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%version v6 - made on 29.10.2014, after proof corrections. Published online  in Math. Z. Very minor changes. 
%version v5 - made on 08.09.2014, to appear in Math. Z. Minor changes. Explained why the stratification exists.
%version v4 - made on 01.04.2014 to submit to a journal. The proof that the singular support is nilpotent omitted.
%version v3 - made on 23.03.2014 to put on the arXiv. Lemma in the Preliminaries corrected, and character of H added. For that we passed to distributions on G.
%version v2 - made on 24.11.2013 to put on the arXiv. Preliminaries extended.
%version v1 - made on 19.7.2013 to put on homepage and send to colleagues.
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\begin{document}

\author{Avraham Aizenbud}
%\address{Avraham Aizenbud, Faculty of Mathematics
%and Computer Science, The Weizmann Institute of Science POB 26,
%Rehovot 76100, ISRAEL.}
\address{Avraham Aizenbud,
Faculty of Mathematics and Computer Science, Weizmann
Institute of Science, POB 26, Rehovot 76100, Israel }
\email{aizenr@gmail.com}
\urladdr{http://www.wisdom.weizmann.ac.il/~aizenr/}
\author{Dmitry Gourevitch}
\address{Dmitry Gourevitch, Faculty of Mathematics and Computer Science, Weizmann
Institute of Science, POB 26, Rehovot 76100, Israel }
\email{dimagur@weizmann.ac.il}
\urladdr{http://www.wisdom.weizmann.ac.il/~dimagur}


\keywords{Vanishing of distributions, spherical spaces, multiplicity, Shalika, Bessel function}
\subjclass[2010]{20G05, 22E45, 46F99}
%
%\classification{22E,22E45,20G05,20G25,46F99}
%
%20-xx  Group theory and generalizations 20Cxx Representation
%theory of groups 20C99 None of the above, but in this section
%
%20Gxx Linear algebraic groups (classical groups) 20G05
%Representation theory 20G25 Linear algebraic groups over local
%fields and their integers
%
%
%22-xx  Topological groups, Lie groups 22Exx Lie groups 22E45
%Representations of Lie and linear algebraic groups over real
%fields: analytic methods {For the purely algebraic theory, see
%20G05}
%
%
%46-xx  Functional analysis 46Fxx Distributions, generalized
%functions, distribution spaces 46F10 Operations with distributions
%
%14-xx  Algebraic geometry 14Lxx Algebraic Groups 14L24 Geometric
%invariant theory [See also 13A50] 14L30 Group actions on varieties
%or schemes (quotients) [See also 13A50, 14L24]

\date{\today}
\title{Vanishing of certain equivariant distributions on spherical spaces}
\maketitle
%\tableofcontents
\begin{abstract}
\DimaA{
We prove vanishing of $\fz$-eigen distributions on a split real reductive group which change according to  a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup.}

This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties.

As an application we prove non-vanishing of \DimaA{spherical} Bessel functions.
\end{abstract}
\section{Introduction}\label{sec:intro}

\subsection{Main results}

In this paper we prove the following generalization of Shalika's result \cite[\S 2, Proposition 2.10]{Shal}.
\begin{introtheorem}\label{thm:main}
\DimaA{Let $G$ be a split real reductive group and $H$ be its spherical subgroup. Let $U$ be the unipotent radical of a Borel subgroup $B$ of $G$. Let $\psi$ be a non-degenerate character of $U$ and $\chi$ be a character of $H$. Let $Z$ be the complement to the union of open $B\times H$-double cosets in $G$. Let $\fz$ be the center of the universal enveloping algebra of the Lie algebra $\g$ of $G$.

Then there are no non-zero $\fz$-eigen $(U\times H,\psi\times \chi)$-equivariant distributions supported on $Z.$}
\end{introtheorem}

This result in the group case (\cite[\S 2, Proposition 2.10]{Shal}) was crucial in the proof of Shalika's multiplicity one theorem.

Our proof begins by applying the technique used by Shalika. However, this technique was not enough for this generality and we had to complement it by using integrability of the singular support, as in \cite{AGAMOT}.



Theorem \ref{thm:main} provides a new tool for the study of the multiplicities of the irreducible quotients of the quasi-regular representation of $G$ on Schwartz functions on $G/H$, see \S\S \ref{subsec:qr} below for more details.


