Tempered homogeneous spaces %(II)
Yves Benoist and Toshiyuki Kobayashi
{ \noindent Let $G$ be a semisimple real Lie group with finite center and $H$ a connected closed subgroup. We establish a geometric criterion which detects whether the representation of $G$ in $L^2(G/H)$ is tempered.
temperedII.tex
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\begin{document}
\title{
Tempered homogeneous spaces %(II)
}
\author{Yves Benoist and Toshiyuki Kobayashi
}
\date{}
\maketitle
\begin{abstract}{
\noindent Let $G$ be a semisimple real Lie group with finite center  and $H$ a connected closed subgroup.
We establish a geometric criterion which detects whether  the  representation  of $G$
in $L^2(G/H)$ is tempered.
}\end{abstract}
{\footnotesize \tableofcontents}


%1
\section{Introduction}
\label{secintrod}

This article 
is the sequel of our paper \cite{BeKoI} dealing with  harmonic analysis 
on homogeneous spaces $G/H$ of semisimple Lie groups $G$. 
In the first paper \cite{BeKoI} we studied the regular representation 
of $G$ in $L^2(G/H)$ when both $G$ and $H$ are semisimple groups.
The main result of \cite{BeKoI} is a geometric criterion which detects whether  
the representation of $G$ in $L^2(G/H)$ is tempered. 
The aim of the present paper is to extend this geometric criterion 
to the whole generality that $H$ is an arbitrary closed connected subgroup.


In this introduction we will discuss the following questions:\\
- Why do we care about tempered representations of semisimple Lie groups?\\
- What is our temperedness criterion for the homogeneous space $G/H$?\\
- What are the main ideas and ingredients in the proof of this criterion? 



%11
\subsection{Tempered representations}
\label{sectemrep}
\hspace*{0em}\let\thefootnote\relax\footnote{\hspace*{-2em}
Key words: Lie groups, homogeneous spaces, tempered 
representations, matrix coefficients,
unitary representations\\
\textit{MSC (2010):}\enspace
Primary 22E46; 
Secondary 43A85,
22F30. }
Let $G$ be a semisimple Lie group with finite center
and $K$ a maximal compact subgroup of $G$.
Understanding unitary representations of $G$
in Hilbert spaces is a major topic 
of research since the beginning of the $20^{\rm th}$ century.
Its history includes the works of Cartan, Weyl, Gelfand,
Harish-Chandra, Helgason, Langlands, Vogan and many others.
The main motivations came from quantum physics, analysis and number theory.

Among unitary representations of $G$ a smaller class called 
{\it tempered representations} plays a crucial role. 
Let us recall why they are so useful.\\
- By definition  tempered representations are those 
which are weakly contained 
in the regular representation of $G$ in $L^2(G)$.
Therefore a unitary representation of $G$ is tempered 
if and only if its disintegration into irreducible 
unitary representations involves only 
tempered representations.\\
%Moreover, the irreducible tempered representations 
%of $G$ are exactly the 
%irreducible unitary representations of $G$ 
%which occur in the desintegration of 
%the regular representation of $G$ in $L^2(G)$.\\
- Tempered representations are those for which $K$-finite matrix coefficients belong to $L^{2+\eps}(G)$ for all $\eps>0$. This definition does not look so enlightening
but equivalently these matrix coefficients are bounded by an explicit 
multiple of an explicit spherical function $\Xi$, see \cite{CHH}.\\ 
%called the Harish-Chandra function.\\ 
- Classification of irreducible tempered representations of $G$
was accomplished by
Knapp and Zuckerman in \cite{KnZu82},
while non-tempered irreducible unitary representations 
have not yet been completely understood.\\
- Tempered representations are a cornerstone of the Langlands classification 
of admissible irreducible representations of $G$
in \cite{Lan89}, see also \cite{Kn01}.\\
- Irreducible tempered representations $\pi$
can be characterized in term of the {\it leading exponents}: these  exponents
must be a positive linear combination of negative roots, see \cite[Chap.~8]{Kn01}.\\
- One can also characterize them in terms of the 
{\it distribution character of $\pi$}: this  character must
be a tempered distribution on $G$, 
see \cite[Chap.~12]{Kn01}. \\
- Tempered representations are closed under induction, restriction,
tensor product, and direct integral of unitary representations. \\
- The Kirillov--Kostant orbit methods works fairly well for tempered representations,
 see \cite{HaTo} for example. 


 



%12
\subsection{The regular representation
in $L^2(G/H)$}
\label{secregrep}


One of the most studied representations of $G$ are
the natural unitary representations of $G$ in $L^2(G/H)$
where 
$H$ is  a  closed subgroup of $G$.
When $H$ is unimodular, the space $G/H$ is implicitly endowed with 
a $G$-invariant measure and $G$ acts naturally by translation on 
$L^2$-functions. When $H$ is not unimodular,
the representation of $G$ in $L^2(G/H)$ involves an extra factor \eqref{eqnregrec} 
and is nothing but the (unitarily) induced representation
$\Ind_H^G({\bf 1})$. 

