On Euler characteristic of equivariant sheaves
Alexander Braverman
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\ell$ be another prime number. O.~Gabber and F.~Loeser proved that for any algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$ on $T$ the Euler characteristic $\chi(\calF)$ is non-negative. We conjecture that the same result holds for any perverse sheaf $\calF$ on a reductive group $G$ over $k$ which is equivariant with respect to the adjoint action. We prove the conjecture when $\calF$ is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in $G$. From this we deduce that the conjecture holds for $G$ of semi-simple rank 1.
math0202165.tex
%------------------------------------------------------------
%----------    This is LaTeX 2e -----------------------------
%------------------------------------------------------------

%Style section
\documentclass[reqno,12pt]{amsart}
\usepackage{amscd,amssymb}

%file: latexformat.tex

%page formatting

\textwidth 6in
%\textheight 8.5in
\oddsidemargin 0.25in
\evensidemargin 0.25in
\pagestyle{plain}
%\pagenumbering{alph}
%\pagestyle{myheadings}


%Declaration section
\theoremstyle{plain}
\newtheorem{Thm}[subsection]{Theorem}
\newtheorem{Cor}[subsection]{Corollary}
\newtheorem{Lem}[subsection]{Lemma}
\newtheorem{Prop}[subsection]{Proposition}
\newtheorem{Conj}[subsection]{Conjecture}

\theoremstyle{definition}
\newtheorem{Def}[subsection]{Definition}

\theoremstyle{remark}
\newtheorem{notation}{Notation}
\renewcommand{\thenotation}{}
\newtheorem{Rem}[subsection]{Remark}

\newtheorem{exs}{Examples}
\renewcommand{\theexs}{}

\newtheorem{note}{Note}
\renewcommand{\thenote}{}

\newtheorem{conj}{Conjecture}

%Commandsection
\errorcontextlines=0
\numberwithin{equation}{section}
\renewcommand{\rm}{\normalshape}





%Labeling macros
\newif\ifShowLabels
\ShowLabelstrue
\newdimen\theight
\def\TeXref#1{%
        \leavevmode\vadjust{\setbox0=\hbox{{\tt
                \quad\quad  {\small \rm #1}}}%
        \theight=\ht0
        \advance\theight by \lineskip
        \kern -\theight \vbox to
        \theight{\rightline{\rlap{\box0}}%
        \vss}%
        }}%

\ShowLabelsfalse% comment this out if labels should be printed

%Section titles that can be referenced
\renewcommand{\sec}[2]{\section{#2}\label{S:#1}%
        \ifShowLabels \TeXref{{S:#1}} \fi}
\newcommand{\ssec}[2]{\subsection{#2}\label{SS:#1}%
        \ifShowLabels \TeXref{{SS:#1}} \fi}

\newcommand{\sssec}[2]{\subsubsection{#2}\label{SSS:#1}%
        \ifShowLabels \TeXref{{SSS:#1}} \fi}


%Referencing sections and declarations
\newcommand{\refs}[1]{Section ~\ref{S:#1}}
\newcommand{\refss}[1]{Section ~\ref{SS:#1}}
\newcommand{\refsss}[1]{Section ~\ref{SSS:#1}}
\newcommand{\reft}[1]{Theorem ~\ref{T:#1}}
\newcommand{\refl}[1]{Lemma ~\ref{L:#1}}
\newcommand{\refp}[1]{Proposition ~\ref{P:#1}}
\newcommand{\refc}[1]{Corollary ~\ref{C:#1}}
\newcommand{\refd}[1]{Definition ~\ref{D:#1}}
\newcommand{\refr}[1]{Remark ~\ref{R:#1}}
\newcommand{\refe}[1]{\eqref{E:#1}}
\newcommand{\refco}[1]{Conjecture ~\ref{Co:#1}}

% New environments for declarations that can be referenced
\newenvironment{thm}[1]%
        { \begin{Thm} \label{T:#1}  \ifShowLabels \TeXref{T:#1} \fi }%
        { \end{Thm} }

\renewcommand{\th}[1]{\begin{thm}{#1} \sl }
\renewcommand{\eth}{\end{thm} }

\newenvironment{lemma}[1]%
        { \begin{Lem} \label{L:#1}  \ifShowLabels \TeXref{L:#1} \fi }%
        { \end{Lem} }
\newcommand{\lem}[1]{\begin{lemma}{#1} \sl}
\newcommand{\elem}{\end{lemma}}


\newenvironment{propos}[1]%
        { \begin{Prop} \label{P:#1}  \ifShowLabels \TeXref{P:#1} \fi }%
        { \end{Prop} }
\newcommand{\prop}[1]{\begin{propos}{#1}\sl }
\newcommand{\eprop}{\end{propos}}

\newenvironment{corol}[1]%
        { \begin{Cor} \label{C:#1}  \ifShowLabels \TeXref{C:#1} \fi }%
        { \end{Cor} }
\newcommand{\cor}[1]{\begin{corol}{#1} \sl }
\newcommand{\ecor}{\end{corol}}

\newenvironment{defeni}[1]%
        { \begin{Def} \label{D:#1}  \ifShowLabels \TeXref{D:#1} \fi }%
        { \end{Def} }
\newcommand{\defe}[1]{\begin{defeni}{#1} \sl }
\newcommand{\edefe}{\end{defeni}}

\newenvironment{remark}[1]%
        { \begin{Rem} \label{R:#1}  \ifShowLabels \TeXref{R:#1} \fi }%
        { \end{Rem} }
\newcommand{\rem}[1]{\begin{remark}{#1}}
\newcommand{\erem}{\end{remark}}

\newenvironment{conjec}[1]%
        { \begin{Conj} \label{Co:#1}  \ifShowLabels \TeXref{Co:#1} \fi }%
        { \end{Conj} }
\renewcommand{\conj}[1]{\begin{conjec}{#1} \sl }
\newcommand{\econj}{\end{conjec}}

\newcommand{\eq}[1]%
        { \ifShowLabels \TeXref{E:#1} \fi
           \begin{equation} \label{E:#1} }
\newcommand{\eeq}{ \end{equation} }

\newcommand{\prf}{ \begin{proof} }
\newcommand{\epr}{ \end{proof} }




%------------------------------------------------------------
%----------- This is def.latex2e ---------------------------
%------------------------------------------------------------

%Greek letters
\newcommand\alp{\alpha}         \newcommand\Alp{\Alpha}
\newcommand\bet{\beta}
\newcommand\gam{\gamma}         \newcommand\Gam{\Gamma}
\newcommand\del{\delta}         \newcommand\Del{\Delta}
\newcommand\eps{\varepsilon}
\newcommand\zet{\zeta}
\newcommand\tet{\theta}         \newcommand\Tet{\Theta}
\newcommand\iot{\iota}
\newcommand\kap{\kappa}
\newcommand\lam{\lambda}                \newcommand\Lam{\Lambda}
\newcommand\sig{\sigma}         \newcommand\Sig{\Sigma}
\newcommand\vphi{\varphi}
\newcommand\ome{\omega}         \newcommand\Ome{\Omega}