\subsection{Non-vanishing of spherical Bessel functions}
Another application of Theorem \ref{thm:main} is to the study of spherical Bessel distributions and functions.
\begin{defn}
Let $G$ be a split real reductive group, and $H \subset G$ be a spherical subgroup.
Let $(\pi,V)$ be a (smooth) irreducible admissible representation of $G$. Let $\phi$ be an \DimaA{$(H,\chi)$-equivariant} continuous functional on $V$ and $v$ be  a $(U,\psi)$-equivariant continuous functional on the contragredient representation $\tilde V$. Define the \emph{spherical Bessel distribution} by
$$\xi_{v,\phi}(f):=\langle  v, \pi^*(f) \phi \rangle. $$

\DimaB{Let $Z$ be the complement to the union of open $B\times H$-double cosets in $G$.}
Define the \emph{spherical Bessel function} to be the restriction $j_{v,\phi}:=\xi_{v,\phi}|_{\DimaA{G} - Z}$.
\end{defn}
It is well-known that $j_{v,\phi}$ is a smooth function.
\DimaB{One can also deduce this from Lemma \ref{lem:WF} below, since it implies that the singular support of $j_{v,\phi}$ is the zero section, which implies (by \cite[Theorem 8.5]{Hor}) that $j_{v,\phi}$ is smooth.}


Theorem \ref{thm:main} easily implies the following corollary.
\begin{introcor}\label{cor:SpherChar}
Suppose that \DimaA{$v$ and $\phi$ are non-zero.} Then $j_{v,\phi} \neq 0$.
\end{introcor}

\DimaA{
\subsection{Non-archimedean analogs}$\,$\\
Over non-archimedean fields, the universal enveloping algebra does not act on the distributions. However, the Bernstein's center $\End_{G \times G}(\Sc(G))$
does act.
In \cite{AGS_Z} we study this action in details. In \cite{AGK} we prove, using \cite[Theorem A]{AGS_Z},  analogs of Theorem \ref{thm:main}
and Corollary \ref{cor:SpherChar} for non-archimedean fields of characteristic zero. These analogs are somewhat weaker for general spherical pairs, but are of the same strength for the group case and for Galois symmetric pairs.
The group case of the non-archimedean counterpart of Corollary \ref{cor:SpherChar} was proven before in \cite[Appendix A]{LM}.

}
\subsection{Relation with multiplicities in regular representations of symmetric spaces}\label{subsec:qr}

Let $(G,H)$ be a symmetric pair of real reductive groups. Suppose that $G$ is quasi-split and let $B\subset G$ be a Borel subgroup. Let $k$ be the number of open $B$-orbits on $G/H$.

Theorem \ref{thm:main} can be used  to study the following conjecture.

\begin{introconj}\label{conj:mult}
Let $(\pi,V)$ be a (smooth) irreducible admissible representation of $G$. Then the dimension of the space \DimaA{$(V^*)^{H}$ of $H$-invariant} continuous functionals on $V$ is at most $k$. In particular, any complex reductive symmetric pair is a Gelfand pair.
\end{introconj}

We suggest to divide this conjecture into two cases
\begin{itemize}
\item $\pi$ is non-degenerate, i.e. $\pi$ has a non-zero continuous $(U,\psi)$-equivariant functional for some non-degenerate character $\psi$ of $U$.
\item $\pi$ is degenerate.
\end{itemize}

In the first case, Conjecture \ref{conj:mult} follows from the next one.

\begin{introconj}
 Let $U$ be the unipotent radical of $B$ and let $\psi$ be a non-degenerate character of $U$. Let $\fz$ be the center of the universal enveloping algebra of the Lie algebra $\g$ of $G$. Let $\lambda$ be a character of $\fz$.

Then the dimension of the space of $(\fz,\lambda)$-eigen $(U,\psi)$-equivariant distributions on $G/H$ does not exceed $k$.
\end{introconj}

We believe that  Theorem \ref{thm:main} can be useful in approaching this conjecture, since it allows to reduce the study of distributions to the union of open $B$-orbits.

