% $Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.4 2004/10/07 20:53:08 tantau Exp $
%\texhash
\documentclass{beamer}

% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.



% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


\mode<presentation>
{
  \usetheme{Warsaw}
  % or ...
  % or whatever (possibly just delete it)
  \setbeamercovered{invisible}
}

%\usetheme{Boadilla}


\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever
\usepackage{xypic}
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
%\newtheorem{theorem*}{Theorem}
%\newtheorem*{lemma}{Lemma}
\newtheorem*{proposition}{Proposition}
\newtheorem*{remark}{Remark}
%\newtheorem*{example}{Example}
%\newtheorem*{theorem}{Theorem}
%\newtheorem*{definition}{Definition}
\newtheorem*{notation}{Notation}
%\newtheorem*{result}{Result}
%\newtheorem*{property}{}
%\newtheorem*{corollary}{Corollary}
%\newtheorem*{construction}{Construction}
%\newtheorem*{case}{Case}
\newtheorem*{conjecture}{Conjecture}
\newtheorem*{setting}{Setting}
\newtheorem*{flowchart}{Flowchart}

\newcommand{\R}{{\mathbb R}}
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\newcommand{\Dist}{{\mathcal{D}}}
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\definecolor{DarkGreen}{rgb}{0,0.5,0}

%\newcommand{\A}{\mathbb{A}}
\title % (optional, use only with long paper titles)
{Invariant Distributions and Gelfand Pairs}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{A. Aizenbud and D. Gourevitch}
% - Use the \inst{?} command only if the authors have different
%   affiliation.


\institute[Weizmann Institute of Science] % (optional, but mostly needed)
{}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[] % (optional)
%{August 14, 2008}

%\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
%
%\pgfdeclareimage[height=0.5cm]{fiatslug}{fiatslug}
%\logo{\pgfuseimage{fiatslug}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%\AtBeginSubsection[]
%{
%  \begin{frame}<beamer>
%    \frametitle{Outline}
%    \tableofcontents[currentsection,currentsubsection]
%  \end{frame}
%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

\begin{frame}
  \titlepage
\href{http://www.wisdom.weizmann.ac.il/~aizenr/}{$$
http://www.wisdom.weizmann.ac.il/\sim aizenr/ $$}
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
%  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Part I: Overview}
%
%\subsection{Gelfand Pairs}
\begin{frame}
    \frametitle{\emph{Gelfand Pairs and distributional criterion}}
%    \framesubtitle{\emph{models of representations of compact groups}}

\begin{definition}
A pair of groups $(G \supset H)$ is called a \textbf{Gelfand pair}
if for any irreducible "admissible" representation $\rho$ of $G$
%
$$dim Hom_{H}(\rho,\cc) \leq 1.$$
\end{definition}
\pause

\begin{theorem}[Gelfand-Kazhdan,...]
Let $\sigma$ be an involutive anti-automorphism of $G$ (i.e.
$\sigma(g_1g_2) = \sigma(g_2)\sigma(g_1)$) and $\sigma^2=Id$ and assume $\sigma(H)=H$.\\
Suppose that $\sigma(\xi)=\xi$ for all bi $H$-invariant
distributions $\xi$ on $G$. Then $(G,H)$ is a Gelfand pair.
\end{theorem}

\end{frame}

\begin{frame}
  \frametitle{Strong Gelfand Pairs}
\thispagestyle{empty}
%  \framesubtitle{(of modular forms)}
  % - A title should summarize the slide in an understandable fashion
  %   for anyone how does not follow everything on the slide itself.
\begin{definition}
A pair of groups $(G , H)$ is called a \textbf{strong Gelfand
pair} if for any irreducible "admissible" representations $\rho$
of $G$ and $\tau$ of $H$
$$dim Hom_{H}(\rho|_H,\tau) \leq 1.$$
\end{definition}
\pause

\begin{proposition}
The pair $(G,H)$ is a strong Gelfand pair if and only if the pair
$(G \times H, \Delta H)$ is a Gelfand pair.
\end{proposition}
%Proof ?
\pause

\begin{corollary}
Let $\sigma$ be an involutive anti-automorphism of $G$ s.t.
$\sigma(H)=H$. Suppose $\sigma(\xi)=\xi$ for all distributions
$\xi$ on $G$ invariant with respect to conjugation by $H$. Then
$(G,H)$ is a strong Gelfand pair.
\end{corollary}