%Caligraphic roman letters
\newcommand\calA{{\mathcal{A}}}
\newcommand\calB{{\mathcal{B}}}
\newcommand\calC{{\mathcal{C}}}
\newcommand\calD{{\mathcal{D}}}
\newcommand\calE{{\mathcal{E}}}
\newcommand\calF{{\mathcal{F}}}
\newcommand\calG{{\mathcal{G}}}
\newcommand\calH{{\mathcal{H}}}
\newcommand\calI{{\mathcal{I}}}
\newcommand\calJ{{\mathcal{J}}}
\newcommand\calK{{\mathcal{K}}}
\newcommand\calL{{\mathcal{L}}}
\newcommand\calM{{\mathcal{M}}}
\newcommand\calN{{\mathcal{N}}}
\newcommand\calO{{\mathcal{O}}}
\newcommand\calP{{\mathcal{P}}}
\newcommand\calQ{{\mathcal{Q}}}
\newcommand\calR{{\mathcal{R}}}
\newcommand\calS{{\mathcal{S}}}
\newcommand\calT{{\mathcal{T}}}
\newcommand\calU{{\mathcal{U}}}
\newcommand\calV{{\mathcal{V}}}
\newcommand\calW{{\mathcal{W}}}
\newcommand\calX{{\mathcal{X}}}
\newcommand\calY{{\mathcal{Y}}}
\newcommand\calZ{{\mathcal{Z}}}



%Bold roman letters
\newcommand\bfa{{\mathbf a}}            \newcommand\bfA{{\mathbf A}}
\newcommand\bfb{{\mathbf b}}            \newcommand\bfB{{\mathbf B}}
\newcommand\bfc{{\mathbf c}}            \newcommand\bfC{{\mathbf C}}
\newcommand\bfd{{\mathbf d}}            \newcommand\bfD{{\mathbf D}}
\newcommand\bfe{{\mathbf e}}            \newcommand\bfE{{\mathbf E}}
\newcommand\bff{{\mathbf f}}            \newcommand\bfF{{\mathbf F}}
\newcommand\bfg{{\mathbf g}}            \newcommand\bfG{{\mathbf G}}
\newcommand\bfh{{\mathbf h}}            \newcommand\bfH{{\mathbf H}}
\newcommand\bfi{{\mathbf i}}            \newcommand\bfI{{\mathbf I}}
\newcommand\bfj{{\mathbf j}}            \newcommand\bfJ{{\mathbf J}}
\newcommand\bfk{{\mathbf k}}            \newcommand\bfK{{\mathbf K}}
\newcommand\bfl{{\mathbf l}}            \newcommand\bfL{{\mathbf L}}
\newcommand\bfm{{\mathbf m}}            \newcommand\bfM{{\mathbf M}}
\newcommand\bfn{{\mathbf n}}            \newcommand\bfN{{\mathbf N}}
\newcommand\bfo{{\mathbf o}}            \newcommand\bfO{{\mathbf O}}
\newcommand\bfp{{\mathbf p}}            \newcommand\bfP{{\mathbf P}}
\newcommand\bfq{{\mathbf q}}            \newcommand\bfQ{{\mathbf Q}}
\newcommand\bfr{{\mathbf r}}            \newcommand\bfR{{\mathbf R}}
\newcommand\bfs{{\mathbf s}}            \newcommand\bfS{{\mathbf S}}
\newcommand\bft{{\mathbf t}}            \newcommand\bfT{{\mathbf T}}
\newcommand\bfu{{\mathbf u}}            \newcommand\bfU{{\mathbf U}}
\newcommand\bfv{{\mathbf v}}            \newcommand\bfV{{\mathbf V}}
\newcommand\bfw{{\mathbf w}}            \newcommand\bfW{{\mathbf W}}
\newcommand\bfx{{\mathbf x}}            \newcommand\bfX{{\mathbf X}}
\newcommand\bfy{{\mathbf y}}            \newcommand\bfY{{\mathbf Y}}
\newcommand\bfz{{\mathbf z}}            \newcommand\bfZ{{\mathbf Z}}

%Capital roman double letters
\newcommand\QQ{\mathbb{Q}}
\newcommand\WW{\mathbb{W}}
\newcommand\EE{\mathbb{E}}
\newcommand\RR{\mathbb{R}}
\newcommand\TT{\mathbb{T}}
\newcommand\YY{\mathbb{Y}}
\newcommand\UU{\mathbb{U}}
\newcommand\II{\mathbb{I}}
\newcommand\OO{\mathbb{O}}
\newcommand\PP{\mathbb{P}}
\renewcommand\AA{\mathbb{A}}
\renewcommand\SS{\mathbb{S}}
\newcommand\DD{\mathbb{D}}
\newcommand\FF{\mathbb{F}}
\newcommand\GG{\mathbb{G}}
\newcommand\HH{\mathbb{H}}
\newcommand\JJ{\mathbb{J}}
\newcommand\KK{\mathbb{K}}
\newcommand\LL{\mathbb{L}}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\XX{\mathbb{X}}
\newcommand\CC{\mathbb{C}}
\newcommand\VV{\mathbb{V}}
\newcommand\BB{\mathbb{B}}
\newcommand\NN{\mathbb{N}}
\newcommand\MM{\mathbb{M}}


  %Euler Fraktur letters
\newcommand\grA{{\mathfrak{A}}} \newcommand\gra{{\mathfrak{a}}}
\newcommand\grB{{\mathfrak{B}}} \newcommand\grb{{\mathfrak{b}}}
\newcommand\grC{{\mathfrak{C}}} \newcommand\grc{{\mathfrak{c}}}
\newcommand\grD{{\mathfrak{D}}} \newcommand\grd{{\mathfrak{d}}}
\newcommand\grE{{\mathfrak{E}}} \newcommand\gre{{\mathfrak{e}}}
\newcommand\grF{{\mathfrak{F}}} \newcommand\grf{{\mathfrak{f}}}
\newcommand\grG{{\mathfrak{G}}} \newcommand\grg{{\mathfrak{g}}}
\newcommand\grH{{\mathfrak{H}}} \newcommand\grh{{\mathfrak{h}}}
\newcommand\grI{{\mathfrak{I}}} \newcommand\gri{{\mathfrak{i}}}
\newcommand\grJ{{\mathfrak{J}}} \newcommand\grj{{\mathfrak{j}}}
\newcommand\grK{{\mathfrak{K}}} \newcommand\grk{{\mathfrak{k}}}
\newcommand\grL{{\mathfrak{L}}} \newcommand\grl{{\mathfrak{l}}}
\newcommand\grM{{\mathfrak{M}}} \newcommand\grm{{\mathfrak{m}}}
\newcommand\grN{{\mathfrak{N}}} \newcommand\grn{{\mathfrak{n}}}
\newcommand\grO{{\mathfrak{O}}} \newcommand\gro{{\mathfrak{o}}}
\newcommand\grP{{\mathfrak{P}}} \newcommand\grp{{\mathfrak{p}}}
\newcommand\grQ{{\mathfrak{Q}}} \newcommand\grq{{\mathfrak{q}}}
\newcommand\grR{{\mathfrak{R}}} \newcommand\grr{{\mathfrak{r}}}
\newcommand\grS{{\mathfrak{S}}} \newcommand\grs{{\mathfrak{s}}}
\newcommand\grT{{\mathfrak{T}}} \newcommand\grt{{\mathfrak{t}}}
\newcommand\grU{{\mathfrak{U}}} \newcommand\gru{{\mathfrak{u}}}
\newcommand\grV{{\mathfrak{V}}} \newcommand\grv{{\mathfrak{v}}}
\newcommand\grW{{\mathfrak{W}}} \newcommand\grw{{\mathfrak{w}}}
\newcommand\grX{{\mathfrak{X}}} \newcommand\grx{{\mathfrak{x}}}
\newcommand\grY{{\mathfrak{Y}}} \newcommand\gry{{\mathfrak{y}}}
\newcommand\grZ{{\mathfrak{Z}}} \newcommand\grz{{\mathfrak{z}}}



