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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.} 

%\email{aizenr@yahoo.com}
\title[Schwartz functions on Nash manifolds]{Schwartz functions on Nash
manifolds and applications to representation theory}
\maketitle
\centerline{Joint with Dmitry Gourevitch and Eitan Sayag} 
\centerline{arXiv:0704.2891 [math.AG], arxiv:0709.1273 [math.RT]}

$ $\\

Let us start with the following motivating example. Consider the
circle $S^1$, let $N \subset S^1$ be the north pole and denote
$U:=S^1 \setminus N$. Note that $U$ is diffeomorphic to $\R$ via
the stereographic projection. Consider the space $\cD(S^1)$ of
distributions on $S^1$, that is the space of continuous linear
functionals on the \Fre space $C^{\infty}(S^1)$. Consider the
subspace $\cD_{S^1}(N) \subset \cD(S^1)$ consisting of all
distributions supported at $N$. Then the quotient $ \cD(S^1) /
\cD_{S^1}(N)$ will not be the space of distributions on $U$.
However, it will be the space $\Sc^*(U)$ of Schwartz distributions
on $U$, that is continuous functionals on the \Fre space $\Sc(U)$
of Schwartz functions on $U$. In this case, $\Sc(U)$ can be
identified with $\Sc(\R)$ via the stereographic projection.

The space of Schwartz functions on $\R$ is defined to be the space
of all infinitely differentiable functions that rapidly decay at
infinity together with all their derivatives, i.e. $x^nf^{(k)}$ is
bounded for any $n,k$.

In this talk we extend the notions of Schwartz functions and
Schwartz distributions to a larger geometric realm.

%\subsection{}
As we can see, the definition is of algebraic nature. Hence it
would not be reasonable to try to extend it to arbitrary smooth
manifolds. However, it is reasonable to extend this notion to
smooth algebraic varieties. Unfortunately, sometimes this is not
enough. For example, a connected component of real algebraic
variety is not always an algebraic variety. By this reason we
extend this notion to smooth semi-algebraic manifolds. They are
called \textbf{Nash manifolds}.

For any Nash manifold $M$, we will define the spaces $\G(M)$,
$\T(M)$ and $\Sc(M)$ of \textbf{generalized Schwartz
functions}\footnote{Here we distinguish between the (similar)
notions of a generalized function and a distribution. They can be
identified by choosing a measure. Without fixing a measure, a
smooth function defines a generalized function but not a
distribution.}, \textbf{tempered functions} and \textbf{Schwartz
functions} on $M$. Informally, $\T(M)$ is the ring of functions
that have no more than polynomial growth together with all their
derivatives, $\G(M)$ is the space of generalized functions with no
more than polynomial growth and $\Sc(M)$ is the space of functions
that decay together with all their derivatives faster than any
inverse power of a polynomial.

As in the classical case, in order to define generalized Schwartz
functions, we have to define Schwartz functions first. Both
$\G(M)$ and $\Sc(M)$ are modules over $\T(M)$.

The triple $\Sc(M)$, $\T(M)$, $\G(M)$ is analogous to
$C_c^{\infty}(M)$, $C^{\infty}(M)$ and $C^{-\infty}(M)$ but it has
additional nice properties as we will see later.

We will show that for $M=\R^n$, $\Sc(M)$ is the space of classical
Schwartz functions and $\G(M)$ is the space of classical
generalized Schwartz functions. For compact Nash manifold $M$,
$\Sc(M)=\T(M)=C^{\infty}(M)$.

\subsection*{Main results}

%In this subsection we summarize the main results of the paper.
\begin{result}
Let $M$ be a Nash manifold and $Z\subset M$ be a closed Nash
submanifold. Then the restriction maps $\T(M) \rightarrow \T(Z)$
and $\Sc(M) \rightarrow \Sc(Z)$ are onto.
\end{result}

\begin{result} \label{mainresult}
Let $M$ be a Nash manifold and $U \subset M$  be a semi-algebraic
open subset. Then a Schwartz function on $U$ is the same as a
Schwartz function on $M$ which vanishes with all its derivatives
on $M \setminus U$.
\end{result}

This theorem tells us that extension by zero $\Sc(U)\rightarrow
\Sc(M)$ is a closed imbedding, and hence restriction morphism
$\G(M)\rightarrow \G(U)$ is onto.
\\Classical generalized functions do not have this property. This was our
main motivation for extending the definition of Schwartz
functions.