%\subsection{Structure of the paper}

\subsection{Acknowledgements}
We thank Erez Lapid for pointing out to us the application to non-vanishing of spherical Bessel functions. \RamiA{We also thank Vladimir Hinich, Omer Offen and Eitan Sayag for useful discussions.}

D.G. was partially supported by ISF grant 756/12, ERC grant 291612, and a Minerva foundation grant.
A.A. was partially supported by NSF grant DMS-1100943 and ISF grant 687/13;
\section{Preliminaries}

\subsection{Conventions}

\begin{itemize}
\item By an algebraic manifold we mean a smooth real algebraic variety.
% \item For a manifold $M$, a closed submanifold $N\subset M$ and a point $x\in N$ we denote by $CN_{N,x}^M$ the conormal space to $N$ at $x$ in $M$.
\item We will use capital Latin letters to denote Lie groups and the corresponding Gothic letters to denote their Lie algebras.
\item Let a Lie group $G$ act on a smooth manifold $M$. For a vector $v\in \fg$ and a point $x \in M$ we will denote by $vx\in T_xM$ the image of $v$ under the differential of the action map $g \mapsto gx$. Similarly, we will use the notation $\fh x$, for any subspace $\fh \subset \fg$.
\item We denote by $G_x$ the stabilizer of $x$ in $G$ and by $\fg_x$ its Lie algebra.
% we will denote
\end{itemize}

\subsection{Tangential and transversal differential operators}\label{subsec:trans}

In this subsection we shortly review the method of \cite[\S 2]{Shal}. For a more detailed description see \cite[\S\S 2.1]{JSZ}.

\begin{defn}
Let $M$ be a smooth manifold and $N$ be a smooth submanifold.
\begin{itemize}
\item A vector field $v$ on $M$ is called \emph{tangential} to $N$ if for every point $p\in N, \, v_p\in T_pN$ and \emph{transversal} to $N$ if for every point $p\in N, \, v_p\notin T_pN$.

\item A differential operator $D$ is called \emph{tangential} to $N$ if every point $p\in N$ has an open neighborhood $U_x \subset N$ such that $D|_{U_x}$ is a finite sum of differential operators of the form \DimaB{$\phi V_1\cdot \dots \cdot V_r$}, where $\phi$ is a smooth function on $U_x, \, r\geq 0$, and $V_i$ are vector fields on $U_x$ tangential to $U_x \cap N$.
\end{itemize}
\end{defn}

\begin{lem}[cf. the proof of {\cite[Proposition 2.10]{Shal}}]\label{lem:TransTan}
Let $M$ be a smooth manifold and $N$ be a smooth submanifold. Let $D$ be a differential operator on $M$ tangential to $N$ and $V$ be a vector field  on $M$ transversal to $N$. Let $\eta$ be a distribution on $M$ supported in $N$ such that $D\eta=V\eta$. Then $\eta=0$.
\end{lem}

\subsection{Singular support}\label{subsec:SS}

Let $M$ be an algebraic manifold and $\eta$ be a distribution on $M$.
The \emph{singular support} of $\eta$ is defined to be the singular support of the D-module generated by $\eta$ and denoted $SS(\eta) \subset T^*M$.

We will shortly review the properties of the singular support that are most important for this paper.
For more detailed overview we refer the reader to \cite[\S\S 2.3 and Appendix B]{AGAMOT}.

%We will use the following corollary of Gabber's integrability theorem
\begin{notn} For a point $x\in M$
\begin{itemize}
\item we denote by $SS_x(\eta)$ the fiber of $x$ under the natural projection $SS(\eta)\to M$,
\item for a submanifold $N \subset M$ we denote by $CN_N^M\subset T^*M$ the conormal bundle to $N$ in $M$, and by $CN_{N,x}^M$ the conormal space at $x$ to $N$ in $M$.
\end{itemize}
\end{notn}

%\Dima{
\begin{lem}[{See e.g. \cite[Fact 2.3.9 and Appendix B]{AGAMOT}}] \label{Ginv}
Let an algebraic group $G$ act on   $M$. Suppose that $\eta$ is $G$-equivariant. Then
$$SS(\eta) \subset \{(x,\phi) \in T^*M \, | \, \forall \alpha \in
\g, \, \phi(\alpha(x)) =0\}=\bigcup_{x\in M}CN_{Gx}^M.$$
\end{lem}