\newcommand\nek{,\ldots,}
\newcommand\sdp{\times \hskip -0.3em {\raise 0.3ex
\hbox{$\scriptscriptstyle |$}}} % semidirect product

%words in roman font

\newcommand\ad{\operatorname{ad}}
\newcommand\area{\operatorname{area}}
\newcommand\Aug{\operatorname{Aug}}
\newcommand\Aut{\operatorname{Aut}}
\newcommand\Ad{\operatorname{Ad}}
\newcommand\Char{\operatorname{Char}}
\newcommand\Cl{\operatorname{Cl}}
\newcommand\cf{{\rm \,cf\,}}
\newcommand\Cone{\operatorname{Cone}}
\newcommand\cont{\operatorname{cont}}
\newcommand\codim{\operatorname{codim}}
\newcommand\conv{\operatorname{conv}}
\newcommand\Conv{\operatorname{Conv}}
\newcommand\const{\operatorname{const}}
\newcommand\Const{\operatorname{Const}}
\newcommand\Det{\operatorname{Det}}
\newcommand\diag{\operatorname{diag}}
\newcommand\diam{\operatorname{diam}}
\newcommand\Diam{\operatorname{Diam}}
\newcommand\dist{\operatorname{dist}}
\newcommand\Dom{\operatorname{Dom}}
\newcommand\dom{\operatorname{dom}}
\newcommand\End{\operatorname{End\,}}
\newcommand\Ext{\operatorname{Ext}}
\newcommand\esssup{\operatorname{ess\ sup}}
\newcommand\Ran{{\rm Ran}}
\newcommand\RANK{\operatorname{rank}}
\newcommand\Geo{\operatorname{Geo}}
\newcommand\GL{\operatorname{GL}}
\newcommand\Gr{\operatorname{Gr}}
\newcommand\gl{{\bf gl}}
\newcommand\grad{\operatorname {\rm grad}}
\newcommand\Hom{\operatorname {Hom}}
\newcommand\RHom{\operatorname {RHom}}
\newcommand\id{\operatorname{id}}
\newcommand\Id{\operatorname{Id}}
\newcommand\im{\operatorname {im}}
\newcommand\IM{\operatorname{Im}}
\newcommand\Ind{\operatorname{Ind}}
\newcommand\ind{\operatorname{ind}}
\newcommand\Inf{\operatorname{Inf}}
\newcommand\Int{\operatorname{Int}}
\newcommand\Min{\operatorname{Min}}
\newcommand\MOD{\operatorname{mod}}
\newcommand{\Mor}{\operatorname{Mor}}
\newcommand{\Ob}{\operatorname{Ob\,}}
\newcommand\ord{\operatorname{ord}}
\newcommand\Ka{\operatorname{Ka}}
\newcommand\Ker{\operatorname{Ker}}
\newcommand\PGL{{\rm PGL}}
\newcommand\PGSp{{\rm PGSp}}
\newcommand\Plt{\operatorname{Plt}}
\newcommand\proj{\operatorname{proj}}
\newcommand\Proj{\operatorname{Proj}}
\newcommand\res{\rm res}
%\newcommand\rank{\rm rank}
\newcommand\rk{\operatorname{rk}}
\newcommand\Range{\operatorname{Range}}
\newcommand\RE{\operatorname{Re}}
\newcommand\Res{\operatorname{Res}}
\newcommand\rot{\operatorname{rot}}
\newcommand\Max{\operatorname{Max}}
\newcommand\Maximum{\operatorname{Maximum}}
\newcommand\Minimum{\operatorname{Minimum}}
\newcommand\Minimize{\operatorname{Minimize}}
\newcommand\Perv{\operatorname{Perv}}
\newcommand\Prob{\operatorname{Prob}}
\newcommand\pr{\operatorname{pr}}
\newcommand\sech{\rm sech}
\newcommand\sgn{\operatorname{sgn}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand\SL{{\rm SL}}
\newcommand\Sbm{\operatorname{Sbm}}
\newcommand\SO{{\rm SO}}
\newcommand\SPAN{\operatorname{span}}
\newcommand\spec{{\rm spec}}
\newcommand\Spec{\operatorname{Spec}}
\newcommand\supess{\operatorname{sup\ ess}}
\newcommand\supp{\operatorname{supp}}
\newcommand\Supp{\operatorname{Supp}}
\newcommand\Sup{\operatorname{Sup}}
\newcommand\Sym{\operatorname{Sym}}
\newcommand\tr{\operatorname{tr}}
\newcommand\Tr{\operatorname{Tr}}
\newcommand\Tor{\operatorname{Tor}}
\newcommand\Var{\operatorname{Var}}
\newcommand\Vol{\operatorname{Vol}}
%\newcommand\vol{\operatorname{vol}}

