\documentclass{beamer}
% $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

% 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[all]{xy}
\usepackage{xypic}
\usepackage{times}
\usepackage{ulem}
\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*{question}{Question}
%\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}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Dist}{{\mathcal{D}}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Sc}{{\mathcal S}}
\newcommand{\Fre}{{Fr\'{e}chet \,}}


\newcommand{\Matrix}[4]{ \left( \begin{array}{cc}  #1 & #2 \\  #3 & #4 \\ \end{array} \right) }
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\Cat}[1]{ {\mathfrak #1} }
\newcommand{\Lie}[1]{ {\mathfrak #1} }
\newcommand{\filt}[1]{ {\mathcal #1} }
\newcommand{\scheme}[1]{\underline{ \mathbf{#1}}}
\newcommand{\alg}[1]{\mathbf{#1}}
\newcommand{\mc}[1]{\mathcal #1}
\newcommand{\Hom}{ {\mbox{Hom}} }
\newcommand{\Ext}{ {\mbox{Ext}} }
\newcommand{\Ind}{ {\mbox{Ind}} }
% \newcommand{\proof}{ {\sc Proof:} }
% \newcommand{\qed}{\begin{flushright}Q.E.D \end{flushright}}
\newcommand{\nats}{\mathbb N}
\newcommand{\ints}{\mathbb Z}
\newcommand{\rats}{\mathbb Q}
\newcommand{\reals}{\mathbb R}
\newcommand{\complex}{\mathbb C}
\newcommand{\quats}{\mathbb H}
\newcommand{\octs}{\mathbb O}
\newcommand{\hecke}{\mathcal H}
\newcommand{\fourier}{\mathcal F}
\newcommand{\FF}{\mathbb F}
\newcommand{\Gp}{\mathbb G}
\newcommand{\adeles}{\mathbb A}
%\newcommand{\eps}{\varepsilon}
\newcommand{\To}{\longrightarrow}
\newcommand{\BX}{\mathbf{B}(X)}
\newcommand{\A}{\mathcal{A}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\DOP}{\mathbb{D}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\Schwarz}{\mathcal{S}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\ZU}{\mathcal{Z}}
\newcommand{\e}{\textbf{e}}
\newcommand{\Orb}{\mathcal{O}}
\newcommand{\Period}{\mathcal{P}}
\newcommand{\isom}{\cong}
\newcommand{\gK}{( \Lie{g}_\complex, K)}
\newcommand{\J}{\mathcal J}
\newcommand{\g}{{\mathfrak{g}}}




\newcommand{\speh}{\operatorname{Speh}}
\newcommand{\aaa}{\mathfrak{a}}
\newcommand{\res}{\operatorname{res}}
\newcommand{\mix}{\mathcal{M}}
\newcommand{\tr}{\operatorname{Tr}}
\newcommand{\www}{\mathfrak{W}}
\newcommand{\ob}{\operatorname{ ob}}
\newcommand{\bes}{\mathfrak{B}}
\newcommand{\eisen}{\mathcal{E}}
\newcommand{\rest}{\lvert}
\newcommand{\sg}{\mathfrak{S}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\disc}{\Delta}
\newcommand{\modulus}{\delta}
\newcommand{\I}{\mathbb{I}}
\newcommand{\eps}{\varepsilon}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\Ht}{H}
\newcommand{\diff}{\mathfrak{d}}
\newcommand{\triv}{{\bf 1}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\inv}{\operatorname{inv}}
%\newcommand{\Ind}{\operatorname{Ind}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\match}[1]{\mathop{\leftrightarrow}\limits^{#1}}
\newcommand{\der}[1]{\mathfrak{D}^{(#1)}}
%\newcommand{\abs}[1]{\left|{#1}\right|}
%\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\sprod}[2]{\left\langle#1,#2\right\rangle}