\DimaA{We will need the following well-known lemma}

\begin{lem} \label{lem:nilp}
Let $G$ be a real reductive group, $\cN \subset \fg^*$ be the nilpotent cone and $\fz$ be the center of the universal enveloping algebra $\cU(\g)$.
Let $\xi$ be a $\fz$-eigen distribution on $G$. Identify $T^*G$ with $G\times \fg^*$ using the right action.
Then $SS(\xi)\subset G \times \cN$.
\end{lem}
% This lemma is well-known but we will prove it here for the benefit of the reader.
% \begin{proof}
%
%
% Consider the standard filtrations on $\cU(\fg)$ and on the ring of differential operators $D(G)$. Consider $\fg$ as the space of left-invariant vector fields on $G$. Then the natural map $\RamiA{i:}\cU(\fg)\to D(G)$ is a morphism of filtered algebras.
% \RamiA{We have a commutative diagram
%  $$\xymatrix{\parbox{30pt}{$Gr\cU(\fg)$}\ar@{->}^{Gr(i)}[r]\ar@{<-}_{\pi_U}[d] &
%  \parbox{20pt}{$Gr D(G)$}\ar@{<-}_{\pi_D}[d]\\
%  %
%  \parbox{20pt}{$S(\fg)$ }\ar@{->}^{\bar i}[r] &
%  \parbox{40pt}{$\cO(T^*G)$,}}
%  $$
%
% Where $\bar i$ and ${\pi_U}$ are the algebra homomorphisms which extend the natural embeddings $\g \to Gr\cU(\fg)$ and $\g \to \cO(T^*G)$, and $\pi_D$ is the algebra homomorphism which extends the natural embedding
% of vector fields on $G$ into  $D(G)$. By  the PBW theorem the vertical maps are isomorphisms. This implies that $Gr(i)$ is an embedding, and thus the filtration on $\cU(\g)$ is the one induced from   $D(G)$ using the embedding $i$. Therefore we have the following commutative diagram
%
% $$  \xymatrix{ \cU(\fg)\ar@{->}^{i}[r]\ar@{->}^{\sigma_{U}}[d] &
% D(G)\ar@{->}^{\sigma_{D}}[d]\\
% %
% \parbox{30pt}{$Gr\cU(\fg)$}\ar@{->}^{Gr(i)}[r]\ar@{<-}_{\pi_U}[d] &
%  \parbox{20pt}{$Gr D(G)$}\ar@{<-}_{\pi_D}[d]\\
%  %
%  S(\fg)\ar@{->}^{\bar i}[r] &
% \cO(T^*G),}
% $$
% where ${\sigma_{U}}$ and $\sigma_D$ are the (nonlinear) symbol maps. Note that the map ${\bar i}$ is a section of the restriction map $r:\cO(T^*G)\to \cO(T_e^*G)\cong\cO(\g^*)\cong S(\g).$
% }

% the natural morphism $ S(\fg)\to gr(\cU(\fg))$ is an isomorphism.  Thus we have the following commutative diagram
%
% % $$\xymatrix{\parbox{20pt}{$\cU(\fg)$}\ar@{->}^{i}[r]\ar@{->}^{\sigma}[d] &
% % \parbox{20pt}{$D(G)$}\ar@{->}^{\sigma}[d]\\
% % %
% % \parbox{20pt}{$S(\fg)$ $gr(\cU(\fg))$}\ar@{->}^{Gr(i)}[r] &
% % \parbox{40pt}{$\cO(T^*G)$}\ar@/^1pc/@{->}[l]^{p|_{T^*_{e}(G)}\mapsot p}}
% % $$
%
% $$  \xymatrix{ &\cU(\fg)\ar@{->}^{i}[r]\ar@{->}^{\sigma}[d] &
% D(G)\ar@{->}^{\sigma}[d]\\
% %
% S(\fg) \ar@{->}^{}[r]& gr(\cU(\fg))\ar@{->}^{gr(i)}[r] &
% \cO(T^*G)\ar@/^1pc/@{->}[ll]^{p|_{T^*_{e}(G)}\mapsot p}}
% $$
%
%where ${\sigma}$ is the (nonlinear) symbol map.

% In order to prove the lemma it is enough to show that $SS_{e}(\xi)\subset \cN$.
% Note that $\cN$ is the zero set the ideal $I\subset S(\fg)=\cO(\fg^*)$ generated by all homogeneous non-constant $\fg$-invariant polynomials. We have to show that for any homogeneous \RamiA{$p \in S(\g)^\g$ of degree $d>0,$}  there exists \RamiA{non-constant} $u\in \fz$ such that \RamiA{$r(\pi_D^{-1}(\sigma_D(i(u))))=p$, or equivalently (in view of the above) that $\sigma_{U}(u)=\pi_U(p)$. Let $s:S(\g)\to\cU(\g)$ be the symmetrization map.   It is easy to see that $\sigma^{d}_{U}(s(p))=\pi_U(p)$, where $\sigma^{d}_{U}$ denotes the $d$'s symbol.  Since $\pi_U$ is an isomorphism this implies that $\sigma^{d}_{U}(s(p)) \neq 0$ and thus $\sigma_{U}(s(p))=\sigma^{d}_{U}(s(p))=\pi_U(p)$. This implies the assertion.
% %  $$\xymatrix{\parbox{30pt}{$Gr\cU(\fg)$}\ar@{->}^{Gr(i)}[r]\ar@{<-}_{\pi_U}[d] &
% %  \parbox{20pt}{$Gr D(G)$}\ar@{<-}_{\pi_D}[d]\\
% %  %
% %  \parbox{20pt}{$S(\fg)$ }\ar@{->}^{\bar i}[r] &
% %  \parbox{40pt}{$\cO(T^*G)$,}}
% %  $$
%
%
% }
% \end{proof}
%}