%overlined math alphabet
\newcommand\oa{{\overline{a}}}
\newcommand\oA{{\overline{A}}}
\newcommand\ob{{\overline{b}}}
\newcommand\oB{{\overline{B}}}
\newcommand\oc{{\overline{c}}}
\newcommand\oC{{\overline{C}}}
\newcommand\oD{{\overline{D}}}
\newcommand\od{{\overline{d}}}
\newcommand\oE{{\overline{E}}}
\renewcommand\oe{{\overline{e}}}
\newcommand\of{{\overline{f}}}
\newcommand\oF{{\overline{F}}}
\newcommand\og{{\overline{g}}}
\newcommand\oG{{\overline{G}}}
\newcommand\oh{{\overline{h}}}
\newcommand\oH{{\overline{H}}}
\newcommand\oI{{\overline{I}}}
\newcommand\oj{{\overline{j}}}
\newcommand\oJ{{\overline{J}}}
\newcommand\ok{{\overline{k}}}
\newcommand\oK{{\overline{K}}}
\newcommand\oL{{\overline{L}}}
\newcommand\om{{\overline{m}}}
\newcommand\oM{{\overline{M}}}
\newcommand\oN{{\overline{N}}}
\newcommand\oO{{\overline{O}}}
\newcommand\oo{{\overline{o}}}
\newcommand\op{{\overline{p}}}
\newcommand\oP{{\overline{P}}}
\newcommand\oq{{\overline{q}}}
\newcommand\oQ{{\overline{Q}}}
\newcommand\OR{{\overline{r}}}
\newcommand\oS{{\overline{S}}}
\newcommand\os{{\overline{s}}}
\newcommand\ot{{\overline{t}}}
\newcommand\oT{{\overline{T}}}
\newcommand\ou{{\overline{u}}}
\newcommand\oU{{\overline{U}}}
\newcommand\ov{{\overline{v}}}
\newcommand\oV{{\overline{V}}}
\newcommand\ow{{\overline{w}}}
\newcommand\oW{{\overline{W}}}
\newcommand\ox{{\overline{x}}}
\newcommand\oX{{\overline{X}}}
\newcommand\oy{{\overline{y}}}
\newcommand\oY{{\overline{Y}}}
\newcommand\oz{{\overline{z}}}
\newcommand\oZ{{\overline{Z}}}

%overlined Greek alphabet
\newcommand\oalp{{\overline{\alpha}}}
\newcommand\obet{{\overline{\bet}}}
\newcommand\odel{{\overline{\del}}}
\newcommand\oDel{{\overline{\Del}}}
\newcommand\ocup{{\overline{\cup}}}
\newcommand\ovarphi{{\overline{\varphi}}}
\newcommand\ochi{{\overline{\chi}}}
\newcommand\oeps{{\overline{\eps}}}
\newcommand\oeta{{\overline{\eta}}}
\newcommand\ogam{{\overline{\gam}}}
\newcommand\okap{{\overline{\kap}}}
\newcommand\olam{{\overline{\lambda}}}
\newcommand\oLam{{\overline{\Lambda}}}
\newcommand\omu{{\overline{\mu}}}
\newcommand\onu{{\overline{\nu}}}
\newcommand\oOme{{\overline{\Ome}}}
\newcommand\ophi{\overline{\phi}}
\newcommand\oPhi{{\overline{\Phi}}}
\newcommand\opi{{\overline{\pi}}}
\newcommand\oPsi{{\overline{\Psi}}}
\newcommand\opsi{{\overline{\psi}}}
\newcommand\orho{{\overline{\rho}}}
\newcommand\osig{{\overline{\sig}}}
\newcommand\otau{{\overline{\tau}}}
\newcommand\otet{{\overline{\theta}}}
\newcommand\oxi{{\overline{\xi}}}
\newcommand\oome{\overline{\ome}}
\newcommand\opart{{\overline{\partial}}}


%underlined math alphabet
\newcommand\ua{{\underline{a}}}
\newcommand\ub{{\underline{b}}}
\newcommand\uc{{\underline{c}}}
\newcommand\uD{{\underline{D}}}
\newcommand\uk{{\underline{k}}}
\newcommand\ue{{\underline{e}}}
\newcommand\uj{{\underline{j}}}
\newcommand\ul{{\underline{l}}}
\newcommand\uL{{\underline{L}}}
\newcommand\uo{{\underline{o}}}
\newcommand\uO{{\underline{O}}}
\newcommand\uP{{\underline{P}}}
\newcommand\uQ{{\underline{Q}}}
\newcommand\um{{\underline{m}}}
\newcommand\uM{{\underline{M}}}
\newcommand\un{{\underline{n}}}
\newcommand\us{{\underline{s}}}
\newcommand\ut{{\underline{t}}}
\newcommand\uu{{\underline{u}}}
\newcommand\uv{{\underline{v}}}
\newcommand\uV{{\underline{V}}}
\newcommand\ux{{\underline{x}}}
\newcommand\uX{{\underline{X}}}
\newcommand\uy{{\underline{y}}}
\newcommand\uz{{\underline{z}}}

%underline Greek alphabet
\newcommand\ualp{{\underline{\alp}}}
\newcommand\ubet{{\underline{\bet}}}
\newcommand\uchi{{\underline{\chi}}}
\newcommand\udel{{\underline{\del}}}
\newcommand\uell{{\underline{\ell}}}
\newcommand\ueps{{\underline{\eps}}}
\newcommand\ueta{{\underline{\eta}}}
\newcommand\uGam{{\underline{\Gamma}}}
\newcommand\unu{{\underline{\nu}}}
\newcommand\uome{{\underline{\omega}}}
\newcommand\utet{{\underline{\tet}}}
\newcommand\ulam{{\underline{\lam}}}


%math alphabet with hat
\newcommand\hata{{\widehat{a}}}
\newcommand\hatA{{\widehat{A}}}
\newcommand\hatb{{\widehat{b}}}
\newcommand\hatc{{\widehat{c}}}
\newcommand\hatC{{\widehat{C}}}
\newcommand\hatB{{\widehat{B}}}
\newcommand\hatD{{\widehat{D}}}
\newcommand\hate{{\widehat{e}}}
\newcommand\hatE{{\widehat{E}}}
\newcommand\hatf{{\widehat{f}}}
\newcommand\hatF{{\widehat{F}}}
\newcommand\hatg{{\widehat{g}}}
\newcommand\hatG{{\widehat{G}}}
\newcommand\hath{{\widehat{h}}}
\newcommand\hatH{{\widehat{H}}}
\newcommand\hati{{\hat{i}}}
\newcommand\hatI{{\hat{I}}}
\newcommand\hatj{{\widehat{j}}}
\newcommand\hatJ{{\widehat{J}}}
\newcommand\hatk{{\widehat{k}}}
\newcommand\hatK{{\widehat{K}}}
\newcommand\hatL{{\widehat{L}}}
\newcommand\hatm{{\widehat{m}}}
\newcommand\hatM{{\widehat{M}}}
\newcommand\hatn{{\widehat{n}}}
\newcommand\hatN{{\widehat{N}}}
\newcommand\hatp{{\widehat{p}}}
\newcommand\hatP{{\widehat{P}}}
\newcommand\hatr{{\widehat{r}}}
\newcommand\hatR{{\widehat{R}}}
\newcommand\hatq{{\widehat{q}}}
\newcommand\hatQ{{\widehat{Q}}}
\newcommand\hatT{{\widehat{T}}}
\newcommand\hatu{{\widehat{u}}}
\newcommand\hatU{{\widehat{U}}}
\newcommand\hatV{{\widehat{V}}}
\newcommand\hatv{{\widehat{v}}}
\newcommand\hatw{{\widehat{w}}}
\newcommand\hatW{{\widehat{W}}}
\newcommand\hatx{{\widehat{x}}}
\newcommand\hatX{{\widehat{X}}}
\newcommand\haty{{\widehat{y}}}
\newcommand\hatY{{\widehat{Y}}}
\newcommand\hatZ{{\widehat{Z}}}
\newcommand\hatz{{\widehat{z}}}