\subsection*{Schwartz sections of Nash bundles}

Similar notions will be defined for Nash bundles, i.e. smooth
semi-algebraic bundles.

For any Nash bundle $E$ over $M$ we will define the spaces
$\G(M,E)$, $\T(M,E)$ and $\Sc(M,E)$ of generalized Schwartz,
tempered and Schwartz sections of $E$.

As in the classical case, a generalized Schwartz function is not
exactly a functional on the space of Schwartz functions, but a
functional on Schwartz densities, i.e. Schwartz sections of the
bundle of densities.

Therefore, we will define generalized Schwartz sections by
$\G(M,E)=(\Sc(M,\widetilde {E}))^*$, where $\widetilde
{E}=E^*\otimes D_M$ and $D_M$ is the bundle of densities on M.

Let $Z \subset M$ be a closed Nash submanifold, and $U=M \setminus
Z$. Result \ref{mainresult} tells us that the quotient space of
$\G(M)$ by the subspace $\G(M)_Z$ of generalized Schwartz
functions supported in $Z$ is $\G(U)$. Hence it is useful to study
the space $\G(M)_Z$. As in the classical case, $\G(M)_Z$ has a
filtration by the degree of transversal derivatives of delta
functions. The quotients of the filtration are generalized
Schwartz sections over $Z$ of symmetric powers of normal bundle to
$Z$ in $M$, after a twist.

This result can be extended to generalized Schwartz sections of
arbitrary Nash bundles.

\subsection*{Restricted topology and sheaf properties}

Similarly to algebraic geometry, the reasonable topology on Nash
manifolds to consider is a topology in which open sets are open
semi-algebraic sets. Unfortunately, it is not a topology in the
usual sense of the word, it is only what is called
\textbf{restricted topology}. This means that the union of an
infinite number of open sets does not have to be open. The only
open covers considered in the restricted topology are finite open
covers.

The restriction of a generalized Schwartz function (respectively
of a tempered function) to an open subset is again a generalized
Schwartz (respectively a tempered function). This means that they
form pre-sheaves. We will show that they are actually sheaves,
which means that for any finite open cover $M = \bigcup \limits
_{i=1}^n U_i$, a function $\alpha$ on $M$ is tempered if and only
if $\alpha|_{U_i}$ is tempered for all $i$. It is of course not
true for infinite covers. We denote the sheaf of generalized
Schwartz functions by $\G_M$ and of the sheaf of tempered
functions by $\T_M$. By result \ref{mainresult}, $\G_M$ is a
flabby sheaf.

Similarly, for any Nash bundle $E$ over $M$ we will define the
sheaf $\T_M^E $ of tempered sections and the sheaf $\G_M^E$ of
generalized Schwartz sections.

As we have mentioned before, Schwartz functions behave similarly
to compactly supported smooth functions. In particular, they
cannot be restricted to an open subset, but can be extended by
zero from an open subset. This means that they do not form a
sheaf, but an object dual to a sheaf, a so-called cosheaf. We
denote the cosheaf of Schwartz functions by $\Sc_M$.
%we will show that if $U$ is an open (semi-algebraic) subset of
%$M$, the extension by zero of a Schwartz function on $U$ is a
%Schwartz function on $M$.
%This property shows us the structure of
%cosheaf on Schwartz functions on $M$. Cosheaf is a notion dual to
%the notion of sheaf, i.e. instead of restriction to open subset we
%have extension from open subset.
We will prove that $\Sc_M$ is actually a cosheaf and not just
pre-cosheaf by proving a Schwartz version of the partition of
unity theorem. Similarly, for any Nash bundle $E$ over $M$ we will
define the cosheaf $\Sc_M^E$ of Schwartz sections.

