\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*{Task}{Task}

%\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{\N}{{\mathcal N}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Sc}{{\mathcal S}}
%\newcommand{\Dist}{{\mathcal{D}}}
\newcommand{\Dist}{\Sc^*}
\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{\Fou}{\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{\h}{{\mathfrak{h}}}




\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}}
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\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{\Rami}[1]{{{#1}}}

%\newcommand{\A}{\mathbb{A}}
\title % (optional, use only with long paper titles)
{Generalized Harish-Chandra descent}
%
%%\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{Aim}
\Rami{
    \begin{Task}
    Let a reductive group $G$ act on an affine variety $X$ and let $\chi$ be a character of $G$.
   \pause
   We want to prove:
    %We should prove that the space of equvariant distributions venish:
    $$\Sc^*(X)^{G,\chi}=0.$$
    \end{Task}
\pause
Applications: \pause Representation theory, Harmonic analysis,  \pause Gelfand pairs, \pause trace formula,   relative trace formula, ...
%\begin{itemize}
%\item
\pause
 \center{\bf Necessary condition:}
\center{Close orbits do not carry equivariant distributions} \pause \\
 $\Updownarrow$ \\
  $\chi|_{G_a} \neq 1$ for any semi simple $a \in X$ (i.e. $a \in X$ with closed orbit $G a$)
  }

%\end{itemize}
\end{frame}

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

\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{Frobenius descent}

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

%$$\overset{X_z \subset X}{\underset{z \, \in \, Z}{\downarrow \quad \downarrow}}$$
  \pause
\begin{theorem}[Bernstein, Baruch, ...]
%$ $\\
Let $\psi:X \to Z$ be a map. \\
Let  $G$ 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{Luna's slice theorem}
\Rami{
\pause
\begin{theorem}[Luna]
Let a reductive group $G$ act on a smooth affine algebraic variety
$X$. Let $a \in X$ be a semi-simple point. Then there exist an invariant (etale) neighborhood $U$ of $Ga$ with an equvariant projection $p:U \to Ga$ s.t. the fiber $p^{-1}(a)$ is $G$-isomorphic to an (etale) neighborhood of $0$ in the normal space $N_{Ga,a}^X$
\end{theorem}
}
\begin{figure}[htp]
\includegraphics[height=35mm]{LunaSlice3a.jpg}
\end{figure}


% Uncomment !! ??
\end{frame}

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

\begin{theorem}[A.-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}
\Rami{
\begin{frame}
\frametitle{Conclusions}
\pause
We reduce to the following
    \begin{Task}
    Let a reductive group $G$ act (linear) on an linear space $V$ and let $\chi$ be a character of $G$.
   \pause
    We should prove that $$\Sc^*(V)^{G,\chi}=0$$
    \end{Task}
\begin{itemize}
\pause
\item We may assume $V^G=0$
\pause
\item Let $p:V\to V//G:=spec(O(V)^G)$
\pause
\item Let $\N(V):= p^{-1}(p(0))
\pause
=\{x \in V| \overline{Gx} \ni 0\}$
\pause
\item Let $\mathcal{R}(V):=V-\N(V)$
\pause
\center {by induction we may assume:}
$$\Sc^*(\mathcal{R}(V))^{G,\chi}=0.$$
\end{itemize}
\end{frame}
}

\begin{frame}
    \frametitle{Stratification}

\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, $\Dist_X(Z)^{G,\chi} \cong \Dist(Z)^{G,\chi}$.\\ \pause
For smooth manifolds, there is a slightly more complicated statement which takes into account transversal derivatives:
\pause
$$gr_k(\Sc^*_X(Z))= \Sc^*(Z, Sym^k(CN_{Z}^X) )$$
\end{frame}