%Greek alphabet with hat
\newcommand\hatalp{{\widehat{\alpha}}}
\newcommand\hatdel{{\widehat{\delta}}}
\newcommand\hatDel{{\widehat{\Delta}}}
\newcommand\hatbet{{\widehat{\beta}}}
\newcommand\hateps{{\hat{\eps}}}
\newcommand\hatgam{{\widehat{\gamma}}}
\newcommand\hatGam{{\widehat{\Gamma}}}
\newcommand\hatlam{{\widehat{\lambda}}}
\newcommand\hatmu{{\widehat{\mu}}}
\newcommand\hatnu{{\widehat{\nu}}}
\newcommand\hatOme{{\widehat{\Ome}}}
\newcommand\hatphi{{\widehat{\phi}}}
\newcommand\hatPhi{{\widehat{\Phi}}}
\newcommand\hatpi{{\widehat{\pi}}}
\newcommand\hatpsi{{\widehat{\psi}}}
\newcommand\hatPsi{{\widehat{\Psi}}}
\newcommand\hatrho{{\widehat{\rho}}}
\newcommand\hatsig{{\widehat{\sig}}}
\newcommand\hatSig{{\widehat{\Sig}}}
\newcommand\hattau{{\widehat{\tau}}}
\newcommand\hattet{{\widehat{\theta}}}
\newcommand\hatvarphi{{\widehat{\varphi}}}
\newcommand\hatZZ{{\widehat{\ZZ}}}


%roman with widetilde

\newcommand\tilA{{\widetilde{A}}}
\newcommand\tila{{\widetilde{a}}}
\newcommand\tilB{{\widetilde{B}}}
\newcommand\tilb{{\widetilde{b}}}
\newcommand\tilc{{\widetilde{c}}}
\newcommand\tilC{{\widetilde{C}}}
\newcommand\tild{{\widetilde{d}}}
\newcommand\tilD{{\widetilde{D}}}
\newcommand\tilE{{\widetilde{E}}}
\newcommand\tilf{{\widetilde{f}}}
\newcommand\tilF{{\widetilde{F}}}
\newcommand\tilg{{\widetilde{g}}}
\newcommand\tilG{{\widetilde{G}}}
\newcommand\tilh{{\widetilde{h}}}
\newcommand\tilk{{\widetilde{k}}}
\newcommand\tilK{{\widetilde{K}}}
\newcommand\tilj{{\widetilde{j}}}
\newcommand\tilm{{\widetilde{m}}}
\newcommand\tilM{{\widetilde{M}}}
\newcommand\tilH{{\widetilde{H}}}
\newcommand\tilL{{\widetilde{L}}}
\newcommand\tilN{{\widetilde{N}}}
\newcommand\tiln{{\widetilde{n}}}
\newcommand\tilO{{\widetilde{O}}}
\newcommand\tilP{{\widetilde{P}}}
\newcommand\tilp{{\widetilde{p}}}
\newcommand\tilq{{\widetilde{q}}}
\newcommand\tilQ{{\widetilde{Q}}}
\newcommand\tilR{{\widetilde{R}}}
\newcommand\tilr{{\widetilde{r}}}
\newcommand\tilS{{\widetilde{S}}}
\newcommand\tils{{\widetilde{s}}}
\newcommand\tilT{{\widetilde{T}}}
\newcommand\tilt{{\widetilde{t}}}
\newcommand\tilu{{\widetilde{u}}}
\newcommand\tilU{{\widetilde{U}}}
\newcommand\tilv{{\widetilde{v}}}
\newcommand\tilV{{\widetilde{V}}}
\newcommand\tilw{{\widetilde{w}}}
\newcommand\tilW{{\widetilde{W}}}
\newcommand\tilX{{\widetilde{X}}}
\newcommand\tilx{{\widetilde{x}}}
\newcommand\tily{{\widetilde{y}}}
\newcommand\tilY{{\widetilde{Y}}}
\newcommand\tilZ{{\widetilde{Z}}}
\newcommand\tilz{{\widetilde{z}}}

%Greek alphabet with widetilde
\newcommand\tilalp{{\widetilde{\alpha}}}
\newcommand\tilbet{{\widetilde{\beta}}}
\newcommand\tildel{{\widetilde{\delta}}}
\newcommand\tilDel{{\widetilde{\Delta}}}
\newcommand\tilchi{{\widetilde{\chi}}}
\newcommand\tileta{{\widetilde{\eta}}}
\newcommand\tilgam{{\widetilde{\gamma}}}
\newcommand\tilGam{{\widetilde{\Gamma}}}
\newcommand\tilome{{\widetilde{\ome}}}
\newcommand\tillam{{\widetilde{\lam}}}
\newcommand\tilmu{{\widetilde{\mu}}}
\newcommand\tilphi{{\widetilde{\phi}}}
\newcommand\tilpi{{\widetilde{\pi}}}
\newcommand\tilpsi{{\widetilde{\psi}}}
\renewcommand\tilome{{\widetilde{\ome}}}
\newcommand\tilOme{{\widetilde{\Ome}}}
\newcommand\tilPhi{{\widetilde{\Phi}}}
\newcommand\tilQQ{{\widetilde{\QQ}}}
\newcommand\tilrho{{\widetilde{\rho}}}
\newcommand\tilsig{{\widetilde{\sig}}}
\newcommand\tiltau{{\widetilde{\tau}}}
\newcommand\tiltet{{\widetilde{\theta}}}
\newcommand\tilvarphi{{\widetilde{\varphi}}}
\newcommand\tilxi{{\widetilde{\xi}}}
\newcommand{\obfp}{{\overline \bfp}}


\newcommand\twolongrightarrow{\ \hbox{$\longrightarrow\hskip -17pt
\longrightarrow$}\ }

\newcommand\x{\times}
\newcommand\ten{\otimes}

\renewcommand{\>}{\rangle}
\newcommand{\<}{\langle}


\newcommand{\ra}{\rangle}
\newcommand{\la}{\langle}

\newcommand\qlb{{\overline \QQ}_l}
\newcommand\fr{\text{Fr}}
\newcommand\frw{\fr_w}
\newcommand\gln{{\bf GL}(n)}
\newcommand\wt{\widetilde}
\newcommand\chl{\calK_{\calL}}
\newcommand\ch{\text{ch}}
\newcommand\fq{\FF_q}