\newcommand{\ff}{\mathbb{F}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bc}{\operatorname{bc}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\uu}{\mathcal{U}}
%\newcommand{\hecke}{\mathcal{H}}
\newcommand{\pp}{\mathcal{P}}
\newcommand{\oo}{\mathcal{O}}
\newcommand{\ch}{\operatorname{ch}}
\newcommand{\sss}{\mathcal{S}}
\newcommand{\ww}{\mathcal{W}}
\newcommand{\bb}{\mathcal{B}}
\newcommand{\ii}{\mathcal{I}}
\newcommand{\ccc}{\mathcal{C}}
\newcommand{\nm}{\operatorname{Nm}}
\newcommand{\bs}{\backslash}
%\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\kk}{\mathfrak{K}}
\newcommand{\zz}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\cc}{\mathbb{C}}
\newcommand{\qq}{\mathbb{Q}}
\newcommand{\rr}{\mathbb{R}}
\newcommand{\aut}{\mathcal{A}}
\newcommand{\Supp}{\mathrm{Supp}}
%\newcommand{\g}{{\mathfrak{g}}}
\newcommand{\proofend}{\hfill$\Box$\smallskip}
\definecolor{DarkGreen}{rgb}{0,0.5,0}
\newcommand{\Rami}[1]{{\color{red}{#1}}}

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


\institute[Massachusetts Institute of Technology] % (optional, but mostly needed)
{Massachusetts Institute of Technology}
% - 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
%{Joint with D. Gourevitch} \\$ $ \\ $ $ \\
\center{\url{http://math.mit.edu/~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{Examples}
%\thispagestyle{empty}
%\Rami{
\begin{example}[Fourier Series]
\begin{itemize}
\pause
\item $L^2(S^1)=\bigoplus span \{e^{inx}\}$
\pause
\item $g e^{inx}=\chi(g) e^{inx}$
\end{itemize}
\end{example}
\pause

\begin{example}[Spherical Harmonics]
\begin{itemize}
%\item $S^2=SO_3/SO_2$
\pause
\item $L^2(S^2)=\bigoplus_m H^m$
\pause
\item $H^m=span_i \{y_i^m\}$
are irreducible representations of $O_3$
\end{itemize}
\end{example}
\pause
%}
\begin{example}
Let $X$ be a finite set. Let the symmetric group $Perm(X)$ act on $X$. Consider the space $F(X)$ of complex valued functions on $X$ as a representation of $Perm(X)$. Then it decomposes to direct sum of \textbf{distinct} irreducible representations.
\end{example}
\pause
\end{frame}

\begin{frame}
    \frametitle{Gelfand Pairs}
\thispagestyle{empty}
\begin{definition}
A pair of compact topological groups $(G \supset H)$ is called a
\textbf{Gelfand pair} if the following equivalent conditions hold:
\begin{itemize}
\pause
\item $L^2(G/H)$ decomposes to direct sum of \textbf{distinct} irreducible representations of $G$.
\pause
\item for any irreducible representation $\rho$ of $G$
$dim \rho^H \leq 1.$
\pause
\item for any irreducible representation $\rho$ of $G$
$dim Hom_{H}(\rho,\cc) \leq 1.$
\pause
\item the algebra of bi-$H$-invariant functions on $G$, $C(H \!\setminus \!G/H)$, is commutative w.r.t. convolution.
\end{itemize}
\end{definition}
\end{frame}
\begin{frame}
    \frametitle{Strong Gelfand Pairs}
\begin{definition}
A pair of compact topological groups $(G \supset H)$ is called a \textbf{strong Gelfand pair}
if one of the following equivalent conditions is satisfied:
\pause
\begin{itemize}
\item the pair $(G \times H \supset \Delta H)$ is a Gelfand pair
\pause
\item for any irreducible representations $\rho$
of $G$ and $\tau$ of $H$
$$dim Hom_{H}(\rho|_H,\tau) \leq 1.$$
\pause
\item the algebra of  Ad$(H)$-invariant functions on $G$, $C(G//H)$, is commutative w.r.t. convolution.
\end{itemize}
\end{definition}

\end{frame}

\begin{frame}
\frametitle{{Some classical applications}}
\begin{itemize}
\item Harmonic analysis.\\
\pause
$(SO(3,\R),SO(2,\R)$ is a Gelfand pair -\\ spherical harmonics.
\pause
\item Gelfand-Zeitlin basis:\\
\pause
$(S_{n},S_{n-1})$ is a strong Gelfand pair -\\ basis for irreducible representations of $S_n$\\
\pause
The same for $O(n,\R)$ and $U(n,\R)$.
\pause
\item Classification of representations:\\
\pause
$(GL(n,\R),O(n,\R))$ is a Gelfand pair - \\ the irreducible representations of $GL(n,\R)$ which have an $O(n,\R)$-invariant vector are the same as characters of the algebra $C(O(n,\R) \! \setminus \! GL(n,\R)  \!  / \! O(n,\R)$.\\
\pause
The same for the pair $(GL(n,\C),U(n))$.
\end{itemize}
\end{frame}