\end{frame}


\begin{frame}
  \frametitle{Results}
\thispagestyle{empty} Local fields of characteristic zero:
\itemize{
\item Archimedean: $\R$ and $\C$
\item Non-archimedean(p-adic): $\Q_p$ and its finite extensions.}\\
\pause
%\begin{small}
\begin{tabular}{|c|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &Field& By \\
\hline
$(GL_{n+1},GL_{n})$& &A.-G.-Sayag, van-Dijk\\
\cline{1-1} \cline{3-3}$(O(V \oplus F),O(V))$& &van-Dijk-Bossmann-Aparicio,\\
 &any&A.-G.-Sayag\\
 %\pause
\cline{1-1} \cline{3-3}$(GL_n(E), GL_n(F))$& &Flicker, A.-G.\\
%\pause
\cline{1-1} \cline{3-3}$(GL_{n+k},GL_{n} \times GL_k)$& &Jacquet-Rallis, A.-G.\\
\cline{1-1} \cline{2-3}$(O_{n+k},O_n \times O_k)$&$\C$&A.-G.\\
%\pause
\cline{1-1}
$(GL_n,O_n)$& &\\
\cline{1-1}   \cline{2-3}
$(GL_{2n},Sp_{2n})$&$F \neq \R$&Heumos - Rallis, Sayag\\
%\pause
\hline

$(GL_{n+1},GL_{n})$ strong& $\R$,$\C$ &Aizenbud-Gourevitch \\
\cline{2-3}
 & &Aizenbud-Gourevitch- \\
\cline{1-1}
$(O(V \oplus F),O(V))$ strong&p-adic&-Rallis-Schiffmann\\
\cline{1-1}
$(U(V \oplus F),U(V))$ strong& &\\
\hline
%\pause
%\cline{1-1} \cline{2-3}

\end{tabular}
%\end{small}
\end{frame}


\begin{frame}
\frametitle{Distributions on smooth manifolds and $\ell$-spaces}

\begin{notation}
Let $M$ be a smooth manifold. We denote by $C_c^{\infty}(M)$ the
space of smooth compactly supported functions on $M$. We denote by
$\Dist(M):=(C_c^{\infty}(M))^*$ the space of distributions on $M$.
Sometimes we will also consider the space $\Sc^*(M)$ of Schwartz
distributions on $M$.
\end{notation}
\pause

\begin{definition}
An $\ell$-space is a Hausdorff locally compact totally
disconnected topological space. For an  $\ell$-space $X$ we denote
by $\Sc(X)$ the space of compactly supported locally constant
functions on $X$. We let $\Sc^*(X):=\Dist(X):=\Sc(X)^*$ be the
space of distributions on $X$.
\end{definition}

\end{frame}


\begin{frame}
\frametitle{Distributions supported in a closed subset}
%
For a closed subset $Z \subset X$ we denote by $\Dist_X(Z)$ the
space of distributions on $X$ supported in $Z$.

\begin{proposition}
Let $Z\subset X$ be a closed subset and $U:=X - Z$. Then we have
the exact sequence
$$0 \to \Dist_X(Z) \to \Dist(X) \to \Dist(U).$$
\end{proposition}
\pause
For $\ell$-spaces, $\Dist_X(Z) \cong \Dist(Z)$.\\
\pause For smooth manifolds, $\Dist_X(Z)$ has an infinite
filtration whose factors are $\Dist(Z,Sym^k(CN_Z^X))$, where
$Sym^k(CN_Z^X)$ denote symmetric powers of the conormal bundle to
$Z$.

\end{frame}

%\subsection[periods of automorphic forms]{periods of automorphic forms}
\begin{frame}
    \frametitle{Geometric conditions}
\begin{setting} Let $G$ be an algebraic group over a local field
$F$. Let $H$ be a closed algebraic subgroup. Let $\sigma: G \to G$
be an antiinvolution.

We want to show that every $H\times H$ invariant distribution on
$G$ is $\sigma$-invariant.\end{setting} \pause

A necessary condition for that is : \\
\textcolor{blue}{"$\sigma$ preserves every closed double coset
(which carries $H\times H$ invariant distribution)"}.\\ $ $\\
\pause

Over p-adic fields, it is sufficient (but not necessary) to prove
that $\sigma$ preserves every double coset.
\end{frame}

\begin{frame}
 \frametitle{Reformulation of the problem}
\begin{notation}
Let $\sigma$ act on $H \times H$ by
$\sigma(h_1,h_2):=(\sigma(h_2^{-1}), \sigma(h_1^{-1}))$. Denote
$$\widetilde{H \times H}:= (H \times H) \rtimes \{1,\sigma\}.$$
It has a natural action on $G$. Define a character $\chi$ of
$\widetilde{H \times H}$ by $$\chi(H \times H)=\{1\}, \,
\chi(\widetilde{H \times H}- (H\times H))=\{-1\}.$$
\end{notation}
Now our problem becomes equivalent to $\Dist(G)^{\widetilde{H
\times H},\chi}=0$.
\end{frame}