\newcommand{\bPhi}{\Phi\hskip-7.5pt \Phi}
\newcommand{\balp}{\alp\hskip-7.1pt \alp}
\newcommand{\bpi}{\pi\hskip-7.1 \pi}
\newcommand{\obalp}{{\overline \balp}}
\newcommand{\obpi}{{\overline \bpi}}
\renewcommand{\oPhi}{{\overline \Phi}}
\renewcommand{\gl}{{\bf GL}}
\newcommand{\hgln}{{\hat GL(n,F)}}
\renewcommand{\Id}{\text{Id}}
\newcommand{\Irr}{\operatorname{Irr}}
\newcommand\gd{{\bfG}^{\vee}}
\newcommand\cic{C^{\infty}_c}
\renewcommand\Spec{\operatorname{Spec}}

\newcommand\tbY{{\widetilde \bfY}}
\newcommand\tbf{{\widetilde \bff}}
\newcommand\tbp{{\widetilde \bfp}}
\newcommand\obY{{\overline \bfY}}
\newcommand\obX{{\overline \bfX}}
\newcommand\obf{{\overline \bff}}
\newcommand\obp{{\overline \bfp}}
\renewcommand\oome{{\overline \ome}}
\newcommand\tbphi{{\widetilde \bPhi}}
\newcommand\obPhi{{\overline \bPhi}}
\newcommand\tbG{{\widetilde \bfG}}
\newcommand\obT{{\overline \bfT}_\rho}
\newcommand\arlam{{\overset{\to}\lam}}
\newcommand\tilLam{{\widetilde \Lam}}
\newcommand\tbT{{\widetilde \bfT}}


\renewcommand\top{\text{top}}
\newcommand\obg{{\overline \bfG}_\rho}
\newcommand\dr{d_\rho g}
\newcommand\td{\bfT^{\vee}}
\newcommand\gaf{\bfG(\AA_K)}
\renewcommand\cic{C^{\infty}_c}
\newcommand\phpsi{\bPhi_\psi}
\newcommand\phro{\bPhi_{\rho,\psi}}
\newcommand\phrog{\bPhi_{\bfG,\rho,\psi}}
\newcommand\phrot{\bPhi_{\bfT,\rho,\psi}}
\newcommand\kg{\text{K}_G}
\newcommand\obfT{{\overline \bfT}}
\newcommand\Spr{{\bf Spr}}
\newcommand\tgrg{{\widetilde \grg}}
\newcommand\ocalA{{\overline \calA}}
\newcommand\avgb{{\bf Av_{G/B}}}
\newcommand\tilInd{{\widetilde \Ind}}
\newcommand\tilRes{{\widetilde \Res}}

\renewcommand\o{\omega}
\newcommand\qlbs{(\qlb[1](\frac{1}{2}))}
\newcommand\tilgrg{{\widetilde \grg}}
\begin{document}
\title{On Euler characteristic of equivariant sheaves}
\author{Alexander Braverman}
\address{Department of Mathematics, Harvard University, 1 Oxford st.
Cambridge MA, 02138}
\email{braval\@math.harvard.edu}
\begin{abstract}
Let $k$ be an algebraically closed field of characteristic $p>0$ and let
$\ell$ be another prime number. O.~Gabber and F.~Loeser proved that for any
algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf $\calF$
on $T$ the Euler characteristic $\chi(\calF)$ is non-negative.

We conjecture that the same result holds for any perverse sheaf $\calF$ on a
reductive group $G$ over $k$ which is equivariant with respect to the
adjoint action. We prove the conjecture when $\calF$ is obtained by Goresky-MacPherson extension
from the set of regular semi-simple elements in $G$. From this we deduce that the
conjecture holds  for $G$ of semi-simple rank 1.
\end{abstract}
\maketitle

\sec{}{The main conjecture}

\ssec{}{Notations}
In what follows $k$ denotes an algebraically closed field of characteristic
$p>0$. Let $\ell$ be a prime number different from $p$. For an algebraic
variety $X$ over $k$ we denote by $\calD(X)$ the derived category of
$\ell$-adic sheaves on $X$. Also we denote by $\Perv(X)\subset \calD(X)$ the
subcategory of perverse sheaves. For any $\calF\in\calD(X)$ we denote by
$\chi(\calF)$ the  Euler characteristic of $\calF$, i.e.
\eq{}
\chi(\calF)=\sum (-1)^i\dim H^i_c(X,\calF)=\sum (-1)^i\dim H^i(X,\calF)
\end{equation}
(cf. \cite{La} for the proof of the equality).

Let $T$ be an algebraic torus over $k$. The following theorem is proved in
\cite{GL} (cf. also \cite{FK} for a partial analogue in characteristic zero):
%-------------------------------------------------------------------------------
\th{torus}
Let $\calF\in\Perv(T)$. Then $\chi(\calF)\geq 0$.
\eth
%------------------------------------------------------------------------------------
Let $G$ be a connected reductive algebraic group over $k$. We shall denote by
$G_{rs}\subset G$ the open subspace of regular semi-simple elements.
We denote by $\Perv_G(G)$ the subcategory of $\Perv(G)$ consisting of
perverse sheaves which are equivariant with respect to the adjoint action.
We propose the following generalization of \reft{torus}.
%------------------------------------------------------------------------------------------------------
\conj{main}
Let $\calF\in\Perv_G(G)$. Then $\chi(\calF)\geq 0$.
\econj
\th{main}
Assume that $\calF\in\Perv_G(G)$ is equal to the Goresky-MacPherson
extension of its restriction to $G_{rs}$. Then $\chi(\calF)\geq 0$.
\eth
\cor{}
\refco{main} holds for $G$ of semi-simple rank 1.
\ecor
\prf
Let $\calN$ be the cone of unipotent elements in $G$ and let $Z$ be the
center of $G$. Clearly it is enough to prove \refco{main} only for irreducible sheaves.
However if $\calF\in\Perv_G(G)$ is irreducible then either $\calF$ is equal to the
Goresky-MacPherson extension of its restriction to $G_{rs}$ or it is supported on
$Z\calN$. The former case is covered by \reft{main}; hence it is enough to deal with the
latter.

First of all there exists a connected component $Z'$ of $Z$ such that
$\calF$ is supported on $Z'\calN$. Thus  one of the following
is true:

1) $\calF$ is supported on $Z'$.

2) $\calF$ is equal to $\calF'\boxtimes (\qlb)_\calN[2]$ where $\calF$ is
an irreducible perverse sheaf on $Z'$ and $(\qlb)_{\calN}$ is the constant
sheaf on $\calN$.

3) $\calF=\calF'\boxtimes \calE$ where $\calF'$ is an irreducible perverse
sheaf on $Z'$ and $\calE$ is an irreducible perverse sheaf on $\calN$ whose
restriction to the open orbit is isomorphic to (the only) non-trivial
equivariant irreducible local system on this orbit (this local system is of rank 1 and
its square is isomorphic to the constant sheaf).