\begin{frame}
    \frametitle{{Gelfand trick}}
\thispagestyle{empty}

\begin{figure}[htp]
\includegraphics[height=20mm]{gelfand1.jpg}
% en.wikipedia.org/wiki/Israel_Gelfand
\end{figure}

\begin{proposition}[Gelfand]
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 $\sigma(f)=f$ for all bi $H$-invariant
functions $f \in C(H \!\setminus \!G/H)$. Then $(G,H)$ is a Gelfand pair.
\end{proposition}
\pause
\begin{proposition}[Gelfand]
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 $\sigma(f)=f$ for all Ad$(H)$-invariant
functions $f \in C(G//H)$. Then $(G,H)$ is a strong Gelfand pair.
\end{proposition}
\end{frame}

\begin{frame}
\frametitle{{Sum up}}
%{\small
$$\xymatrix{
%
 \framebox{\parbox{70pt}{Rep. theory:\\ $\forall \rho \dim \rho^H \leq 1$}}\ar@{<=>}[r] &
%
\framebox{\parbox{65pt}{Algebra:\\ $C(H \!\setminus \!G/H)$ is commutative}}\ar@{<=}[r] &
%
\framebox{\parbox{85pt}{"Analysis":\\ $\exists$ anti-involution $\sigma$ s.t. $f = \sigma(f)$\\ $\forall f \in C(H \!\setminus \!G/H)$}}\ar@{<=>}[d] \\
%
& & \framebox{\parbox{85pt}{Geometry:\\ $\exists$ anti-involution $\sigma$\\ that preserves\\ $H$ double cosets}}}$$%}
\end{frame}

\begin{frame}
\frametitle{{Classical examples}}
\begin{tabular}{|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &Anti-involution \\
\hline
$(G \times G, \Delta G)$&  $(g,h) \mapsto (h^{-1},g^{-1})$\\
\hline
%$(S_{n+1},S_{n})$ strong& $g \mapsto g^{-1}$\\
%\hline
%$(O(n+1),O(n))$ strong& \\
%\cline{1-1}
$(O(n+k),O(n) \times O(k))$ &\\
\cline{1-1}
%$(U(n+1),U(n))$ strong& $g \mapsto g^{t}$\\
%\cline{1-1}
%\cline{1-1}
$(U(n+k),U(n) \times U(k))$& $g \mapsto g^{-1}$\\
%\cline{1-1}
\hline
$(GL(n,\R),O(n))$ & $g \mapsto g^t$ \\
%\cline{1-1} $(GL(n,\C),U(n))$ &\\
\hline
$(G,G^{\theta}),$ where &\\ $G$ - Lie group, $\theta$- involution,&$g \mapsto \theta(g^{-1})$ \\ $G^{\theta}$ is compact & \\
\hline $(G,K)$, where&\\ $G$ - is a reductive group,& Cartan anti-involution\\ $K$ - maximal
compact subgroup & \\ \hline
\end{tabular}\end{frame}

\begin{frame}
    \frametitle{{Non compact setting}}
\begin{setting}
In the non compact case we will consider complex \emph{smooth admissible
representations} of \emph{algebraic} \emph{reductive} groups over
\emph{local fields}.
\end{setting}
\pause

%\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{definition}
A local field is a locally compact non-discrete topological field.\\
There are 2 types of local fields of characteristic zero:
\begin{itemize}
\item Archimedean: $\R$ and $\C$
\item non-Archimedean: $\Q_p$ and their finite extensions
\end{itemize}
\end{definition}
\pause
\begin{definition}
A linear algebraic group is a subgroup of $GL_n$ defined by polynomial equations.
\end{definition}
\end{frame}

\begin{frame}
\frametitle{{Reductive groups}}
\begin{examples}%[of reductive groups]
$GL_n$, $O_n$, $U_n$, $Sp_{2n}$,..., semisimple groups,
\end{examples}
\pause
\begin{fact}
Any algebraic representation of a reductive group decomposes to a
direct sum of irreducible representations.
\end{fact}
\pause
\begin{fact}
Reductive groups are unimodular.
\end{fact}
\end{frame}