%\end{document}


\begin{frame}
    \frametitle{First tool: Stratification}

\begin{setting} A group $G$ acts on a
space $X$, and $\chi$ is a character of $G$. We want to show
$\Dist(X)^{G,\chi}=0$.
\end{setting}
\pause

\begin{proposition}
Let $U\subset X$ be an open $G$-invariant subset and $Z:=X -U$.
Suppose that $\Dist(U)^{G,\chi}=0$ and $\Dist_X(Z)^{G,\chi}=0$.
Then $\Dist(X)^{G,\chi}=0.$
\end{proposition}
\pause
\begin{proof}
$0 \to \Dist_X(Z)^{G,\chi} \to \Dist(X)^{G,\chi} \to
\Dist(U)^{G,\chi}.$
\end{proof}
\pause
For $\ell$-spaces, $\Dist_X(Z)^{G,\chi} \cong \Dist(Z)^{G,\chi}$.\\
For smooth manifolds, to show $\Dist_X(Z)^{G,\chi}$ it is enough
to show that $\Dist(Z,Sym^k(CN_Z^X))^{G,\chi}=0$ for any $k$.
\end{frame}

\begin{frame}
    \frametitle{Frobenius reciprocity}

$$\xymatrix{
\parbox{10pt}
{$X_z$}\ar@{->}[d]\ar@{->}[r] & \parbox{10pt}{$X$}\ar@{->}[d]\\
 %
\parbox{5pt}{$z$}\ar@{->}[r] & \parbox{10pt} {$Z$}}$$

%$$\overset{X_z \subset X}{\underset{z \, \in \, Z}{\downarrow \quad \downarrow}}$$

\begin{theorem}[Bernstein, Baruch, ...]
$ $\\Let $\psi:X \to Z$ be a map. \\
Let a $G$ act on $X$ and $Z$ such that $\psi(gx)=g\psi(x)$.\\
Suppose that the action of $G$ on $Z$ is transitive.\\ Suppose
that both $G$ and $Stab_G(z)$ are unimodular. Then
$$\Dist(X)^{G,\chi} \cong
\Dist(X_z)^{Stab_G(z),\chi}.$$
%More generally,
%$$\mathcal{\Sc}^*(X)^{G,\chi} \cong
%\mathcal{\Sc}^*(X_z)^{Stab_G(z),\chi \cdot \Delta_G \cdot
%\Delta_H^{-1}},$$ where $\Delta_G$ and $\Delta_H$ denote modular
%characters.
\end{theorem}
\end{frame}

\begin{frame}
    \frametitle{Reductive groups}

%\begin{definition}
%A \textbf{reductive group} is a closed algebraic subgroup of
%$GL_n$ that does not have normal subgroups consisting of unipotent
%elements (i.e. operators with characteristic polynomial
%$(1-x)^n$). \end{definition}

\begin{example}
$GL_n$, semisimple groups, $O_n$, $U_n$, $Sp_{2n}$,...
\end{example}

\begin{fact}
Any algebraic representation of a reductive group decomposes to a
direct sum of irreducible representations.
\end{fact}

\begin{fact}
Reductive groups are unimodular.
\end{fact}
\end{frame}

\begin{frame}
\frametitle{Luna's slice theorem} \thispagestyle{empty}
%\begin{definition}
We say that $x\in X$ is $G$-semisimple if its orbit is closed.
%\end{definition}

\begin{theorem}[Luna's slice theorem] \label{LocLuna}
Let a reductive group $G$ act on a smooth affine algebraic variety
$X$. Let
$x \in X$ be $G$-semisimple. Then there exist\\
(i) an open $G$-invariant neighborhood $U$ of $Gx$ in $X$ with a
$G$-equivariant retract $p:U \to Gx$ and\\
(ii) a $G_x$-equivariant embedding $\psi:p^{-1}(x) \hookrightarrow
N_{Gx,x}^{X}$ with open image such that $\psi(x)=0$.
\end{theorem}
\begin{figure}[htp]
%\centering
\includegraphics[height=35mm]{LunaSlice2.jpg}
%\caption{Transverse momentum distributions}\label{fig:erptsqfit}
\end{figure}


\end{frame}

\begin{frame}
\frametitle{Generalized Harish-Chandra descent}

\begin{figure}[htp]
\includegraphics[height=35mm]{LunaSlice2.jpg}
\end{figure}

\begin{theorem}%[Aizenbud-Gourevitch] \label{Gen_HC}
Let a reductive group $G$ act on a smooth affine algebraic variety
$X$. Let $\chi$ be a character of $G$. Suppose that for any
$G$-semisimple $x \in X$ we have
$$\Dist(N_{Gx,x}^X)^{G_x,\chi}=0.$$ Then
$\Dist(X)^{G,\chi}=0.$
\end{theorem}