In cases 1 and 2 our result follows immediately from \reft{torus}
(note that $Z'$ is a torus and that in case 2 we have
$\chi(\calF)=\chi(\calF')$).
In case 3 it is known that $H^*(\calN,\calE)=0$, hence
$H^*(G,\calF)=0$, hence $\chi(\calF)=0$.
\epr

\noindent
{\it Remark.} \reft{torus} has an "ideological"
explanation: namely in \cite{GL} O.~Gabber and F.~Loeser construct a
"Mellin transform" functor $\calM:\calD^b(T)\to \calD^b_{coh}(\text{Loc}_T)$ where
$\text{Loc}_T$ denotes the moduli space of tame rank one local systems on $T$ and
$\calD^b_{coh}(\text{Loc}_T)$ is the derived category of quasi-coherent sheaves
on $\text{Loc}_T$. Moreover, for every $\calF\in\Perv(T)$ the complex
$\calM(\calF)$ is actually a sheaf and $\chi(\calF)$ is equal to the generic
rank of $\calM(\calF)$ (hence $\chi(\calF)\geq 0$). We don't know whether a
similar explanation of \refco{main} is possible.

We expect that \refco{main} holds also when $k$ has characteristic 0 and perverse
sheaves are replaced by  holonomic D-modules. When $G$ is a torus and the
D-modules in question have regular singularities a beautiful geometric proof
of \reft{torus} was given in \cite{FK} where the authors explain how to
compute the corresponding Euler characteristics using certain non-compact
generalization of Kashiwara's index theorem. It would be interesting to
generalize the proof of \cite{FK} to the case of arbitrary reductive group. Some results in this
direction have recently been obtained by V.Kiritchenko.
%----------------------------------------------------------------------------------------------------
\sec{}{Proof of \reft{main}}

\ssec{}{The space $\tilG$}Let $\calB$ denote the flag variety of $G$, i.e. the variety of all Borel
subgroups of $G$. Let $\tilG$ denote the variety of pairs $(B\in\calB,g\in
B)$. We have the natural maps $\pi:\tilG\to G$, $\alp:\tilG\to T$ and
$\beta:\tilG\to\calB$. It is easy to see that $\alp$ is smooth and $\pi$ is proper.
Let $d=\dim G-\dim T=2\dim\calB$. Set
$\tilG_{rs}=\pi^{-1}(G_{rs})=\alp^{-1}(T_{rs})$.

\lem{induction}
\begin{enumerate}
\item
Let $\calG\in\Perv(T)$. Assume that $\calG$ is equal to the Goresky-MacPherson extension
of its restriction
on $T_{rs}$.
$\pi_!\alp^*\calG[d](\frac{d}{2})$ is a perverse sheaf which is equal to the Goresky-MacPherson extension
of its restriction to $G_{rs}$.
\item
Every $W$-equivariant structure on a sheaf $\calG\in\Perv(T)$ as above gives rise to
a $W$-action on $\pi_!\alp^*\calG[d](\frac{d}{2})$.
\item
Let $\calF\in\Perv_G(G)$. Assume that $\calF$ is equal to the
Goresky-MacPherson extension of its restriction to $G_{rs}$.
Then there exists a $W$-equivariant sheaf $\calG\in\Perv(T)$ which is equal to the
Goresky-MacPherson extension of its restriction to $T_{rs}$ such that
\eq{fginvariant}
\calF=(\pi_!\alp^*\calG[d](\frac{d}{2}))^W.
\end{equation}
\end{enumerate}
\elem
\prf The first statement of \refl{induction} is well-known (it follows from
the smallness property of $\alp$ -- cf. \cite{CG} and references therein)

It follows from 1 that in order to prove 2 it is enough to construct an action of $W$ on
the restriction of $\pi_!\alp^*\calG[d](\frac{d}{2})$ to $G_{rs}$. Note that $W$ acts
freely on $\tilG_{rs}$ in the natural way and
\eq{quotient}
G_{rs}=\tilG_{rs}/W.
\end{equation}
Moreover, the restriction of $\alp$ to $\tilG_{rs}$ is $W$-equivariant.
Hence $\alp^*\calG[d](\frac{d}{2})|_{\tilG_{rs}}$ is $W$-equivariant and thus
\refe{quotient} implies that $\pi_!\alp^*\calG[d](\frac{d}{2})|_{G_{rs}}$ has
a natural action of $W$.

Let us prove 3. Choose an embedding $i:T\to G$. Let $i_{rs}:T_{rs}\to G$ be
its restriction to $T_{rs}$. Let $\calG_{rs}=i_{rs}^*\calF[-d](-\frac{d}{2})$. It follows from
the $G$-equivariance of $\calF$ that $\calG_{rs}$ is perverse. We let
$\calG$ be its Goresky-MacPherson extension to $T$. We leave the
verification of \refe{fginvariant} to the reader.
\epr
%---------------------------------------------------------------------
\prop{torus-g}
Let $\calF$ be as in
\reft{main} and let $\calG$ be as in \refl{induction}(3). Then
$\chi(\calF)=\chi(\calG)$.
\eprop
Clearly \refp{torus-g} together with \reft{torus} imply \reft{main}.
%----------------------------------------------------------------------------------
\ssec{proof}{Proof of Proposition \refp{torus-g}}
Set $\calH=\alp^*\calG[d](\frac{d}{2})\in\Perv(\tilG)$. Clearly,
$H^*_c(G,\calF)=H^*_c(\tilG,\calH)^W$.

Consider $\alp_!(\calH)$. By the projection formula it is isomorphic
to $\calG\ten \alp_!(\qlb)_{\tilG}[d](\frac{d}{2})$. It is easy to see that the
complex $\alp_!(\qlb)_{\tilG}[d](\frac{d}{2})$ is
constant with fibers isomorphic to $H^*(\calB,\qlb)$. Indeed, let $\del:\calB\x T\to T$
denote the projection to the first multiple.
Then $\alp_!(\qlb)_{\tilG}=\del_!(\beta\x\alp)_!(\qlb)_{\tilG}$. However,
the map $(\beta\x \alp):\tilG\to\calB\x T$ is a locally trivial fibration
(in Zariski topology) with fiber isomorphic to $\AA^d$. Thus
$(\beta\x \alp)_!(\qlb)_{\tilG}\simeq (\qlb)_{\calB\x T}[-d](-\frac{d}{2})$ and
hence
\eq{projection}
\alp_!(\qlb)_{\tilG}\simeq (\qlb)_T\ten H^*(\calB,\qlb)[-d](-\frac{d}{2}).
\end{equation}
This clearly gives rise to the isomorphism
\eq{twosides}
H^*_c(\tilG,\calH)\simeq H^*_c(T,\calG)\ten H^*(\calB,\qlb).
\end{equation}
We can also characterize the isomorphism \refe{projection}
in the following way. To determine it uniquely it is enough to construct it on $T_{rs}$.
Choose as before an embedding
$i:T\to G$. There is a canonical Borel subgroup $B$ containing $i(T)$ (recall that the abstract
Cartan group $T$ comes with a root system with a preferred set of positive roots). Then for every
$t\in T_{rs}$ we have canonical isomorphism
\eq{}
\alp^{-1}(t)\simeq G/T
\end{equation}
(which depends however on the above choice). Clearly
$H^*_c(G/T,\qlb)[d](\frac{d}{2})=H^*(\calB,\qlb)$. Hence we have constructed an isomorphism
between $\alp_!(\qlb)_{\tilG}[d](\frac{d}{2})|_{T_{rs}}$ and the constant complex with
fiber $H^*(\calB,\qlb)$. It is easy to see that this isomorphism does not
depend on the choice of the embedding $T\to G$ and coincides with
\refe{projection}.