\subsection*{Possible applications}

Schwartz functions are used in the representation theory of
algebraic groups. Our definition coincides with Casselman's
definition (cf. \cite{Cas}) for algebraic groups. Our paper allows
to use Schwartz functions in more situations in the representation
theory of algebraic groups, since an orbit of an algebraic action
is a Nash manifold, but does not have to be an algebraic group or
even an algebraic variety.

Generalized Schwartz sections can be used for ``devisage''. We
mean the following. Let $U\subset M$ be an open (semi-algebraic)
subset. Instead of dealing with generalized Schwartz sections of a
bundle on $M$, we can deal with generalized Schwartz sections of
its restriction to $U$ and generalized Schwartz sections of some
other bundles on $M \setminus U$.

For example if we are given an action of an algebraic group $G$ on
an algebraic variety $M$, and a $G$-equivariant bundle $E$ over
$M$, then devisage to orbits helps us to investigate the space of
$G$-invariant generalized sections of $E$.

%One of the implementations of this argument appears in \cite{AGS}.
%There we also use the fact that $\Sc(\R^n)$ is preserved by
%Fourier transform.

\subsection*{Summary} \label{Summary}

To sum up, for any Nash manifold $M$ we define a sheaf $\T_M$ of
algebras on $M$ (in the restricted topology) consisting of
tempered functions, a sheaf $\G_M$ of modules over $\T_M$
consisting of generalized Schwartz functions, and a cosheaf
$\Sc_M$ of modules over $\T_M$ consisting of Schwartz functions.

Moreover, for any Nash bundle $E$ over $M$ we define sheaves
$\T^E_M$ and $\G^E_M$ of modules over $\T_M$ consisting of
tempered and generalized Schwartz sections of $E$ respectively and
a cosheaf $\Sc^E_M$ of modules over $\T_M$ consisting of Schwartz
sections of $E$.

Let us list the main properties of these objects that we will
prove in this paper:

\begin{property2}\label{p1}
Compatibility: For open semi-algebraic subset $U \subset M$,
$\Sc^E_M|_U=\Sc^{E|_U}_U$, $\T^E_M|_U=\T^{E|_U}_U$,
$\G^E_M|_U=\G^{E|_U}_U$.
\end{property2}
%
\begin{property2}\label{p2}
$\Sc(\R ^n)$ = Classical Schwartz functions on $\R ^n$.
\end{property2}
%
\begin{property2}\label{p3}
For compact $M$, $\Sc(M,E) = \T(M,E)=$ $C^{\infty}(M,E)$.
\end{property2}
%
\begin{property2}\label{p4}
$\G_M^{E}=(\Sc^{\widetilde {E} }_M)^*$ , where $\widetilde
{E}=E^*\otimes D_M$ and $D_M$ is the bundle of densities on M.
\end{property2}
%
\begin{property2}\label{p5}
Let $Z \subset M$ be a closed Nash submanifold. Then the
restriction maps  $\Sc(M,E)$ onto $\Sc(Z,E|_Z)$ and $\T(M,E)$ onto
$\T(Z,E|_Z)$.
\end{property2}
%
\begin{property2}\label{p6}
Let $U \subset M$  be a semi-algebraic open subset, then
$$\Sc^E_M(U) \cong \{\phi \in \Sc(M,E)| \quad \phi \text{ is 0 on } M
\setminus U \text{ with all derivatives} \}.$$
\end{property2}
%
\begin{property2} \label{p7}
Let $Z \subset M$ be a closed Nash submanifold. Consider
$\G(M,E)_Z=\{\xi \in \G(M,E) |\xi$ is supported in $Z \}$. It has
a canonical filtration such that its factors are canonically
isomorphic to $\G(Z,{E|_Z \otimes S^i(N_Z^M) \otimes {D_M^*|}_Z
\otimes D_Z})$ where $N_Z^M$ is the normal bundle of $Z$ in $M$
and $S^i$ means i-th symmetric power.
\end{property2}
%
\subsection*{Remarks}
%
\begin{remark}
Harish-Chandra has defined a Schwartz space for every reductive
Lie group. However, Harish-Chandra's Schwartz space does not
coincide with the space of Schwartz functions that we define in
this paper even for the algebraic group $\R^{\times}$.

