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%\title % (optional, use only with long paper titles)
%{Fourier transform of algebraic measures}
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%%{$F$ a $p$-adic field} % (optional)
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{D. Gourevitch}
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%{August 14, 2008}

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\title % (optional, use only with long paper titles)
[Applications of the Bernstein-Kashiwara Theorem]
{Applications of the Bernstein-Kashiwara Theorem}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{ Avraham Aizenbud }
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%   affiliation.


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%{Massachusetts Institute of Technology}

%\myD
\begin{document}


\begin{frame}
\myTP
\center{J. w. Dmitry Gourevitch,  Bernhad Kr\"otz, Andrey Minchenko,}
\center{Gang Liu, Eitan Sayag}

$ $

$ $

\center{\url{http://www.wisdom.weizmann.ac.il/~aizenr/}}

\end{frame}
\begin{frame}
\myFT{Holonomic D-modules and distributions}
\thispagestyle{empty}
 \myP
%\begin{definition}
A $D$-module over a smooth affine algebraic variety $X$ is a module over the ring $D(X)$ of differential operators  on $X$. \myP
%\end{definition}

A $D$-module $M$ given by generators and relations can be thought of as a system of PDE.  A solution of  $M$ is a $D$-module  homomorphism of $M$ to an appropriate space of functions.\myP
\begin{definition}
Let $M$ be a $D$-module over  $X$ with generators $m_{1}\dots m_k$. Define $F_i(D(X))$ to be the space of differential operators of degree $i$ and $F_i(M):=F_i(D(X))(m_{1}\dots m_k)$.  Define $$SS(M):=supp(gr_F(M)) \subset T^*X.$$
\end{definition}
\myP
For a distribution $\xi$ on $X(\R)$ define $$SS(\xi):=SS(D(X)\xi)\myP=\bigcap_{d\xi=0} Zeros(symbol(d)).$$\myP
A  $D$-module (or a distribution) $\xi$ is called holonomic if $$\dim(SS(\xi))=\dim X.$$
\end{frame}

\begin{frame}
%\thispagestyle{empty}
\myFT{Bernstein-Kashiwara theorem}
{\small{
\myP
\begin{thm}[Bernstein, Kashiwara $\sim$1974]
Let $X$ be a  real algebraic manifold. Let $M$ be a holonomic right $D_X$-module. Then $\dim \Hom(M,\Sc^*(X)) < \infty$.
\end{thm}

\myP
\begin{thm}[Bernstein, Kashiwara, Aizenbud, Gourevitch,  Minchenko]
Let $X,Y$ be smooth algebraic varieties and $\cM$  be a family of holonomic $D_X$-modules parameterized by $Y$.  Then $\dim \Hom(\cM_y,\Sc^*(X))   $ is bounded when $y$ ranges over $Y$.
\end{thm}

\myP
\begin{cor}[Aizenbud, Gourevitch,  Minchenko]
Let a real algebraic group $G$ act on a real algebraic manifold $X$ with finitely many orbits. Let $\cE$ be an algebraic $G$-equivariant bundle on  $X$ and $\chi$ be a character of $\g$.  Then,   $$\dim  \Sc^*(X,\cE)^{\g,\chi}< \infty.$$
Moreover, it remains bounded when we change $\chi$ or tensor  $\cE$ with a representation of $\g$ of a  fixed dimension.
\end{cor}
}}
\end{frame}

\begin{frame}
\thispagestyle{empty}
\myFT{Applications for co-invariants}
\myP
\begin{thm}[Aizenbud, Gourevitch, Kr\"otz, Liu]
Let a real algebraic group $G$ act on a real algebraic manifold $X$ with finitely many orbits. Let $\cE$ be an algebraic $G$-equivariant bundle on  $X$ and $\chi$ be a tempered character of $G$.  Then, $$\fg(\Sc(X,\cE)\otimes \chi)\subset \Sc(X,\cE)\otimes \chi$$
is closed and has finite codimension.
\end{thm}
\myP
\begin{cor}
Let $G$ be a  real reductive group, $H$ be a real spherical subgroup, and $\fh$ be the Lie algebra of $H$. Let $\chi$ be a tempered character of $H$. \myP Then for any admissible representation $\pi$ of $G$, $\mathrm{H}_0(\fh,\pi\otimes \chi)$ is separated and is non-degenerately paired with $(\pi^*)^{\fh,-\chi}$. \myP
In particular, the following conj. of Casselman are equivalent
\begin{itemize}
\myIt Automatic continuity: \myP $((\pi^{hc})^*)^{\fh}\cong (\pi^*)^{\fh}$ \myIt Comparison: \myP
$\mathrm{H}_0(\fh,\pi^{hc}) \cong \mathrm{H}_0(\fh,\pi)$
\end{itemize}