%\begin{frame}

%\end{frame}

\begin{frame}
\frametitle{{Smooth representations}}
\begin{definition}
Over Archimedean $F$, by smooth representation $V$ we mean a complex \Fre representation $V$ such that for any $v\in V$ the map $G \to V$ defined by $v$ is smooth.
\end{definition}
\pause
\begin{definition}
Over non-Archimedean $F$, by smooth representation $V$ we mean a complex linear representation $V$ such that for any $v\in V$
there exists an open compact subgroup $K <G$ such that $Kv= v$.
\end{definition}
\end{frame}

\begin{frame}
    \frametitle{{Distributions}}

\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 will consider the space
$(C_c^{\infty}(M))^*$ 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 $\Sc^*(X):=\Sc(X)^*$ be the
space of distributions on $X$.
\end{definition}
\end{frame}

\begin{frame}
    \frametitle{{Gelfand Pairs}}
%    \framesubtitle{{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 $G$
%
$$dim Hom_{H}(\rho,\cc) \cdot dim Hom_{H}(\widetilde{\rho},\cc) \leq 1$$
usually, this implies that
$$dim Hom_{H}(\rho,\cc) \leq 1.$$
\end{definition}
\end{frame}

\begin{frame}
\frametitle{{Gelfand-Kazhdan distributional criterion}}
\begin{figure}[htp]
\includegraphics[height=20mm]{GelfandKazhdan.JPG}
\end{figure}

\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 $H$

$$dim Hom_{H}(\rho,\tau) \cdot dim Hom_{H}(\widetilde{\rho},\widetilde{\tau}) \leq 1$$
usually, this implies that $dim Hom_{H}(\rho,\tau) \leq 1.$
\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}
\thispagestyle{empty}
%\frametitle{{Sum up}}

{\small
$$\xymatrix{
%
 \framebox{\parbox{90pt}{Rep. theory:\\ $\forall \rho \dim \rho^H \leq 1$}}\ar@{<=>}[r] &
%
\framebox{\parbox{79pt}{Algebra:\\ $C(H \!\setminus \!G/H)$\\ is commutative}}\ar@{<=}[r] &
%
\framebox{\parbox{79pt}{"Analysis":\\ $\exists$ $\sigma$ s.t. $f = \sigma(f)$ $\forall f \in C(H \!\setminus \!G/H)$}}\ar@{<=>}[d] \\
%
Compact \,\, case& & \framebox{\parbox{79pt}{Geometry:\\ $\exists$ $\sigma$ that preserves $H$ double cosets}}}$$%

%\begin{tabular}{|c|} \hline
%
%\end{tabular}

$\underline{\quad
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\quad\quad\quad\quad}$

\pause
$$\xymatrix{
%
 \framebox{\parbox{90pt}{Rep. theory:\\ $\dim Hom_H(\rho,\C) \leq 1$ \\ $\forall \rho$}}\ar@/^2pc/@{<=}[rr] &
%
\framebox{\parbox{40pt}{\sout{Algebra}}} &
%
\framebox{\parbox{79pt}{Analysis:\\ $\exists$ $\sigma$ s.t. $\xi = \sigma(\xi)$ $\forall \xi \in \Sc^*( G)^{H \times H}$}}\ar@{<=}[d]^{p-adic} \\
%
Non-compact \,\, case& \framebox{\parbox{79pt}{Geometry:\\ $\exists$  $\sigma$ that preserves\\ closed\\ $H$ double cosets}}\ar@{<=}[ur]\ar@{<=}[r] & \framebox{\parbox{79pt}{Geometry:\\ $\exists$  $\sigma$ that preserves\\ $H$ double cosets}}  }$$}%\ar@{<=}[ur]\ar@{<=}[r]}
 %\pause
%
%$$\xymatrix{\framebox{\parbox{85pt}{Representation theory\\ $\forall \rho \dim \rho^H \leq 1.$}}\ar@{<=>}[r] & \framebox{\parbox{85pt}{\centeline{Algebra}}} & \framebox{\parbox{85pt}{Analysis:\\ $\exists$ anti-involution $\sigma$ s.t. $\xi = \sigma(\xi) \,\, \forall \xi \in \Sc^*( G)^{H \times H$}}}\ar@{\overset{p-adic}{<=}}[r] & \framebox{\parbox{85pt}{Geometry\\ $\exists$ anti-involution $\sigma$ s.t. $HgH = H\sigma(g)H \,\, \forall g \in G$}}\\ & & & \framebox{\parbox{85pt}{Geometry\\ $\exists$ anti-involution $\sigma$ s.t. $HgH=\sigma(HgH) \,\, \forall$ closed double cosets}}\ar@{<=}[u]\ar@{=>}[ul]}$$
\end{frame}

