The quantity $\\Omega$ is the opacity (or optical density) of the interaction of the incoming hadrons. This suppression factor of a hard process accompanied by a rapidity gap depends not only on the probability of the initial state survive, but is sensitive to the spatial distribution of partons inside the incoming hadrons, and thus on the dynamics of the whole diffractive part of the scattering matrix.\n\nFor our purpose, we consider two theoretical estimates for the suppression factor. The first one is the work of Ref. \\cite{KKMR} (labeled KMR), which considers a two-channel eikonal model that embodies pion-loop insertions in the pomeron trajectory, diffractive dissociation and rescattering effects. The survival probability is computed for single, central and double diffractive processes at several energies, assuming that the spatial distribution in impact parameter space is driven by the slope $B$ of the pomeron-proton vertex. We will consider the results for single diffractive processes with $2B=5.5$ GeV$^{-2}$ (slope of the electromagnetic proton form factor) and without $N^*$ excitation, which is relevant to a forward proton spectrometer (FPS) measurement. Thus, we have $<\\!|S|^2\\!>_{\\mathrm{KMR}}=0.15$ for $\\sqrt{s}=1.8$ TeV (Tevatron) and $<\\!|S|^2\\!>_{\\mathrm{KMR}}=0.09$ for $\\sqrt{s}=14$ TeV (LHC).\n\nThe second theoretical estimate for the gap factor is from Ref. \\cite{GLMrev} (labeled GLM), which considers a single channel eikonal approach. We take the case where the soft input is obtained directly from the measured values of $\\sigma_{tot}$, $\\sigma_{el}$ and hard radius $R_{H}$. Then, one has $<\\!|S|^2\\!>_{\\mathrm{GLM}}=0.126$ for $\\sqrt{s}=1.8$ TeV (Tevatron) and $<\\!|S|^2\\!>_{\\mathrm{GLM}}=0.081$ for $\\sqrt{s}=14$ TeV (LHC). We quote Ref. \\cite{GLMrev} for a detailed comparison between the two approaches and further discussions on model dependence of inputs and consideration of multi-channel calculations.\nIt should be stressed that our particular choice by KMR and GLM (single channel)  models is in order to indicate the uncertainty (model dependence) of the soft interaction effects. It is worth to mention that some implementations of GLM model include the results of a two or three channel calculation for  $<\\!|S|^2\\!>$, which are considerably smaller than the one channel result \\cite{GLMrev}.\n\n\n\n\n\\section{Results and Discussion}\n\n\\begin{table}\n\\caption{\\label{tab:table1} Data versus model predictions for diffractive $W^{\\pm}$ hadroproduction (cuts $E_{T_{\\mathrm{min}}}=20$ GeV and $x_{\\tt I\\! P}<0.1$). }\n\\begin{ruledtabular}\n\\begin{tabular}{lccr}\n$\\sqrt{s}$  & Rapidity  & Data (\\%) & Estimate (\\%)\\\\\n\\hline\n1.8 TeV &    $|\\eta_e|<1.1$  & $1.15\\pm 0.55$ \\cite{CDF}  & $0.715\\pm 0.045$\\\\\n1.8 TeV &    $|\\eta_e|<1.1$  & $1.08\\pm 0.25$ \\cite{D0} & $0.715\\pm 0.045$\\\\\n1.8 TeV &    $1.5<|\\eta_e|<2.5$  & $0.64\\pm 0.24$  \\cite{D0} & $1.7\\pm 0.875$\\\\\n1.8 TeV &     Total $W\\rightarrow e\\nu $ & $0.89\\pm 0.25$ \\cite{D0} & $0.735\\pm 0.055$ \\\\\n14 TeV &    $|\\eta_e|<1$  & ---  & $31.1\\pm 1.6$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nIn the following, we present our predictions for hard diffractive production of\nW's and Z's based on the previous discussion. These predictions are compared\nwith experimental data from Refs.~\\cite{CDF,D0} in Tables I and II. In addition, estimations for the LHC are presented. In the numerical calculations, we have used the new H1 parameterizations for the diffractive pdf's \\cite{H1diff}. The `H1 2006 DPDF Fit A' was considered and one verifies that the results are not quite sensitive  to a replacement by `H1 2006 DPDF Fit B'. For the usual pdf's in the proton (anti-proton) we have considered the updated MRST2004F4 parameterization \\cite{mrst2004f4}, which is a four-fixed-flavour version of the standard MRST2004 parton distributions. As the larger uncertainty comes from the gap survival factor, the error in the predictions correspond to the theoretical band for $<\\!|S|^2\\!