\end{frame}

\begin{frame}
\frametitle{A stronger version}

Let $V$ be an algebraic finite dimensional representation over $F$
of a reductive group $G$.
\begin{itemize}
\item $Q(V):=(V/V^G).$
Since $G$ is reductive, there is a canonical splitting $V= Q(V)
\oplus V^G$.
%
\item $\Gamma(V):= \{v \in Q(V)| \overline{Gv} \ni 0\}.$
%
\item $R(V):= Q(V) - \Gamma(V).$
\end{itemize}
\pause
\begin{theorem} \label{Strong_HC_Cor}
Let a reductive group $G$ act on a smooth affine variety $X$. Let
$\chi$ be a character of $G$. Suppose that for any $G$-semisimple
$x \in X$ such that \textcolor{blue}{
$$\Dist(R(N_{Gx,x}^X))^{G_x,\chi}=0$$} we have
\textcolor{red}{$$\Dist(Q(N_{Gx,x}^X))^{G_x,\chi}=0.$$} Then
\textcolor{DarkGreen}{$\Dist(X)^{G,\chi}=0$}.
\end{theorem}
\end{frame}

\begin{frame}
  \frametitle{Fourier transform}
%\thispagestyle{empty}
%
Let $V$ be a finite dimensional vector space over $F$ and $B$ be a
non-degenerate quadratic form on $V$. Let $\widehat{\xi}$ denote
the Fourier transform of $\xi$ defined using $B$.
\begin{proposition}
Let $G$ act on $V$ linearly and preserving $B$. Let $\xi \in
\Sc^*(V)^{G,\chi}$.  Then $\widehat{\xi} \in \Sc^*(V)^{G,\chi}.$
\end{proposition}

\end{frame}

\begin{frame}
  \frametitle{Fourier transform and homogeneity}
\thispagestyle{empty}
\begin{itemize}
\item We call a distribution $\xi \in \Sc^*(V)$ \textbf{abs-homogeneous of
degree} $d$ if for any $t \in F^{\times}$,
%
$$ h_t(\xi) = u(t) |t|^{d} \xi, $$
where $h_t$ denotes the homothety action on distributions and $u$
is some unitary character of $F^{\times}$.
\end{itemize}
\pause
\begin{theorem} [Jacquet, Rallis, Schiffmann,...] \label{NonArchHom}
Assume $F$ is \textbf{non-archimedean}. Let $\xi \in
\Sc^*_{V}(Z(B))$ be s. t. $\widehat{\xi} \in \Sc^*_{V}(Z(B))$.
Then $\xi$ is abs-homogeneous of degree $\frac{1}{2}dimV$.
\end{theorem}
\pause
\begin{theorem} [archimedean homogeneity] \label{ArchHom}
Let $F$ be any local field. Let $L \subset \Sc^*_{V}(Z(B))$ be a
non-zero linear subspace s. t. $\forall \xi \in L $ we have
$\widehat{\xi} \in L$ and $B \xi \in L$.

Then there exists a non-zero distribution $\xi \in L$ which is
abs-homogeneous of degree  $\frac{1}{2}dimV$ or of degree
$\frac{1}{2}dimV +1$.
\end{theorem}
\end{frame}

\begin{frame}
\frametitle{Localization principle}

$$\xymatrix{
\parbox{10pt}
{$X_y$}\ar@{->}[d]\ar@{->}[r] & \parbox{10pt}{$X$}\ar@{->}[d]\\
 %
\parbox{5pt}{$y$}\ar@{->}[r] & \parbox{10pt} {$Y$}}$$

\begin{theorem}[Aizenbud-Gourevitch-Sayag] \label{LocPrin2}
Let a reductive group $G$ act on a smooth affine variety $X$. Let
$Y$ be an algebraic variety and $\phi:X \to Y$ be an algebraic
$G$-invariant map. Let $\chi$ be a character of $G$. Suppose that
for any $y \in Y$ we have $\Dist_{X}(X_y)^{G,\chi}=0$. Then
$\Dist(X)^{G,\chi}=0$.
\end{theorem}
\pause For $\ell$-spaces, a stronger version of this principle was
proven by J. Bernstein 30 years ago.
\end{frame}


\begin{frame}
  \frametitle{Symmetric pairs}
\begin{itemize}
\item A \textbf{symmetric pair} is a triple $(G,H,\theta)$ where $H
\subset G$ are reductive groups, and $\theta$ is an involution of
$G$ such that $H = G^{\theta}$.
\item We call $(G,H,\theta)$ \textbf{connected} if $G/H$ is Zariski connected.
\item Define an antiinvolution
$\sigma :G \to G$ by $\sigma(g):=\theta(g^{-1})$.
\end{itemize}