\begin{thm}[Integrability theorem, cf. \cite{Gab,GQS,KKS, Mal}]
 The $SS(\eta)$ is a coisotropic subvariety of  $T^*M$.
\end{thm}

This theorem implies the following corollary (see \cite[\S 3]{Aiz} for more details).

\begin{cor}\label{cor:WCI}
Let $N\subset M$ be a closed algebraic submanifold. Suppose that  $\xi$  is supported in $N$. Suppose further that for any $x \in N$, we have $CN_{N,x}^M \nsubseteq SS_x(\eta)$. Then $\eta=0$.
\end{cor}



\section{Proof of Theorem \ref{thm:main}}

\subsection{Sketch of the proof}

% We try to follow Shalika's method (see \S\S\ref{subsec:trans}). However, it gives only partial results and we have to complement it by analysing the  singular support (see \S\S\ref{subsec:SS}).

\DimaA{We decompose $G$ into $B\times H$-double cosets. Each double coset} $\cO$ we decompose $\cO=\cO_s \cup \cO_c$ in a certain way.
 We prove the required vanishing \DimaA{coset by coset},  using Shalika's method (see \S\S\ref{subsec:trans}) for $\cO_s$ and singular support analysis (see \S\S\ref{subsec:SS}) for $\cO_c$.

\subsection{Notation and lemmas}


\begin{notation}$\,$
\begin{itemize}
\item Fix a \DimaB{maximal} torus $T\subset B$ and let $\ft \subset \fb$ denote the corresponding Lie  algebras. Let $\Phi$ denote the root system, $\Phi^+$ denote the set of positive roots and $\Delta\subset \Phi^+$ denote the set of simple roots. For $\alp\in \Phi$ let $\g_{\alpha} \subset \g$ be the corresponding root space.
 \item Let $C\in \fz$ denote the Casimir element.

\item We choose $E_{\alp}\in \fg_{\alp}$, for any $\alp \in \Phi$ such that $C=\sum_{\alp \in \Phi^+} E_{-\alp}E_{\alp}+D$, where $D$ is in the universal enveloping algebra of the Cartan subalgebra $\ft$.
%\item For $x\in X$ let $\Delta_x$ be the set of all simple roots $\al$ such that $\g_{-\al}x \nsubseteq    \fb x$.
%\DimaB{
\item \DimaB{Let $V:= \sum_{\alp \in \Delta} d\psi(E_\alp)E_{-\alp} \in \fg.$\\ Note that $C\xi=(V+D)\xi$ for any $(U,\psi)$-equivariant distribution $\xi$ on $G$.}

\item Let $\cO\subset G$ be a \RamiB{stable} $B\times H$-{double coset\RamiB{, i.e the set of real points of the complexification of a double coset}. Consider the left action of \RamiB{$G$ on itself} and define} $$\cO_c:=\left \{x \in \cO \, |\, Vx  \in T_x\cO \right \}\text{ and }\cO_s:=\cO\setminus \cO_c.$$
\end{itemize}
\end{notation}
\RamiB{
\begin{rem}
By definition $\cO_c$ is a Zariski closed real algebraic subvariety of $\cO$, as such  we can stratify it  by smooth locally closed  real algebraic subvarieties of $\cO$.
\end{rem}
}

%This theorem follows from the next two lemmas.
We will need the following lemmas, that will be proved in subsequent subsections.
\DimaA{
\begin{lemma}\label{lem:WF}
Let $x\in G$. Let $\xi$ be a $\fz$-eigen $(U\times H,\psi \times \chi)$ equivariant distribution on $G$. Then $SS_x(\xi)\subset CN_{BxH,x}$.
\end{lemma}


\begin{lemma}\label{lem:cOProp}
Let $\cO\subset Z$ be a $B\times H$-double coset. Then $\cO_s \neq \emptyset$.
\end{lemma}
}
% I think that our proof works for any spherical variety.