\begin{frame}
  \frametitle{Fourier transform}
\thispagestyle{empty}
%

\begin{figure}[htp]
\includegraphics[height=25mm]{Fourier.jpg}
\end{figure}

Let $V$ be a finite dimensional vector space over $F$ and $Q$ be a
non-degenerate quadratic form on $V$. Let $\widehat{\xi}$ denote
the Fourier transform of $\xi$ defined using $Q$.
\pause
\begin{proposition}
Let $G$ act on $V$ linearly and preserving $Q$. Let $\xi \in
\Sc^*(V)^{G,\chi}$.  Then $\widehat{\xi} \in \Sc^*(V)^{G,\chi}.$
\end{proposition}
\pause
\Rami{
We reduce to the following
    \begin{Task}
%    to prove:
    $$(\Sc^*_{V}(\N(V)) \cap \Fou(\Sc^*_{V}(\N(V))))^{G,\chi}=0 $$
    \end{Task}
}
\end{frame}

\begin{frame}
  \frametitle{Fourier transform and homogeneity}
\thispagestyle{empty}
\pause
\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 $\xi \in
\Sc^*_{V}(Z(Q))$ be s.t. $\widehat{\xi} \in \Sc^*_{V}(Z(Q))$. Then
$\xi$ is abs-homogeneous of degree $\frac{1}{2}dimV$.
\end{theorem}
\pause
\begin{theorem} [Archimedean homogeneity -- A., Gourevitch ] \label{ArchHom}
Let $F$ be any local field. Let $L \subset \Sc^*_{V}(Z(Q))$ be a
non-zero linear subspace s. t. $\forall \xi \in L $ we have
$\widehat{\xi} \in L$ and $Q \xi \in L$.

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{Singular Support and Wave Front Set}
\pause
To a distribution $\xi$ on $X$ one assigns two subsets of $T^*X$.
\pause
\begin{tabular}{|c|c|} \hline
%%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Singular Support & Wave front set\\ (=Characteristic variety)&\\
\hline
Defined using D-modules & Defined using Fourier transform\\
\hline
Available only in the & Available in both cases\\
Archimedean case &\\
\hline
\end{tabular}
\pause
In the non-Archimedean case we define the singular support to be the Zariski closure of the wave front set.

\end{frame}
\begin{frame}
  \frametitle{Properties and the Integrability Theorem}

Let $X$ be a smooth algebraic
variety.
\begin{itemize}\item Let $\xi \in \Sc^*(X)$.  Then $\overline{\Supp(\xi)}_{Zar} =
p_X(SS(\xi))$, where $p_X:T^*X \to X$ is the projection. \pause
\item Let an algebraic group $G$ act on $X$. Let $\xi \in \Sc^*(X)^{G, \chi}$. Then
$$SS(\xi) \subset \{(x,\phi) \in T^*X \, | \, \forall \alpha \in
\g \quad \phi(\alpha(x)) =0\}.$$ \pause
\item Let $V$ be a linear space. Let $Z \subset V^*$ be a closed
subvariety, invariant with respect to homotheties. Let $\xi \in
\Sc^*(V)$. Suppose that $\Supp(\widehat{\xi}) \subset Z$. Then
$SS(\xi) \subset V \times Z$. \pause
\item  Integrability theorem:\\
Let $\xi \in \Sc^*(X)$. Then $SS(\xi)$ is
(weakly) coisotropic.
\end{itemize}


\end{frame}
\begin{frame}
  \frametitle{Coisotropic varieties}
\pause
\begin{definition}
Let $M$ be a smooth algebraic variety and $\omega$ be a symplectic
form on it.
Let $Z\subset M$ be an algebraic subvariety. We call it {\bf $M$-coisotropic} if the following equivalent conditions hold.\\
\begin{itemize}
\pause
\item At every smooth point $z \in Z$ we have  $T_zZ \supset (T_zZ)^{\bot}$. Here, $(T_zZ)^{\bot}$ denotes the orthogonal complement
to $T_zZ$ in $T_zM$ with respect to $\omega$.
\pause
\item The ideal sheaf of regular functions that vanish on $\overline{Z}$ is closed under Poisson bracket. \\
\end{itemize}

%If there is no ambiguity, we will call $Z$ a coisotropic variety.
\end{definition}
\pause
\begin{itemize}
\item Every non-empty coisotropic subvariety of $M$ has
dimension at least $\frac{\dim M}{2}$.
\end{itemize}