Let $W$ act on the left hand side by means
of the identification $H^*_c(\tilG,\calH)=H^*_c(G,\pi_!\calH)$ (note that
by \refl{induction} the group $W$ acts on $\pi_!\calH$).
Let $W$ also act on the right hand side by means of the tensor product of
the $W$-action on $H^*_c(T,\calG)$ coming from the $W$-equivariant structure
on $\calG$ and the natural $W$-action on $H^*(\calB,\qlb)$.
%-----------------------------------------
\lem{w}The isomorphism \refe{twosides}
is also an isomorphism of $W$-modules with respect to the above actions.
\elem
\prf
Let $Z=G\underset{T/W}\x T$ be the image of $\tilG$ in $G\x T$ under the
map $\gam=\pi\x \alp$. This is a closed subvariety of $G\x T$.
It is invariant with respect to the $W$ action on the second multiple.
Let $Z_{rs}$ denote the set of "regular semi-simple" elements of $Z$
(i.e. the set of all elements of $Z$ whose projection to $Z$ is regular
semi-simple).

Let $\calK=\gam_!(\calH)$. Then $\calK$ is equal to the Goresky-MacPherson extension
of its restriction to $Z_{rs}$. Since $\gam$ induces an isomorphism
$\tilG_{rs}\simeq Z_{rs}$ and since the restriction of $\calH$ to $\tilG_{rs}$ is
$W$-equivariant it follows that $\calK$ also has a natural $W$-equivariant structure
as a perverse sheaf on $G\x T$ (where the $W$-action is as before on the
second multiple). Thus $W$ acts naturally on $H^*_c(G\x T,\calK)$.
We have the natural isomorphism $H^*_c(\tilG,\calH)\simeq
H^*_c(G\x T,\calK)$ and by the definition the action of $W$ on
$H^*_c(\tilG,\calH)$ introduced before \refl{w} corresponds to the
action of $W$ on $H^*_c(G\x T,\calK)$ introduced above.

Consider, on the other hand, the complex $p_!(\calK)$ where $p:G\x T\to T$
denotes the natural projection. Clearly, we have $p_!(\calK)=\alp_!(\calH)\simeq
\calG\ten H^*(\calB,\qlb)$. On the other hand, since $p$
commutes with $W$ it follows that $p_!(\calK)$ admits a natural
$W$-equivariant structure. To prove \refl{w} it is enough to show that
under the identification $p_!(\calK)\simeq \calG\ten
H^*(\calB,\qlb)[d](\frac{d}{2})$ this structure is equal to the tensor
product of the original $W$-equivariant structure on $\calG$ with the
natural $W$-action on $H^*(\calB,\qlb)$. Since every perverse cohomology
of $p_!(\calK)$ is equal to the Goresky-MacPherson extension of its
restriction to $T_{rs}$ it follows that it is enough to check this equality
only on $T_{rs}$. This, however, follows immediately from the description of
the $W$-equivariant structure on
$\alp_!(\qlb)_{\tilG}[d](\frac{d}{2})|_{\tilG_{rs}}$ given in \refss{proof}
and the following observation:

Choose as before an embedding $T\to G$.
Let $N(T)$ denote the normalizer of $T$ in $G$. Then $W=N(T)/T$ acts
on $G/T$ and hence on $H^*_c(G/T,\qlb)$. By identifying
$H^*_c(G/T,\qlb)[d](\frac{d}{2})$ with $H^*(\calB,\qlb)$ we get an
action of $W$ on the latter. This is the standard $W$-action on
$H^*(\calB,\qlb)$.
\epr

It follows from \refl{w} that $\chi(\calF)$ is equal to the Euler characteristic of
$(H^*_c(\calG)\ten H^*(\calB,\qlb))^W$.
However, it is known that $H^*(\calB,\qlb)$ lives only in even degrees and
when we forget the grading it is isomorphic to the regular representation $\qlb[W]$ of
$W$. Hence
\begin{align*}
\chi(\calF)=\chi((H^*_c(T,\calG)\ten& H^*(\calB,\qlb)[d])^W)=\\
&\chi((H^*_c(T,\calG)\ten \qlb[W])^W)=\chi(H^*_c(T,\calG))=\chi(\calG)
\end{align*}
which finishes the proof.
%--------------------------------------------------------------------------------
\ssec{}{Acknowledgements}
The author is grateful to F.~Loeser for an illuminating discussion on the subject
held in the summer of 2001, to V.~Kiritichenko for making her results available to the
author before publication and to M.~Kapranov for explaining the contents of
\cite{FK}.
\begin{thebibliography}{JSPS}

\bibitem{CG}
N.~Chriss and V.~Ginzburg,
{\em Representation theory and complex geometry}, Birkh\"auser Boston, Inc., Boston, MA, 1997.


\bibitem{FK}
J.~Franecki and M.~Kapranov
{\em The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties},
Duke Math. Journal {\bf 104} (2000), 171-180.


\bibitem{GL}
O.~Gabber and F.~Loeser,
{\em Faisceaux pervers $l$-adiques sur un tore},  Duke Math. J. {\bf 83}
(1996), no. 3, 501--606


\bibitem{La}
G.~Laumon,
{\em Comparaison de caract\'eristiques d'Euler-Poincar\'e en cohomologie $l$-adique. (French)},
C. R. Acad. Sci. Paris S\'er. I Math. {\bf 292} (1981), no. 3, 209--212.

\end{thebibliography}

\end{document}

math0202165.extracted_bib
\begin{thebibliography}{JSPS}

\bibitem{CG}
N.~Chriss and V.~Ginzburg,
{\em Representation theory and complex geometry}, Birkh\"auser Boston, Inc., Boston, MA, 1997.


\bibitem{FK}
J.~Franecki and M.~Kapranov
{\em The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties},
Duke Math. Journal {\bf 104} (2000), 171-180.


\bibitem{GL}
O.~Gabber and F.~Loeser,
{\em Faisceaux pervers $l$-adiques sur un tore},  Duke Math. J. {\bf 83}
(1996), no. 3, 501--606


\bibitem{La}
G.~Laumon,
{\em Comparaison de caract\'eristiques d'Euler-Poincar\'e en cohomologie $l$-adique. (French)},
C. R. Acad. Sci. Paris S\'er. I Math. {\bf 292} (1981), no. 3, 209--212.

\end{thebibliography}