%Harish-Chandra gave a different definitions of Schwartz functions,
%generalized Schwartz functions and tempered functions on Lie
%groups (cf. \cite{HC}).
%
%These definitions are not equivalent to our definition, even for
%the algebraic group $\R^{\times}$.
%Our definition does not coincide with Harish-Chandra's definition
%(cf. \cite{HC}) for Lie groups, because a Lie group isomorphism
%which is not a Nash diffeomorphism maps Harish-Chandra's Schwartz
%functions to Harish-Chandra's Schwartz functions but will not map
%our Schwartz functions to our Schwartz functions.
\end{remark}
%
\begin{remark}
There is a different approach to the concept of Schwartz
functions. Namely, if $M$ is embedded as an open subset in a
compact manifold $K$ then one can define the space of Schwartz
functions on $M$ to be the space of all smooth functions on $K$
that vanish outside $M$ together with all their derivatives. This
approach is implemented in \cite{CHM}, \cite{KS}, \cite{Mor} and
\cite{Pre}. In general, this definition depends on the embedding
into $K$. Our results show that for Nash manifolds $M$ and $K$ it
coincides with our definition and hence does not depend on the
embedding.
\end{remark}
%

\begin{remark}
After the completion of this project we found out that many of the
properties of Schwartz functions on affine Nash manifolds have
been obtained already in \cite{dCl}.
\end{remark}

%\begin{remark}
%In fact, tempered functions can be defined in terms of Schwartz
%functions or in terms of generalized Schwartz functions by
%$$\T(M,E) = \{\alpha \in
%C^{\infty}(M,E)|\,\,\, \alpha \Sc(M)\subset \Sc(M,E)\} =$$
%$$\quad \quad \quad =\{\alpha \in C^{\infty}(M,E) |\,\,\, \alpha \G(M)\subset
%\G(M,E)\},$$ but the proof is rather technical and lies out of the
%scope of this paper.
%\end{remark}
%
%\begin{remark}
%Throughout the whole paper ``smooth'' means infinitely
%differentiable and ``bounded'' means bounded in absolute value,
%unless stated otherwise.
%\end{remark}
\subsection*{An application to representation theory}

Using the theory of Schwartz functions we showed the following
theorem:
\begin{theorem}
Let $F$ be either the field of real or complex numbers and
consider the standard imbedding $\mathrm{GL}_n(F) \hookrightarrow
\mathrm{GL}_{n+1}(F)$. We consider the two-sided action of
$\mathrm{GL}_n(F) \times \mathrm{GL}_n(F)$ on
$\mathrm{GL}_{n+1}(F)$ defined by
%$(g_1,g_2)h:=j(g_1)hj(g_2^{-1})$
$(g_1,g_2)h:=g_1 hg_2^{-1}$. Then any $\mathrm{GL}_n(F) \times
\mathrm{GL}_n(F)$ invariant distribution on $\mathrm{GL}_{n+1}(F)$
is invariant with respect to transposition.
\end{theorem}
This theorem has the following corollary in representation theory.

\begin{theorem}.
Let $(\pi,E)$ be an irreducible admissible continuous
representation of $\mathrm{GL}_{n+1}(F)$ on a Hilbert space $E$.
Then
\begin{equation}\label{dim1}
dim Hom_{\mathrm{GL}_n(F)}(E^{\infty},\cc) \leq 1
\end{equation}
\end{theorem}

Clearly, the last theorem implies in particular that (\ref{dim1})
holds for unitary irreducible representations of
$\mathrm{GL}_{n+1}(F)$. That is, the pair
$(\mathrm{GL}_{n+1}(F),\mathrm{GL}_{n}(F))$ is a generalized
Gelfand pair in the sense of \cite{vD}.

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Journal Mathematics Vol.\textbf{XL}, No2 , 285-320 (1989)
%\bibitem[Cas2]{Cas2} Casselman,W: {\it Bruhat Filtration}.
%Unpublished
\bibitem[CHM]{CHM} Casselman, William; Hecht, Henryk; Mili\v ci\'c, Dragan: \\
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\end{document}