%\begin{lemma}\label{lem:trans}
%Let $x \in X$. Then the action map  $\oplus_ {\al \in \Delta_x}\g_{-\al} \to T_xX/  \fb x $ is injective.
%\end{lemma}
%For proof see \S\S\ref{subsec:PfLemTrans}.

%\subsection{Proof of Theorem \ref{thm:main}}\label{subsec:PfMain}



\subsection{Proof of Theorem \ref{thm:main}}

%\begin{proof}[Proof of Theorem \ref{thm:main}]

Suppose that there exists a non-zero $\fz$-eigen $(U,\psi)$- equivariant distribution $\xi$  supported on $Z$.

For any \RamiB{stable} {$B\times H$-double coset $\cO \subset G$}, stratify $\cO_{c}$ to a union of smooth locally closed varieties $\cO_c^i$.

The collection \DimaA{$$\{\cO_c^i \, | \, \cO \text{ is a }B\times H-\text{double coset}\}\cup \{\cO_s \, | \,\cO\text{ is a } B\times H-\text{double coset}\}$$} is a stratification of \DimaA{$G$}. Reorder this collection to a sequence $\{S_i\}_{i=1}^N$ of smooth locally closed subvarieties of \DimaA{$G$} such that $U_k:=\bigcup_{i=1}^k S_i$ is open in \DimaA{$G$} for any $1\leq k \leq N$.

Let $k$ be the maximal integer such that $\xi|_{U_{k-1}}=0$. Let $\eta:=\xi|_{U_{k}}$.
We will now show that $\eta=0$, which leads to a contradiction.
\begin{enumerate}[{Case}  1.]
\item $S_{k}=\cO_s$ for some \DimaA{double coset} $\cO$.

Recall that we have the following decomposition of the Casimir element $$C=\sum_{\alp \in \Phi^+} E_{-\alp}E_{\alp}+D$$ Since $\eta$ is  $\fz$-eigen and $(U,\psi)$-equivariant, we have, for some scalar $\lambda$,
%\RamiQ{Need to decide: $\psi(\alpha)$ vs. $\psi(E_\alpha)$ vs. $d\psi(E_\alpha)$.}
$$\lambda \eta=C\eta=\sum_{\alp \in\Phi^+} E_{-\alp}E_{\alp}\eta+D\eta=\sum_{\alp \in \Phi^+} E_{-\alp}d\psi(E_\alp)\eta +D\eta=%\sum_{\alp \in \Delta} E_{-\alp}d\psi(E_\alp)\eta +D\eta=
\DimaB{V\eta}+D\eta$$



% Let $\cO$ be a $B$-orbit of maximal dimension that intersects the support of $\xi$.
% %Define $\cO_s:=\{x \in \cO \, |\, \sum E_{-\alp}\psi(E_\alp)x \notin T_xBx
% % \}$ and $\cO_c:=\cO\setminus \cO_s$.
% $\Supp(\xi)$ intersects $\cO_s$.
%
% Let $Y$ be the union of all $B$-orbits in $X$ of dimension $\leq \dim \cO$, and $$W:=(X \setminus Y) \cup \cO_{s}.$$ Then $\eta:=\xi|_{W}$ is non-zero and is supported in the closed submanifold $\cO_{s}$.


%Let $C\in \fz$ denote the Casimir element. Then one can choose $E_{\alp}\in
%\fg_{\alp}$, for any $\alp \in \Phi$ such that $C=\sum_{\alp \in \Phi^+}
%E_{-\alp}E_{\alp}+D$, where $D$ is in the universal enveloping algebra of
% the Cartan subalgebra $\ft$. Since $\eta$ is  $\fz$-eigen and $(U,\psi)$
%  equivariant, we have, for some scalar $\lambda$,
%$$\lambda \eta=C\eta=\sum_{\alp \in\Phi^+} E_{-\alp}E_{\alp}\eta+D\eta=\sum_{\alp \in \Phi^+} E_{-\alp}\psi(\alp)\eta +D\eta=\sum_{\alp \in \Delta} E_{-\alp}\psi(\alp)\eta +D\eta$$

%Let $x\in \cO_s$.
Let %$V:= \sum_{\alp \in \Delta} d\psi(E_\alp)E_{-\alp}$ and
$D':=\lambda Id-D$.
We have $V\eta=D'\eta$, and it is easy to see that $D'$ is tangential to $\cO_s$,
%Lemmas \ref{lem:out} and \ref{lem:trans} imply that
and $V$ is transversal to $\cO_s$. Now, Lemma \ref{lem:TransTan} implies $\eta=0$ which is a contradiction.