\end{frame}

\begin{frame}
  \frametitle{Symmetric Gelfand pairs}
\begin{itemize}
\item A symmetric pair $(G,H,\theta)$ is called \textbf{good} if
$\sigma$ preserves all closed $H \times H$ double cosets.
\end{itemize}
\pause
\begin{proposition} \label{ComplexGood}
Any connected symmetric pair over $\C$ is good.
\end{proposition}
\pause
\begin{conjecture}
Any good symmetric pair is a Gelfand pair.
\end{conjecture}
\pause To check that a symmetric pair is Gelfand
\begin{enumerate}
\item Prove that it is good
\pause
\item Prove that there are no equivariant distributions supported on the
singular set in the Lie algebra $\g$. \label{regularity} \pause
\item Compute all the "descendants" of the pair and prove
(\ref{regularity}) for them.
\end{enumerate}
\end{frame}

\begin{frame}
  \frametitle{Results for $GL$ and $U$}

$$\xymatrix{
%
 & \framebox{\parbox{124pt}{$ (U(V \oplus W),U(V) \times U(W))$}}\ar@{->}[d]\ar@{->}[dl]
\\
%
 \framebox{\parbox{70pt}{$(GL(V),U(V))$}} & \framebox{\parbox{100pt}{$(GL_{n+k},GL_n \times GL_k )$}}\ar@{->}[d]\\
%
 &
\framebox{\parbox{88pt}{$\,(GL_n(E),GL_n(F))$}}} $$

\begin{corollary}
The pairs $(GL_n(E),GL_n(F))$ and $(GL_{n+k},GL_n \times GL_k )$
are Gelfand pairs.
\end{corollary}

\end{frame}

\begin{frame}
  \frametitle{Results for $O$}

$$ \xymatrix{ \framebox{\parbox{125pt}{$(O(V \oplus W),O(V) \times O(W))$}}\ar@{->}[d]\\
%
 \framebox{\parbox{70pt}{$(U(V_E),O(V))$}}\ar@{->}[d]\\
%
\framebox{\parbox{70pt}{$(GL(V), O(V))$}}}$$

\begin{corollary}
For $F=\C$, the pairs $(O(V \oplus W),O(V) \times O(W))$ and
$(GL(V), O(V))$ are Gelfand pairs.
\end{corollary}
%
\end{frame}

\begin{frame}
  \frametitle{Results for non-symmetric pairs}

Let $F$ be a \textbf{p-adic} field. Then the following pairs are
\textbf{strong} Gelfand pairs

$$\xymatrix{
%
 \framebox{\parbox{85pt}{$ (O(V \oplus F),O(V)) $}}\ar@{->}[d]\\
%
\framebox{\parbox{85pt}{$ (U(V \oplus F),U(V)) $}}\ar@{->}[d]\\
%
\framebox{\parbox{85pt}{\quad $(GL_{n+1},GL_n)$}}} $$

%\begin{figure}[htp]
%\centering
%\includegraphics[height=300mm, width = 380mm]{LunaSlice2.jpg}
%\caption{Transverse momentum distributions}\label{fig:erptsqfit}
%\end{figure}
%\begin{figure}
%  \centering
%  \input{LunaSlice2.jpg}
%%  \caption{}\label{}
%\end{figure}

\end{frame}

\begin{frame}
    \frametitle{Formulation}
%    \framesubtitle{\emph{models of representations of compact groups}}
Let $F$ be a p-adic field of characteristic zero.

\begin{theorem}[Aizenbud-Gourevitch-Rallis-Schiffmann]
Every $GL_n(F)$-invariant distribution on $GL_{n+1}(F)$ is
transposition invariant.
\end{theorem}
\pause

\begin{itemize}
\item $G:=G_n:=GL_n(F)$
\item $\widetilde{G}:=G \rtimes \{1,\sigma\}$
\item Define a character $\chi$ of
$\widetilde{G  }$ by $\chi(G)=\{1\}$, $\chi(\widetilde{G  }-
G)=\{-1\}$.
\end{itemize}
$ $\\ \pause Equivalent formulation:

\begin{theorem}
$\Sc^*(GL_{n+1}(F))^{\widetilde{G  },\chi}=0$.
\end{theorem}


\end{frame}

\begin{frame}
%\frametitle{Reformulation}
\thispagestyle{empty}
%
Equivalent formulation:
\begin{theorem}
$\Sc^*(gl_{n+1}(F))^{\widetilde{G  },\chi}=0$.
\end{theorem}
\pause
%
\begin{itemize}
\item $V:=F^n$
\item $X:=sl(V) \times V \times V^*$
\pause
\item $\widetilde{G  }$ acts on $X$ by\\
$g(A,v,\phi) = (gAg^{-1}, gv, (g^*)^{-1}\phi)$\\
$\sigma(A,v,\phi)=(A^t,\phi^t,v^t)$.
\end{itemize}
\pause
%
Equivalent formulation:
%
\begin{theorem}
$\Sc^*(X)^{\widetilde{G  },\chi}=0$.
\end{theorem}
\pause Reason:
$$g\begin{pmatrix}A_{n\times n} & v_{n \times 1} \\
\phi_{1 \times n} & \lambda \\
\end{pmatrix}g^{-1}= \begin{pmatrix}gAg^{-1} & gv \\
(g^*)^{-1}\phi & \lambda \\
\end{pmatrix} \text{ and }\begin{pmatrix}A & v \\
\phi & \lambda \\
\end{pmatrix}^t= \begin{pmatrix}A^{t} & \phi^{t} \\
v^t & \lambda \\
\end{pmatrix}$$