\end{cor}
\end{frame}




\begin{frame}
\myFT{Applications to multiplicities}
We reprove the following theorem
\myP
\begin{thm}[Kobayashi, Kr\"otz, Oshima, Schlichtkrull 2013]
Let $G$ be a  real reductive group, {$H$ be a Zariski closed subgroup, and $\fh$ be the Lie algebra of $H$}.
\begin{enumerate}
\myIt  If $H$ is a spherical subgroup \myP (i.e. $HB$ is open for some Borel subgroup $B$) \myP then
%Let $\lambda$ be a character of $\fz(\cU(\fg))$.
there exists $C \in  \mathbb{N}$ such that $\dim (\pi^*)^{\fh,\chi} \leq C$ for any $\pi\in Irr(G)$ {and any character $\chi$ of $\fh$}.
\myIt If $H$ is a real spherical subgroup  \myP(i.e. $HP$ is open for some minimal parabolic subgroup $P$)  \myP then, {for every
%irreducible smooth admissible \Fre  representation of moderate growth
irreducible  admissible   representation
 $\pi\in Irr(G)$, and natural number $n\in \N$ there exists $C_n \in  \mathbb{N}$ such that} for every $n$-dimensional representation $\tau$ of $\fh$ we have  $$\dim \Hom_{\fh}(\pi,\tau) \leq C_n.$$
\end{enumerate}

\end{thm}
\end{frame}

\begin{frame}
%\thispagestyle{empty}

\myFT{Sketch of the proof of Bernstein-Kashiwara theorem}
\begin{itemize}
\myIt Enough to prove for the case $X$ is a vector space.
\myIt Stone von-Neumann: The group $Sp(T^*(X))$ acts on the category of D-modules on $X$ stabilizing $\Sc^*(X)$.
\myIt $\dim SS_b=\dim SS_g $
\myIt For $g\in Sp(T^*(X))$ we have, $g(SS_b(M))=SS_b(gM) $
\myIt $\exists g\in Sp(T^*(X))$ s.t. $g(SS_b(M)) \cap X^* =0$
\myIt This implies that $p:g(SS_b(M)) \to\ X$ is finite.
\myIt This implies that $gM$ is smooth.
 \end{itemize}
\end{frame}


\begin{frame}
%\thispagestyle{empty}
\myFT{Relation with multiplicity}
\myP
\begin{thm}[Aizenbud, Gourevitch,  Minchenko 2015]
Let $G$ be an  algebraic reductive group, $H_1,H_2\subset G$ be spherical subgroups. The following system of equations on a distribution $\xi$ on $G$ is holonomic:
\begin{itemize}
\myIt $\xi$ is left $H_1$ invariant
\myIt $\xi$ is right $H_2$ invariant
\myIt $\xi$ is eigen w.r.t. the center $\fz(u(\g))$ of the universal enveloping algebra of the Lie algebra of $G$.
\end{itemize}

\end{thm}
\end{frame}

\begin{frame}
\myFT{The spherical character}
\myP
\begin{definition}
 Let $(\pi,V)$ be an admissible representation of $G(\R)$ and  $v_1\in (V^*)^{H_1}$, $v_2\in (\tilde V^*)^{H_2}$. Define the  spherical character of $\pi$ w.r.t. $v_{1}$ and $v_{2}$ by: $$\langle \xi,f \rangle:=\langle\pi^*(f)v_1,v_2\rangle.$$
\end{definition}
\myP
\begin{cor}
 A spherical character of admissible representation w.r.t. pair of spherical groups is holonomic distribution. In particular
\end{cor}
\myP
\begin{cor}[Aizenbud, Gourevitch,  Minchenko, Sayag]
For any local field $F$, any spherical character of  an admissible representation of $G(F)$ is smooth in a Zariski open dense set.
\end{cor}

\end{frame}

\begin{frame}
\myFT{Relation with Hausdorffness}
\thispagestyle{empty}
\textbf{Theorem:}
If $\#X/G<\infty$ then $\fg\Sc(X)\subset \Sc(X)$
is closed and has finite codimension.
\myP

\textbf{Proof:}
\myP
\begin{lemma}[Aizenbud, Gourevitch, Kr\"otz, Liu]
$H_*(\g, \Sc(G/H))$ are finite dimensional (and Hausdorff).
\end{lemma}
\myP
Assume that $X=U\cup Z$ is a union of an open orbit and a closed one. \myP It is enough to prove that $\g (\Sc(X)/\Sc(Z)) \subset \Sc(X)/\Sc(Z)$  is closed and of finite co-dimension. \myP Let $V:=(\Sc(X)/\Sc(Z))$. \myP The Borel's lemma and the lemma above implies that $V$  is an inverse  limit (with epimorphisms) of representations with finite dimensional co-homologies.
\myP
\begin{lemma}
Such  inverse limit commutes with homologies.
\end{lemma}
\myP
On the other hand Bernstein-Kashiwara theorem implies that  $\dim (V^*)^\g\leq \Sc^*(X)^\g<\infty$.
\myP