>$. In the theoretical expressions of previous section  only the interaction of pomerons (emitted by protons) with antiprotons (protons in LHC case) are computed, that means events with rapidity gaps on the side from which antiprotons come from. The experimental rate is for both sides, that is events with a rapidity gap on the proton or antiproton side. Therefore, we have multiplied the theoretical prediction by a factor 2 in order to compare it with data.\n\nLet us start by the diffractive $W$ production. In order to illustrate our investigation, in Fig. \\ref{fig:1} we present the rapidity distribution of the electron (dot-dashed lines) and positron (solid lines) generated in both inclusive and diffractive $W^{\\pm}$ hadroproduction in Tevatron for $\\sqrt{s}=1.8$ TeV. The diffractive cross sections are not corrected by gap survival factor and they are given by Eq. (\\ref{dsw}). In this case, the diffractive production rate is approximately 7 \\% (using the cut $|\\eta|<1$) being very large compared to the Tevatron data. When considering the gap survival probability correction, the values are in better agreement with data. When considering central $W$ boson fraction, $-1.1<\\eta_e<1.1$ (cuts of CDF and D0 \\cite{CDF,D0}), we obtain a diffractive rate of 0.67 \\% using the KMR estimate for $<\\!|S|^2\\!>$, whereas it reaches 0.76\\% for the GLM estimate. The average rate considering the theoretical band for the gap factor is then $R_W= 0.715\\pm 0.045$ \\%. This result is consistent with the experimental central values $R_W^{\\mathrm{CDF}}=1.15$ \\% and $R_W^{\\mathrm{D0}}=1.08$ \\%. The agreement would be better whether the sub-leading reggeon contribution is added, which was not considered in present calculation. In Ref. \\cite{CFL}, it was shown that its introduction considerably enhances the diffractive ratio in the Tevatron regime. Considering the forward $W$ fraction, $1.5<|\\eta_e|<2.5$ (D0 cut), one obtains $R_W=0.83$ \\% for KMR and $R_W=2.58$ \\% for GLM, with an averaged value of $R_W= 1.7\\pm 0.875$ \\%. In this case, our estimate is larger than the central experimental value $R_W^{\\mathrm{D0}}=0.64$ \\%. For the total $W\\rightarrow e\\nu$ we have $R_W=0.68$ \\% for KMR and $R_W=0.79$ \\% for GLM  and the mean value $R_W= 0.735 \\pm 0.055$ \\%, which is in agreement with data and consistent with a large forward contribution. Finally, we estimate the diffractive ratio for the LHC energy, $\\sqrt{s}=14$ TeV. In this case we extrapolate the pdf's in proton and diffractive pdf's in Pomeron to that kimenatical region. This procedure introduces somewhat additional uncertainties in the theoretical predictions. We take the conservative cuts $|\\eta_e|<1$, $E_{T_{\\mathrm{min}}}=20$ GeV for the detected lepton and $x_{\\tt I\\! P}<0.1$. We find $R_W=32.7$ \\% for KMR gap survival probability factor and $R_W=29.5$ \\% for GLM, with a mean value of $R_W^{\\mathrm{LHC}}=31.1\\pm 1.6 $ \\%. This means that the diffractive contribution reaches one third, or even more, of the inclusive hadroproduction even when multi-Pomeron scattering corrections are taken into account. The reason of this enhancement is the increasingly large diffractive cross section. The results presented above are summarized in Table I. The experimental errors have been summed into quadrature.\n\n\\begin{table}\n\\caption{\\label{tab:table2} Data versus model predictions for diffractive $Z^0$ hadroproduction (cuts $E_{T_{\\mathrm{min}}}=16$ GeV and $x_{\\tt I\\! P}<0.1$).}\n\\begin{ruledtabular}\n\\begin{tabular}{lccr}\n$\\sqrt{s}$  & Rapidity  & Data (\\%) & Estimate (\\%)\\\\\n\\hline\n1.8 TeV &   Total $Z\\rightarrow e^+e^-$  & $1.44\\pm 0.80$ \\cite{D0}  & $0.71\\pm 0.05$\\\\\n14 TeV &    Total $Z\\rightarrow e^+e^-$  & ---  & $30.26\\pm 1.41$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nNow, we present the investigations for the diffractive $Z$ hadroproduction.\nWhen the gap survival factor is not considered the diffractive cross section is given by Eq. (\\ref{dsz}), producing a diffractive rate of 6.2 \\%. This value is once again higher than the Tevatron data by a factor five. When considering the gap survival correction, we verify an agreement with experiment. For the total $Z\\rightarrow e^+e^-$ we obtain a diffractive rate of 0.66 \\% using the KMR estimate for $<\\!