\end{frame}

\begin{frame}
\frametitle{Harish-Chandra descent}

\begin{itemize}
\item  Let $\mathcal{N} \subset
sl_n$ be the cone of nilpotent elements
\item $\Gamma := \{v \in V, \phi
\in V^* \, | \, \phi(v)=0 \}$
\end{itemize}
\pause By Harish-Chandra descent we can assume that any $\xi \in
\Sc^*(X)^{\widetilde{G  },\chi}$ is supported in  $\mathcal{N}
\times \Gamma$. \pause
 \begin{itemize}
\item  $\mathcal{N}_i:= \{a \in \mathcal{N} | \dim Ga \leq i\}  \subset
\mathcal{N}$
\end{itemize}
\pause We prove by descending induction on $i$ that
$\Sc^*(X)^{\widetilde{G  },\chi}=\Sc^*(\mathcal{N}_i \times
\Gamma)^{\widetilde{G  },\chi} $.
\end{frame}

\begin{frame}
   \frametitle{Reduction}

We assume $\Sc^*(X)^{\widetilde{G  },\chi}=\Sc^*(\mathcal{N}_i
\times \Gamma)^{\widetilde{G  },\chi} $.

We want to prove that $\Sc^*(X)^{\widetilde{G
},\chi}=\Sc^*(\mathcal{N}_{i-1} \times \Gamma)^{\widetilde{G
},\chi} $. \pause

 \begin{itemize}
\item  $\nu_{\lambda}(A,v,\phi):=(A+\lambda v \otimes \phi-\frac{\lambda}{n}\phi(v)Id,v,\phi)$
\end{itemize}
\pause

Let $\xi \in \Sc^*(X)^{\widetilde{G  },\chi}$. We know that for
any $\lambda$, $\xi \in \Sc^*(\nu_{\lambda}(\mathcal{N}_i \times
\Gamma))^{\widetilde{G  },\chi}$. \pause
 \begin{itemize}
\item $\widetilde{\mathcal{N}_i}:= \bigcap \limits _{\lambda \in F}  \nu_{\lambda}(\mathcal{N}_i \times \Gamma)$
\end{itemize} \pause

We know that $\xi \in \Sc^*(\widetilde{\mathcal{N}_i}
)^{\widetilde{G  },\chi} .$ \pause

  \begin{itemize}
\item Let $O \subset \mathcal{N}_i - \mathcal{N}_{i-1}$ be an open
orbit. \pause
\item $\widetilde{O}:= (O \times V \times V^*) \cap
\widetilde{\mathcal{N}_i}$ \pause
\item $ \eta := \xi|_{O \times V \times V^*}$.
\end{itemize}
\pause We have to show $\eta = 0$.

\end{frame}

\begin{frame}
   \frametitle{Key Lemma}
It is enough to prove

\begin{lemma}[Key]
Any $\eta \in  \Sc^*(O \times V \times V^*)^{\widetilde{G },\chi}$
such that both $\eta$ and $\widehat{\eta}$ are supported in
$\widetilde{O}$ is zero.
\end{lemma} \pause

Apply Frobenius reciprocity:
$$\xymatrix{
\parbox{10pt}{$\widetilde{O}_A$}\ar@{->}[d]\ar@{->}[r] & \parbox{10pt}{$\widetilde{O} $}\ar@{->}[d]\\
 %
\parbox{10pt}{$A$}\ar@{->}[r] & \parbox{10pt} {$O$}}$$

\begin{itemize}
\item $A \in O$
\item $\widetilde{O}_A:= \{(v,\phi) \in V \times V^* | (A,v,\phi) \in \widetilde{O}\}$
\item Let ${G}_A:= Stab_{G}(A)$ denote the centralizer of $A$.
\item $\widetilde{G}_A:= Stab_{\widetilde{G}}(A)$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Reformulation}

Equivalent formulation:

\begin{lemma}[Key']
Any $\zeta \in  \Sc^*(V \times V^*)^{\widetilde{G}_A,\chi}$  such
that both $\zeta$ and $\widehat{\zeta}$ are supported in
$\widetilde{O}_A$ is zero.
\end{lemma}
\pause
\begin{itemize}
\item $Q_A:= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}$
\end{itemize}

\begin{proposition}
$\widetilde{O}_A \subset Q_A$
\end{proposition}
\pause Now it is enough to prove
\begin{lemma}[Key'']
Any $\zeta \in  \Sc^*(V \times V^*)^{\widetilde{G}_A,\chi}$  such
that both $\zeta$ and $\widehat{\zeta}$ are supported in $Q_A$ is
zero.
\end{lemma}

\end{frame}

\begin{frame}
\frametitle{Reduction to Jordan block}

\begin{proposition}
$Q_{A\oplus B} \subset Q_A \times Q_B$
\end{proposition}
\pause
\begin{proof}