\item $S_{k}\subset \cO_c$ for some orbit $\cO$.

%$\Supp(\xi)\cap \cO_s=\emptyset$.

%
% Let $\cO_c^{i_{0}}$ be a strata of maximal dimension that intersects the support of $\xi$.
%
% Let $Y'$ be the union of all $B$-orbits in $X$ of dimension $\leq \dim \cO$, and let $Y'' \subset \cO$ be the union of all strata $\cO_c^i$ of dimension $\leq \dim \cO_c^{i_0}$. Let $$W':=(X - Y') \cup (\cO-Y'') \cup \cO_c^{i_0}.$$ Then $\eta':=\xi|_{W'}$ is non-zero and is supported in the closed submanifold $\cO^{i_0}_{c}$. In order to get a contradiction we will show that $\eta'=0.$




By Corollary \ref{cor:WCI} it is enough to show that for any  $x \in S_k$ we have
\begin{equation}\label{eq:NonCoIsot}
CN_{S_k,x}^{\DimaA{G}} \nsubseteq SS_x(\eta).
\end{equation}

By Lemma \ref{lem:WF}, $ SS_x(\eta)\subset CN_{\cO,x}^{\DimaA{G}}$. By Lemma \ref{lem:cOProp}, $S_{k}\subsetneq \cO$, thus $CN_{S_{k},x}^{\DimaA{G}} \supsetneq CN_{\cO,x}^{\DimaA{G}}$ which implies \eqref{eq:NonCoIsot}.

% Let $H$ be the stabilizer of $x$ and let $\fh$ be its Lie algebra. Identify %$T_x X$ with $\g/\fh$. Let $\ft_0 :=\{t\in \ft|\sum \psi(\alpha) [t,E_{-\alpha}]\in %\fh+\fb). \}$ It is easy to see that $T_{x}(O_{c})$ is the image of $\ft_0+\mathfrak{u}$ %in $\g/\fh$.
%Identify $T^*_x(X)$ with the orthogonal complement $\fh^\bot \subset \fg$. %\RamiQ{form.} Then $CN_{\cO_c,x}^X=(\ft_0+\mathfrak{u}+\fh)^\bot$. By Lemma %\ref{lem:WF}, $WF_x(\xi)\subset (\ft+\mathfrak{u}+\fh)^\bot$. Thus in order %to proof that  $CN_{O_c,x}^X \nsubseteq WF_x(\xi)$ it is enough to show %that   $\ft_0+\mathfrak{u}+\fh \neq \ft+\mathfrak{u}+\fh$. Since $T_{x}(\cO)$ %is the image of $\ft+\mathfrak{u}$ in $\g/\fh$, this follows from $\cO_c\nsubseteq %\cO$, which in turn follows from Lemma \ref{lem:out}.


%If $\cO_c=\emptyset$ then we are done. Else, fix  $x \in \cO_c$. Let $V$ %be the span of the vectors $E_{-\alpha}$ where $\alpha\in \Delta_x$.

%  Let us first show that $\cO\neq \cO_c$. ??.
%By ?? $\cO\neq \cO_c$.  $T_c$ - subgroup of $T$. Want to show that the condition %on the singular support of $\eta$ is not compatible with the support condition- %compute normal space to $\cO_c$ and show that it gets outside the nilpotent %cone.
\end{enumerate}
\proofend

%\subsection{Proof of Lemma \ref{lem:out}}\label{subsec:PfLemOut}
\DimaA{
\subsection{Proof of Lemma \ref{lem:WF}}\label{subsec:PfLemWF}


\begin{proof}
Let $\fh$ denote the Lie algebra of $H$  and $ad(x)\fh$ denote its conjugation by $x$. Identify $T_x^*G$ with $\fg^*$ using  multiplication by $x^{-1}$ on the right. Then $$CN_{BxH,x}^G= (\ft+\mathfrak{u}+ad(x)\fh)^\bot.$$
Since $\xi$ is $\mathfrak{u}\times \fh$-equivariant, Lemma \ref{Ginv} implies that  $SS_x(\xi)\subset (\mathfrak{u}+ad(x)\fh)^\bot$.
Since $\xi$ is also $\fz$-eigen, Lemma \ref{lem:nilp} implies that $SS_x(\xi)\subset \cN $, where $\cN \subset \fg^*$ is the nilpotent cone. Now we have
$$SS_x(\xi)\subset(\mathfrak{u}+ad(x)\fh)^\bot \cap \cN =(\ft+\mathfrak{u}+ad(x)\fh)^\bot=CN_{BxH,x}^G.$$
\end{proof}