|S|^2\\!>$, whereas it reaches 0.76 \\% for the GLM estimate. The average value gives $R_Z=0.71\\pm 0.05$, which is consistent with the experimental result $R_Z^{\\mathrm{D0}}=1.44+0.61-0.52$. A rough extrapolation to LHC energy gives $R_Z=31.67$ with KMR gap factor and $R_Z=28.85$ for GLM, with a mean value $R_Z^{\\mathrm{LHC}}=30.26\\pm 1.41$. We again consider the conservative cuts $E_{T_{\\mathrm{min}}}=16$ GeV and $x_{\\tt I\\! P}<0.1$. This estimate follows similar trend as for the $W$ case. The results presented above are summarized in Table II. The experimental errors have been summed into quadrature.\n\nOur results can be compared with previous calculations in diffractive boson hadroproduction. For instance, in Refs. \\cite{mara} one uses IS approach with a normalized Pomeron flux \\cite{Goulianos} and the corresponding diffractive pdf's. The data description for the $W$ case is reasonable. However, the calculations are only compared to the CDF \\cite{CDF} data and they are somewhat larger than ours. In Ref. \\cite{CFL} a hard Pomeron flux is considered, i.e. $\\alpha_{\\tt I\\! P}(0)\\simeq 1.4$, and multiple scatterings are taken into account by a Monte Carlo calculation. In addition, for Tevatron energies the reggeon contribution is added. The results are compared only to CDF data \\cite{CDF} for the $W$ production and the description is consistent with experiment. It is interesting the fact that a hard Pomeron flux could mimic the multi-Pomeron suppression  or the effect of normalizing the standard Pomeron flux. Finally, we need call attention to the uncertainty in the determination of the gap survival probability. The estimates considered here (KMR and GLM) are compatible with each other for the case of single diffractive processes. However, recent calculations using one channel eikonal model give larger values for  $<\\!|S|^2\\!>$ \\cite{BH,Luna}. For instance, in Ref. \\cite{Luna} an eikonal QCD model with a dynamical gluon mass (DGM) was considered. Using a gluon mass $m_g=400$ MeV, one obtains $<\\!|S|^2\\!>_{\\mathrm{DGM}}(\\mathrm{Tevatron})=27.6\\pm7.8$~\\%  and\n$<\\!|S|^2\\!>_{\\mathrm{DGM}}(\\mathrm{LHC})=18.2\\pm7.0$~\\%. These values give $R_W (\\sqrt{s}=1.8 \\,\\mathrm{TeV})\\simeq 1.23$~\\% and $R_Z (\\sqrt{s}=1.8 \\,\\mathrm{TeV})\\simeq 1.21$ \\%. This illustrates the size of uncertainty when considering different estimates for the gap probability.\n\nIn summary, we have shown that it is possible to obtain a reasonable\noverall description of hard diffractive hadroproduction of massive gauge bosons by the model based on Regge factorization supplemented by gap survival factor. For the Pomeron model, we take the recent  H1 diffractive parton density functions extracted from their measurement of $F_2^{D(3)}$. The results are directly dependent on the quark singlet distribution in the Pomeron. We did not observe large discrepancy in using the different fit procedure for diffractive pdf's (fit A and B). We estimate the multiple interaction corrections taking the theoretical prediction of distinct multi-channel models, where the gap factor decreases on energy. That is,  $<\\!|S|^2\\!>\\simeq 15-17.5$ \\% for Tevatron energies going down to $<\\!|S|^2\\!>\\simeq 8.1-9$ \\% at LHC energy. We find that the ratio of diffractive to non-diffractive boson production is in good agreement with the CDF and D0 data when considering these corrections. The overall diffractive ratio for $\\sqrt{s}=1.8$ TeV (Tevatron) is of order 1 \\%. In addition, we make predictions which could be compared to future measurements at LHC. The estimates give large rates of diffractive events, reaching values higher than 30~\\% of the inclusive cross section.\n \n\n\\section*{Acknowledgments}\n\nThis work was supported by CNPq, Brazil. M.M. Machado thanks E.G.S. Luna for useful discussions about gap survival probability calculations. The authors thank M. Arneodo and M. Albrow for useful comments. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}