%\begin{frame}
%\thispagestyle{empty}
%%\frametitle{{Sum up}}
%
%{\small
%$$\xymatrix{
%%
% \framebox{\parbox{90pt}{Rep. theory:\\ $\forall \rho \dim \rho^H \leq 1$}}\ar@{<=>}[r] &
%%
%\framebox{\parbox{79pt}{Algebra:\\ $C(H \!\setminus \!G/H)$\\ is commutative}}\ar@{<=}[r] &
%%
%\framebox{\parbox{79pt}{"Analysis":\\ $\exists$ $\sigma$ s.t. $f = \sigma(f)$ $\forall f \in C(H \!\setminus \!G/H)$}}\ar@{<=>}[d] \\
%%
%& \framebox{\parbox{79pt}{Geometry:\\ $\exists$  $\sigma$ that preserves\\ closed\\ $H$ double cosets}}\ar@{<=>}[ur]\ar@{<=>}[r] & \framebox{\parbox{79pt}{Geometry:\\ $\exists$ $\sigma$ that preserves $H$ double cosets}}}$$%
%
%$$\xymatrix{
%%
% \framebox{\parbox{90pt}{Rep. theory:\\ $\dim Hom_H(\rho,\C) \leq 1$ \\ $\forall \rho$}}\ar@/^2pc/@{<=}[rr] &
%%
%\framebox{\parbox{40pt}{\sout{Algebra}}} &
%%
%\framebox{\parbox{79pt}{Analysis:\\ $\exists$ $\sigma$ s.t. $\xi = \sigma(\xi)$ $\forall \xi \in \Sc^*( G)^{H \times H}$}}\ar@{<=}[d]^{p-adic} \\
%%
%& \framebox{\parbox{79pt}{Geometry:\\ $\exists$  $\sigma$ that preserves\\ closed\\ $H$ double cosets}}\ar@{<=}[ur]\ar@{<=}[r] & \framebox{\parbox{79pt}{Geometry:\\ $\exists$  $\sigma$ that preserves\\ $H$ double cosets}}  }$$}%\ar@{<=}[ur]\ar@{<=}[r]}
% %\pause
%%
%%$$\xymatrix{\framebox{\parbox{85pt}{Representation theory\\ $\forall \rho \dim \rho^H \leq 1.$}}\ar@{<=>}[r] & \framebox{\parbox{85pt}{\centeline{Algebra}}} & \framebox{\parbox{85pt}{Analysis:\\ $\exists$ anti-involution $\sigma$ s.t. $\xi = \sigma(\xi) \,\, \forall \xi \in \Sc^*( G)^{H \times H$}}}\ar@{\overset{p-adic}{<=}}[r] & \framebox{\parbox{85pt}{Geometry\\ $\exists$ anti-involution $\sigma$ s.t. $HgH = H\sigma(g)H \,\, \forall g \in G$}}\\ & & & \framebox{\parbox{85pt}{Geometry\\ $\exists$ anti-involution $\sigma$ s.t. $HgH=\sigma(HgH) \,\, \forall$ closed double cosets}}\ar@{<=}[u]\ar@{=>}[ul]}$$
%\end{frame}

\begin{frame}
  \frametitle{Results on Gelfand pairs}
%\thispagestyle{empty} Local fields of characteristic zero:
%\begin{small}
%\center{
%\noindent
%\center{