\( \begin{pmatrix} v\\
w \\
\end{pmatrix} \otimes \begin{pmatrix} \phi & \psi \\
\end{pmatrix} =  \begin{pmatrix}v \otimes \phi & * \\
* & w \otimes \psi \\
\end{pmatrix} \) %\\\\


\(
[\begin{pmatrix}A & 0 \\
0 & B \\
\end{pmatrix}, \begin{pmatrix}X & Y \\
Z & W \\
\end{pmatrix}] = \begin{pmatrix}[A,X] & * \\
* & [B,W] \\
\end{pmatrix}\)
\end{proof}
\pause Hence we can assume that $A=J_n$ is one Jordan block.
\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block}

\begin{align*}
 Q_A &= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}
\end{align*}

\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block}

\begin{align*}
 Q_A &= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}=\\
 &=\{(v,\phi) \in V \times V^*| v \otimes \phi \bot \g_A\}
\end{align*}

\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block}

\begin{align*}
 Q_A &= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}=\\
 &=\{(v,\phi) \in V \times V^*| v \otimes \phi \bot \g_A\} =\\
 &= \{(v,\phi) \in V \times V^*| \phi(Cv)=0  \,\   \forall C \in \g_A\}\\
\end{align*}

\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block}

\begin{align*}
 Q_A &= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}=\\
&=\{(v,\phi) \in V \times V^*| v \otimes \phi \bot \g_A\} =\\
&= \{(v,\phi) \in V \times V^*| \phi(Cv)=0  \,\   \forall C \in \g_A\}= \\
&= \{(v,\phi) \in V \times V^*| \phi(A^iv)=0 \, \forall i \geq 0\}
\end{align*}
\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block}

\begin{align*}
 Q_A &= \{(v,\phi) \in V \times V^*| v \otimes \phi \in [A,gl_n]\}=\\
&=\{(v,\phi) \in V \times V^*| v \otimes \phi \bot \g_A\} =\\
&= \{(v,\phi) \in V \times V^*| \phi(Cv)=0  \,\   \forall C \in \g_A\}= \\
&= \{(v,\phi) \in V \times V^*| \phi(A^iv)=0 \, \forall i \geq 0\}
\subset Z(B)
\end{align*}
where $B(v,\phi):=\phi(v)$.\\
\pause
$\Supp(\zeta), \, \Supp(\widehat{\zeta}) \subset Z(B) \Rightarrow \zeta$ is abs-homogeneous of degree $n$. \\

\end{frame}

\begin{frame}
   \frametitle{Proof for Jordan block }

\begin{itemize}
\item Denote $U:= (V - KerA^{n-1}) \times V^*$
\end{itemize}
\pause We have $$U \cap Q_A \subset V \times 0.$$
%
\pause Hence $\zeta|_U=0$. \pause So $\Supp(\zeta) \subset
KerA^{n-1} \times V^*$.\\ \pause
Similarly, $\Supp(\zeta) \subset KerA^{n-1} \times Ker(A^*)^{n-1}.$\\
\pause
Similarly, $\Supp(\widehat{\zeta}) \subset KerA^{n-1} \times Ker(A^*)^{n-1}.$\\
\pause Hence $\zeta$ is invariant with respect to shifts by
$ImA^{n-1} \times Im(A^*)^{n-1}$. \pause Therefore $$\zeta \in
\Sc^*(KerA^{n-1}/ImA^{n-1} \times
Ker(A^*)^{n-1}/Im(A^*)^{n-1})=\Sc^*(V_{n-2} \times V^*_{n-2}).$$
\pause By induction $\zeta=0$. \proofend
\end{frame}

\begin{frame}
   \frametitle{Summary}
\begin{flowchart}
 \xymatrix{
%
 \parbox{70pt}{$sl(V) \times V \times V^*$}\ar@{->}[r]^{\small \textcolor{magenta}{\quad H.Ch.}}_{\small \textcolor{magenta}{\quad
 descent}}
 & \parbox{30pt}{${\mathcal N}\times \Gamma$}\ar@{->}[r] & \parbox{30pt}
 {${\mathcal N}_i \times
 \Gamma$}\ar@{->}[r]^{\small \textcolor{magenta}{\nu_{\lambda}}}
 & \parbox{15pt}{$\widetilde{{\mathcal N}_i}$}\ar@{->}[d]\\
%
\parbox{15pt}{$Q_{J_n}$}\ar@{->}[d]_{\small \textcolor{magenta}{Fourier \, transform \, and}}^{\small \textcolor{magenta}{homogeneity \,theorem}}&
\parbox{15pt}{$Q_{A}$}\ar@{->}[l]&
\parbox{15pt}{$\widetilde{O}_{A}$}\ar@{->}[l]&
\parbox{10pt}{$\widetilde{O}$}\ar@{->}[l]_{\small \textcolor{magenta}{Frobenius}}^{\small \textcolor{magenta}{reciprocity}}\\
%
\parbox{70pt}{$\quad \quad Q_{J_n} +$\\ Homogeneity}\ar@{->}[r] & \parbox{20pt}{$Q_{J_{n-2}}$}\ar@{->}[r]&
\parbox{10pt}{...}\ar@{->}[r]& \parbox{20pt}{QED}  }
\end{flowchart}