\subsection{Proof of Lemma \ref{lem:cOProp}}\label{subsec:cOProp}
First we need the following lemmas and notation.

\begin{lemma}\label{lem:dense}
Let $K\subset K_{i} \subset G$ for $i=1,\dots,n$ be algebraic subgroups. Suppose that $K_i$ generate $G$. Let $Y$ be a transitive $G$ space. Let $y \in Y$. Assume that $Ky$  is  Zariski dense in  $K_i y$ for each $i$. Then $Ky$  is Zariski dense in  $Y$.
\end{lemma}
\begin{proof}
By induction we may assume that $n=2$. Let $$O_l:=\underbrace{K_1K_2\cdots K_1K_2 }_{l \text{ times}}y.$$ It is enough to prove that for any $l$ the orbit $Ky$ is dense in $O_l$. Let us prove it by induction on $l$. Suppose that we have already proven that $Ky$ is dense in $O_{l-1}$. Then
$$\overline{Ky}=\overline{K_2y}=\overline{K_2Ky}=\overline{K_2O_{l-1}y}.$$
Thus $Ky$  is dense in $K_2O_{l-1}$. Similarly, $K_2O_{l-1}$  is dense in $K_1K_2O_{l-1}=O_l$.
\end{proof}



%For proof see \S\S\ref{subsec:PfLemOut}.



\begin{notn}$\,$
\begin{itemize}
\item Let $Y$ denote the spherical space $G/H$, and $Z'$ denote the image of $Z$ in $Y$.
\item For a simple root $\alp\in \Delta$, denote by $P_{\alp}\subset G$ the parabolic subgroup whose Lie algebra is $\fg_{-\alp}\oplus \fb$.
\end{itemize}
\end{notn}

\begin{lemma}\label{lem:out}
Let $x \in Z'$. Then there exists a simple root $\al \in \Delta$ such that $\g_{-\al}x \nsubseteq    \fb x$.
\end{lemma}

\begin{proof}%[Proof of Lemma \ref{lem:out}]
Assume the contrary. Then for any $\al \in \Delta, \,T_x P_{\al}x=T_xBx$. Thus $Bx$ is Zariski dense in $P_{\al}x$. By Lemma \ref{lem:dense} this implies that $Bx$ is dense in $Y$, which contradicts the condition $x \in Z'$.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:cOProp}]
Note that $\cO_c$ is invariant with respect to the right action of $H$.
Let $\cO'$ denote the image of $\cO$ in $Y$, and let $\cO_c'$ denote  the image of $\cO_c$ in $Y$. Choose $x\in \cO'$ and
let $a:B \to \cO'$ denote the action map. It is enough to show that $a^{-1}(\cO_c')\neq B$.
%$p^{-1}(\cO_c)$ is  a Zariski  closed  subvariety of non-zero codimension.
\begin{multline*}
a^{-1}(\cO_c')=\{b \in B \, | \, \sum \psi(E_{\alp})E_{-\alp}\in \fg_{bx} + \fb\}=
\{b \in B \, | \, \sum \psi(E_{\alp})ad(b^{-1})E_{-\alp} \in \fg_{x} + \fb\}=\\
\{tu \in B \, | \, \sum \psi(E_{\alp})ad(t^{-1})E_{-\alp} \in \fg_{x} + \fb\}=
\{tu \in B \, | \, \sum \psi(E_{\alp})\alp(t)E_{-\alp} \in \fg_{x} + \fb\}
\end{multline*}

%It is now clear that $p^{-1}(\cO_c)$ is Zariski closed, so it is enough to show that $p^{-1}(\cO_c)\neq B$.
By Lemma \ref{lem:out} we can choose $\alp \in \Delta$ such that $\g_{-\al}x \nsubseteq    \fb x$. For any $\eps>0$ there exists $t \in T$ such that $\alp(t)=1$ and $\forall \beta \neq \alp \in \Delta$ we have $|\beta(t)|<\eps$. It is easy to see that for $\eps$ small enough, $t \notin  a^{-1}(\cO_c')$.
\end{proof}
}



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