%\hspace{-5mm}
%\Rami{
\small{
\begin{tabular}{|c|c|c|}
\hline
Pair &p-adic case &real case  \\
\hline
$(G, (N,\psi))$&Gelfand-Kazhdan & Shalika, Kostant\\
\hline
$(GL_n(E), GL_n(F))$& Flicker & \\
%\pause
\cline{1-2} $(GL_{n+k},GL_{n} \times GL_k)$&Jacquet-Rallis & Aizenbud-\\
\cline{1-2} $(O_{n+k},O_n \times O_k)$ over $\C$& $\underline{\quad \quad \quad \quad \quad}$ &Gourevitch\\ %\centerline
%\pause
\cline{1-1}
$(GL_n,O_n)$ over $\C$& &\\
\cline{1-1}   \cline{2-3}
$(GL_{2n},Sp_{2n})$& Heumos-Rallis&Aizenbud-Sayag\\
\hline
$(GL_{2n}, (\begin{pmatrix}
  g & u \\
  0 & g
\end{pmatrix} ,\psi))$ &Jacquet-Rallis & Aizenbud-Gourevitch\\
& & -Jacquet \\
\hline
$(GL_{n}, (\begin{pmatrix}
  SP & u \\
  0 & N
\end{pmatrix} ,\psi))$ & Offen-Sayag & Aizenbud-Offen-Sayag\\
\hline
\end{tabular}}
%}
\begin{itemize}
\item real: $\R$ and $\C$
\item p-adic: $\Q_p$ and its finite extensions.
\end{itemize}
%}
\end{frame}

\begin{frame}
  \frametitle{Results on Gelfand pairs}
\thispagestyle{empty}
\small{
\begin{tabular}{|c|c|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &p-adic & $\mathrm{char} F>0$ &real  \\
\hline
$(G, (N,\psi))$&Gelfand- &Gelfand-& Shalika, Kostant\\
&Kazhdan &Kazhdan& \\
\hline
$(GL_n(E), GL_n(F))$& Flicker &Flicker & \\
%\pause
\cline{1-3} $(GL_{n+k},GL_{n} \times GL_k)$&Jacquet- & A.- Avni-& A.-\\
 &Rallis & Gourevitch& Gourevitch\\
\cline{1-3} $(O_{n+k},O_n \times O_k)$ over $\C$& $\underline{\quad   \quad \quad}$ &$\underline{  \quad \quad \quad}$ &\\ %\centerline
%\pause
\cline{1-1}
$(GL_n,O_n)$ over $\C$& & &\\
\hline
%\cline{1-1}   \cline{2-3}
$(GL_{2n},Sp_{2n})$& Heumos-&Heumos-&A.-\\
& Rallis&Rallis&Sayag\\
\hline $(GL_{2n},( \begin{pmatrix}
  g & u \\
  0 & g
\end{pmatrix}   , \psi))$& Jacquet-& &A.-Gourevitch\\
& Rallis & &-Jacquet\\
\hline
$(GL_{n}, (\begin{pmatrix}
  SP & u \\
  0 & N
\end{pmatrix} ,\psi))$ & Offen-Sayag & Offen-Sayag & A.-Offen-\\
& & & Sayag\\
\hline
\end{tabular}

\begin{itemize}
\item real: $\R$ and $\C$
\item p-adic: $\Q_p$ and its finite extensions.
\item $\mathrm{char} F>0$: $\F_q((t))$
\end{itemize}
}

\end{frame}

\begin{frame}
  \frametitle{Results on strong Gelfand pairs}

\begin{tabular}{|c|c|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &p-adic &$\mathrm{char} F>0$& real  \\
\hline
 & A.- & A.-Avni-&A.-Gourevitch,\\
$(GL_{n+1},GL_{n})$& Gourevitch- &Gourevitch, & Sun-Zhu\\
& Rallis- &Henniart  &\\
\cline{3-4} \cline{1-1}
$(O(V \oplus F),O(V))$ &Schiffmann & & \\
\cline{1-1} %\cline{3-3}
$(U(V \oplus F),U(V))$ && &Sun-Zhu\\
\hline
\end{tabular}