\end{frame}


%\pause
%\cline{1-1} \cline{2-3}
%
%
\begin{frame}
  \frametitle{Symmetric pairs}
\pause
\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}
%\small{
\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}
\Rami{
\begin{Task} %\label{ComplexGood}
\center{$\Sc^*(G)^{H \times H} \subset \Sc^*(G)^{\sigma}$}
\end{Task}
\pause
Necessary condition:
}
\begin{definition}
A symmetric pair $(G,H,\theta)$ is called \textbf{good} if
$\sigma$ preserves all closed $H \times H$ double closets.
\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}

\Rami{
\begin{frame}
\frametitle{How to complete the task?}
\pause
Reformulate our task
\pause
\begin{Task} %\label{ComplexGood}
\pause
Let $\widetilde{H \times H}=H \times H \rtimes \{1,\sigma\}$
\pause
and $\chi:\widetilde{H \times H} \to \C$ \pause
 defaind by $\chi(\widetilde{H \times H}- H \times H)=-1$ \pause
we have to show that \pause
%\begin{equation}\label{t2}
$$\Sc^*(G)^{\widetilde{H \times H},\chi}=0$$
%\end{equation}
\end{Task} %\label{ComplexGood}
\pause
Using Harsh-Chandra Descent it is enough to show that
\pause
\begin{enumerate}
\item The pair $(G,H)$ is good
\pause
\item \label{regularity}
$\Sc^*(\g^\sigma)^{\widetilde{H},\chi}=0$
 provided that $\Sc^*(\mathcal{R}(\g^\sigma))^{\widetilde{H},\chi}=0.$
\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{How to prove regularity?}
\begin{itemize}
\pause
\item it is enough to prove that $(\Sc^*_{\g^\sigma}(\N(\g^\sigma))\cap \Fou(\Sc^*_{\g^\sigma}(\N(\g^\sigma))^{\widetilde{H},\chi}=0$
\pause
\item Let $H'= \widetilde{H} \times F^{\times}$ and $\chi' =\chi \times |\cdot|^{\dim(\g^\sigma)/2(+1)}u$\\
\pause
Using Homoginity theorem it is enough to prove that:
    $(\Sc^*_{\g^\sigma}(\N(\g^\sigma))^{H',\chi'})  \cap \Fou (\Sc^*_{\g^\sigma}(\N(\g^\sigma))^{H',\chi'})=0$
\pause
\item
We call an element $a \in \N(\g^\sigma)$ distinguished if $\h_a$ is nilpotent.
\pause
Using Integrability theorem it is enough to prove that:
$\Sc^*_{\g^\sigma}(O)^{H',\chi'}=0$ for any distinguished orbit $O$
\pause
\item
Using Frobenius descent it is enough to prove that for any distinguished $a$:
\pause
\begin{itemize}
\item $\chi'|_{H'_a} \Delta \neq 1$ -- in the non-Archimidian case
\pause
\item $(N_{Ha,a}^{\g^\sigma})^{H'_a, \chi' \Delta} = 0$ -- in the Archimidian case
\pause
\end{itemize}

\end{itemize}

\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
$(G\times G, \Delta G)$& A.-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 & A.-\\
\cline{1-2} $(O_{n+k},O_n \times O_k)$ & A.-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 &A.-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}
$(e_7 , sl_8)$ & A. &  (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}





\begin{frame}
  \frametitle{Results on Gelfand pairs}
%\thispagestyle{empty} Local fields of characteristic zero:
%\small{
\begin{tabular}{|c|c|c|}
\hline
Pair &p-adic case &real case  \\
\hline
$(GL_n(E), GL_n(F))$& Flicker & \\
%\pause
\cline{1-2} $(GL_{n+k},GL_{n} \times GL_k)$&Jacquet-Rallis & A.-\\
\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&A.-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 strong Gelfand pairs}

\begin{tabular}{|c|c|c|} \hline
%  after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Pair &p-adic & real  \\
\hline
 & A.- &A.-Gourevitch,\\
$(GL_{n+1},GL_{n})$& Gourevitch- & Sun-Zhu\\
 & Rallis- & \\
\cline{3-3} \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{itemize}
\item real: $\R$ and $\C$
\item p-adic: $\Q_p$ and its finite extensions.
\end{itemize}

\pause
\begin{remark}
The results from the last two slides are used to prove splitting of periods of automorphic forms.
\end{remark}
\end{frame}
\end{document}