\end{frame}

\begin{frame}
\frametitle{Orthogonal and unitary groups} %\thispagestyle{empty}
Let $D$ be either $F$ or a quadratic extension of $F$. Let $V$ be
a vector space over $D$. Let < , > be a non-degenerate hermitian
form on $V$. Let $W:=V\oplus D$. Extend < ,
> to $W$ in the obvious way. Consider the embedding of $U(V)$ into
$U(W)$. %\pause
%?? acts by conjugation
\end{frame}

\begin{frame}
   \frametitle{Orthogonal and unitary groups}
Let $D$ be either $F$ or a quadratic extension of $F$. Let $V$ be
a vector space over $D$. Let < , > be a non-degenerate hermitian
form on $V$. Let $W:=V\oplus D$. Extend < ,
> to $W$ in the obvious way. Consider the embedding of $U(V)$ into
$U(W)$. %\pause
%?? acts by conjugation
\begin{theorem}[Aizenbud-Gourevitch-Rallis-Schiffmann]
Every $U(V)$- invariant distribution on $U(W)$ is invariant with
respect to transposition.
\end{theorem}
\end{frame}

\begin{frame}
   \frametitle{Orthogonal and unitary groups}
%\thispagestyle{empty}
Let $D$ be either $F$ or a quadratic extension of $F$. Let $V$ be
a vector space over $D$. Let < , > be a non-degenerate hermitian
form on $V$. Let $W:=V\oplus D$. Extend < , > to $W$ in the
obvious way. Consider the embedding of $U(V)$ into
$U(W)$. %\pause
%?? acts by conjugation
\begin{theorem}
Every $U(V)$- invariant distribution on $U(W)$ is invariant with
respect to transposition.
\end{theorem}
\begin{itemize}
\item $G:=U(V)$
\item $\widetilde{G}:=G \rtimes \{1,\sigma\}$, $\chi$ as before.
\item $X:=su(V) \times V$
\item $\widetilde{G  }$ acts on $X$ by
$g(A,v) = (gAg^{-1}, gv)$,
$\sigma(A,v)=(-\overline{A},-\overline{v}).$
\end{itemize}
\end{frame}

\begin{frame}
   \frametitle{Orthogonal and unitary groups}
\thispagestyle{empty} Let $D$ be either $F$ or a quadratic
extension of $F$. Let $V$ be a vector space over $D$. Let < , > be
a non-degenerate hermitian form on $V$. Let $W:=V\oplus D$. Extend
< ,
> to $W$ in the obvious way. Consider the embedding of $U(V)$ into
$U(W)$. %\pause
%?? acts by conjugation
\begin{theorem}
Every $U(V)$- invariant distribution on $U(W)$ is invariant with
respect to transposition.
\end{theorem}
%
\begin{itemize}
\item $G:=U(V)$
\item $\widetilde{G}:=G \rtimes \{1,\sigma\}$, $\chi$ as before.
\item $X:=su(V) \times V$
\item $\widetilde{G  }$ acts on $X$ by
$g(A,v) = (gAg^{-1}, gv)$,
$\sigma(A,v)=(-\overline{A},-\overline{v}).$
\end{itemize}
Equivalent formulation:
%
%
\begin{theorem}
$\Sc^*(X)^{\widetilde{G  },\chi}=0$.
\end{theorem}
\end{frame}

\begin{frame} \frametitle{Sketch of the proof}

\begin{itemize}
\item  Let $\mathcal{N} \subset
su(V)$ be the cone of nilpotent elements
\item $\Gamma := \{v \in V, <v,v>=0 \}$
\end{itemize}
\pause

By Harish-Chandra descent we can assume that any $\xi \in
\Sc^*(X)^{\widetilde{G  },\chi}$ is supported in  $\mathcal{N}
\times \Gamma$. \pause

\begin{itemize}
\item $\nu_{\lambda}(A,v):=(A+\lambda v \otimes v^t  -  \frac{\lambda}{n}<v,v>Id
,v)$, $\overline{\lambda}=-\lambda$.\\
\pause
\item $\mu_{\lambda}(A,v):=(A+\lambda (v \otimes v^t A + Av \otimes v^t),v)$
\end{itemize}
\pause
\begin{lemma}[Key]
Any $\zeta \in  \Sc^*(V )^{\widetilde{G}_A,\chi}$  such that both
$\zeta$ and $\widehat{\zeta}$ are supported in $Q_A$ is zero.
\end{lemma}

\end{frame}


\end{document}