%\begin{tabular}{|c|c|c|} \hline
%%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%Pair &p-adic case &real case \\
%\hline
%$(GL_{n+1},GL_{n})$ & Aizenbud- &Aizenbud-Gourevitch\\
%& Gourevitch- & Sun-Zhu\\
%\cline{3-3} \cline{1-1}
%$(O(V \oplus F),O(V))$ &Rallis-& \\
%\cline{1-1}
%$(U(V \oplus F),U(V))$ &Schiffmann & Sun-Zhu\\
%\hline
%\end{tabular}
\itemize{
\item real: $\R$ and $\C$
\item p-adic: $\Q_p$ and its finite extensions.
\item $\mathrm{char} F>0$: $\F_q((t))$}
\pause
\begin{remark}
The results from the last two slides are used to prove splitting of periods of automorphic forms.
\end{remark}
\end{frame}
%\pause
%\cline{1-1} \cline{2-3}
%
%
\begin{frame}
\frametitle{Generalized Harish-Chandra descent}

\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 $a \in
X$ s.t. the orbit $Ga$ is closed we have
$$\Dist(N_{Ga,a}^X)^{G_a,\chi}=0.$$ Then $\Dist(X)^{G,\chi}=0.$
\end{theorem}

\begin{figure}[htp]
\includegraphics[height=35mm]{LunaSlice3a.jpg}
\end{figure}

% Uncomment !! ??
\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 $H = G^{\theta}$.
\pause
\item We call $(G,H,\theta)$ \textbf{connected} if $G/H$ is Zariski connected.
\pause \item Define an antiinvolution
$\sigma :G \to G$ by $\sigma(g):=\theta(g^{-1})$.
\end{itemize}

\end{frame}

\begin{frame}
%  \frametitle{Symmetric Gelfand pairs}
  \thispagestyle{empty}
\begin{question} %\label{ComplexGood}
What symmetric pairs are Gelfand pairs?
\end{question}
\pause
For symmetric pairs of rank one this question was studied extensively by van-Dijk, Bosman, Rader and Rallis.
\pause%\begin{remark} %\label{ComplexGood}

%\end{remark}
\begin{definition}
A symmetric pair $(G,H,\theta)$ is called \textbf{good} if
$\sigma$ preserves all closed $H \times H$ double cosets.
\end{definition}
\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
\begin{corollary}
Any connected symmetric pair over $\C$ is a Gelfand pair.
\end{corollary}

\end{frame}

\begin{frame}
\frametitle{Recipe}
To check that a symmetric pair is a Gelfand pair
\begin{enumerate}
\item Prove that it is good
\pause
\item Prove that any $H$-invariant distribution on $g^{\sigma}$ is $\sigma$-invariant provided that this holds outside the cone of nilpotent elements. \label{regularity}
\pause
\item Compute all the "descendants" of the pair and prove
(\ref{regularity}) for them.
\end{enumerate}
\pause
We call the property (\ref{regularity}) regularity. We conjecture that all symmetric pairs are regular. This will imply that any good symmetric pair is a Gelfand pair.
\end{frame}


\begin{frame}
  \frametitle{Regular symmetric pairs}

\begin{tabular}{|c|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &p-adic case by&real case by \\
\hline
$(G\times G, \Delta G)$& Aizenbud-Gourevitch &   \\
\cline{1-2}
$(GL_n(E), GL_n(F))$& Flicker &  \\
%\pause
\cline{1-2} $(GL_{n+k},GL_{n} \times GL_k)$&Jacquet-Rallis & Aizenbud-\\
\cline{1-2} $(O_{n+k},O_n \times O_k)$ & Aizenbud-Gourevitch &Gourevitch\\ %\centerline
%\pause
\cline{1-1}
$(GL_n,O_n)$ & %Aizenbud-Gourevitch
&\\
\cline{1-1}   \cline{2-3}
$(GL_{2n},Sp_{2n})$& Heumos - Rallis &Aizenbud-Sayag\\
%\pause
\hline
$(sp_{2m}, sl_m \oplus \g_a)$& &\\
\cline{1-1}
$(e_6, sp_8)$ & &\\
\cline{1-1}
$(e_6 , sl_6 \oplus sl_2)$ & &Sayag\\
\cline{1-1}
$(e_7 , sl_8)$ & Aizenbud &  (based on\\
\cline{1-1}
$(e_8 , so_{16})$ & & work of Sekiguchi)\\
\cline{1-1}
$(f_4 , sp_6 \oplus sl_2)$ & &\\
\cline{1-1} $(g_2 , sl_2 \oplus sl_2)$& &\\
\hline
\end{tabular}
\end{frame}


\end{document}
% 026584415
%\begin{frame}
%\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{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}