\end{frame}


\end{document}
\begin{frame}
\myFT{Corollaries for homologies}

\begin{thm}[Aizenbud, Gourevitch, Kr\"otz, Liu]
Let a real algebraic group $G$ act on a real algebraic manifold $X$ with finitely many orbits. Let $\cE$ be an algebraic $G$-equivariant bundle on  $X$ and $\chi$ be a tempered character of $G$.  Then, the homology  $\mathrm{H}_0(\g,\Sc(X,\cE)\otimes \chi))$ is separated and is non-degenerately paired with $\Sc^*(X,\cE)^{\g,-\chi}$.
I.e.$$\fg\Sc(X,\cE)\otimes \chi\subset \Sc(X,\cE)\otimes \chi$$
is closed and has finite codimension.
\end{thm}
\myP
\begin{cor}
Let $G$ be a  real reductive group, $H$ be a real spherical subgroup, and $\fh$ be the Lie algebra of $H$. Let $\chi$ be a tempered character of $H$. Then for any admissible representation $\pi$ of $G$, $\mathrm{H}_0(\fh,\pi\otimes \chi)$ is separated and is non-degenerately paired with $(\pi^*)^{\fh,-\chi}$.
\end{cor}\end{frame}



\begin{frame}
\myFT{Geometric formulation}
\begin{thm}[A., Gourevitch,  Minchenko 2015]
Let
\begin{multline*}S=\{g\in G,x \in \g^*|x\in \h_1^\bot, ad(g)(x)\in \h_2^\bot,x \text{ is nilpotent}\}=\\=G \times \cN \cap \bigcup_{g\in G} CN_{H_1gH_2,g}^G\end{multline*}
Then
$$\dim S = \dim G.$$
\end{thm}
\end{frame}
\begin{frame}
\myFT{The group case}
Assume $H_1=H_2=H$, diagonally embedded in  $G=H \times H$. \myP Translating the problem to $H=G/H$ we obtain:
$$S'=\{g\in H,x \in \h^*|Ad(g)(x)=x, x \in \cN_H\}=H  \times \cN_H \cap \bigcup_{g\in H} CN_{ad(G)g,g}^H$$
\myP
passing to the Lie algebra
$$S'=\{g\in \h,x \in \h|[x,g]=0, x \text{ is nilpotent}\}$$
\myP
So
$$S' \subset \bigcup_{x\in \cN_H} CN_{ad(G)x,x}^\h$$
\end{frame}
\begin{frame}
\myFT{Springer resolution and Steinberg theorem}
Let $\cB$ be the flag variety. \myP
$T^*\cB\cong \{B\in\cB,x\in \fb^\bot\}$.
\myP We have a natural map $\mu:T^*\cB \to \cN$.  It is called the Springer resolution.
\myP
\begin{theorem}[Steinberg 1976]
$$
\forall \eta \in \cN \text{ we have }  \dim G_\eta-2\dim\mu^{-1}(\eta)=\rk G.
$$
\end{theorem}


\end{frame}

\begin{frame}
\myFT{Idea of the proof}
\thispagestyle{empty}
\small{
\myP
$$\xymatrixcolsep{5pc}
\xymatrix{
 T^*\BB \times T^*\BB \ar@{->>}[rd]^{\mu \times \mu}  &  & G\times \cN\ar[ld]_{(g,x) \mapsto (x,\Ad^*(g^{-1})x) \quad \quad } \\
  & \cN \times \cN  \ar[d]^{res} & \\
  & \fh_1^* \times \fh_2^* &
 %  &  &
}
$$

\myP
Passing to the fiber of $0\in \fh_1^* \times \fh_2^*$ we get:
\myP
$$
\xymatrixcolsep{5pc}
\xymatrix{
 L_1 \times L_2 \ar@{->>}[rd]  &  & S\ar[ld]\\
  & \cN_{\fh_1} \times \cN_{\fh_2}  &
 }
$$

%\begin{equation}\label{eq:fiberdiag}
%\xymatrixcolsep{5pc}
%\xymatrix{
% L_1 \times L_2 \ar@{->>}[rd]^{\mu'}  &  \\
%  & \cN_{\fh_1} \times \cN_{\fh_2}  \\
% S\ar[ru]^{\alpha'}  &
%}
%\end{equation}
Where ${\cN}_{\fh_i}:=\cN \cap \fh_i^{\bot} \text{ and } L_i:=\{ (B,X) \in T^*\BB \, \vert \, X \in \fh_i^{\bot} \} \myP= \bigcup_{x\in \cB} CN_{H_i x,x}^\cB $.
%\begin{itemize}
%\myIt ${\cN}_{\fh_i}:=\cN \cap \fh_i^{\bot}$
% \myIt $L_i:=\{ (B,X) \in T^*\BB \, \vert \, X \in \fh_i^{\bot} \} \myP= \bigcup_{x\in \cB} CN_{H_i x,x}^\cB $.% is the union of the conormal bundles to the orbits of $H_i$ in $\BB$.
%\end{itemize}

\myP
The estimate on $\dim S$ follows from the Steinberg theorem and:
$$\dim L_i=\dim \cB.$$}
\end{frame}


\end{document}