%In many interesting examples, 
%the whole disintegration of this representation in $L^2(G/H)$
%is completely understood as in
%\cite{HaCh76}, \cite{Osh88}, \cite{Del88}, \cite{BaSc05}, \cite{Kob92}...
The disintegration of $L^2(G/H)$
 is sometimes called {\textit{Plancherel formula}} or $L^2$-harmonic analysis on $G/H$. 
For instance, Harish-Chandra's celebrated Plancherel formula \cite{HaCh76}
 deals with  the case where $H=\{ e\}$.
Another case which attracted a lot of activities in the 
late $20^{\rm th}$ century is the case 
where $G/H$ is a symmetric space,
for which the disintegration of $L^2(G/H)$ is proved 
 up to a classification of (singular) discrete series representations 
 for symmetric spaces \cite{Del98, Osh88}.
The Whittaker Plancherel theorem is the disintegration of $L^2(G/H, \chi)$
 for a unitary character $\chi$ of a maximal unipotent subgroup $H$, \cite{Wallach}.  
Even when $H$ is not unimodular, 
the regular representation of $G$ in $L^2(G/H)$ is still interesting.
For instance, by Bruhat's theory, when $H$ is a parabolic subgroup of $G$, this representation is known to be
a finite sum of irreducible representations of $G$.
The decomposition of the tensor product ({\textit{fusion rule}}) is sometimes equivalent to
 the Plancherel formula for $G/H$ where $H$ is not necessarily unimodular
(see Section \ref{sectensor}).

We refer to  \cite[Intro.]{BeKoI} for more remarks on 
historical developments of the disintegration of the regular representation 
of $G$ in $L^2(G/H)$.
Getting {\textit {a priori}} information on this disintegration 
was one of the main motivation in our research of such a general criterion.
Up to our knowledge there does not exist yet any general theorem involving
simultaneously 
all these regular representations in $L^2(G/H)$ with $H$ connected.
Our temperedness criterion below seems to be the first one in that direction.
\vs



In this series of papers, we address the following question: 
\textsl{What kind of unitary representations occur
in the disintegration of $L^2(G/H)$?} More precisely, 
\textsl{when are all of them tempered?}
\vs


As noted in \cite{BeKoI}, this question has not been completely solved even for
 reductive symmetric spaces $G/H$, because the Plancherel formula 
involves a delicate algebraic problem on 
 discrete series representations for (sub)symmetric spaces
 with singular infinitesimal characters (see Example~\ref{exSp}). %, see \cite{BeKoI}.

We give a geometric necessary and sufficient condition  
on $G/H$   under which all these irreducible unitary representations 
in the Plancherel formula  are tempered, 
or equivalently under which the regular representation of $G$ in $L^2(G/H)$ is tempered.
This criterion was first discovered in our paper \cite{BeKoI}
in the special case where $H$ is a reductive subgroup of $G$.

In the present paper we extend this criterion 
to any  closed subgroup $H$ with finitely many connected components.

Formally the extended criterion is exactly the same as for $H$ reductive.
Here is a short way to state our criterion (See Theorem \ref{thmlghtem}).
\begin{equation}
\label{eqnlghtem}
 L^2(G/H)\; \mbox{\rm is tempered}
\;\;\;\Longleftrightarrow \;\;\;
\rho_{\gs h}(Y)\leq \rho_{\gs g/\gs h}(Y)
\;\;\;\mbox{\rm for all $Y\in \g h$.}
\end{equation}
Here  $\g g$ and $\g h$ are the Lie algebras of $G$ and $H$, 
and, for a $\g h$-module $V$ and $Y\in \g h$, the quantity
$\rho_V(Y)$ is
half the sum of the absolute values of the real part of the eigenvalues of $Y$ in $V$ (see Section \ref{secfunrov}).
We note that our criterion holds beyond  the (real) spherical case (\cite{ko13fm},
 cf.~\cite{SaVe17} for  non archimedean field) 
 so that  the disintegration of $L^2(G/H)$ may involve infinite multiplicities.  
We will give an explicit example of calculations of functions $\rho_V$ in 
Corollary \ref{corslpqr}.

\medskip
Our criterion \eqref{eqnlghtem} detects also whether or not 
$L^2(X)$ is tempered for any real algebraic $G$-variety $X$:
$L^2(X)$ is unitarily equivalent to 
the direct integral of the regular representations for generic orbits,
and one just has to check \eqref{eqnlghtem} at almost all orbits.

%13
\subsection{Strategy of proof}
\label{secstrpro}

The proof of \eqref{eqnlghtem} 
relies on the uniform decay of matrix coefficients
as in \cite{BeKoI}, but the main techniques are  different from those  in \cite{BeKoI}.

 To avoid any confusion we will sometimes say {\it $G$-tempered} 
for {\it tempered as a representation of $G$}.

Dealing with non-unimodular subgroups $H$ and dealing with the finitely many components of $H$ will not be a problem 
because one proves in Corollary \ref{corgh1h2}  the equivalence
\begin{eqnarray*}
\label{eqnlghghh}
L^2(G/H)\; \mbox{\rm is tempered}
&\;\;\;\Longleftrightarrow \;\;\;&
L^2(G/[H,H])\; \mbox{\rm is tempered}\\
&\;\;\;\Longleftrightarrow \;\;\;&
L^2(G/H_e)\; \mbox{\rm is tempered}
\; ,
\end{eqnarray*}
where $[H,H]$ is the derived subgroup and $H_e$ is the identity component of $H$. Therefore the temperedness of $L^2(G/H)$ depends only on $\g h$ and 
we can assume that $\g h=[\g h,\g h]$. Then 
the homogeneous space $G/H$ admits a $G$-invariant Radon measure ${\rm vol}$ and, 
according to Corollary \ref{coralpcol}, the  temperedness of $L^2(G/H)$ means that, for any 
compact subset $C$ of $G/H$, the volume ${\rm vol}(g\, C\cap C)$ is bounded 
by a multiple  of the Harish-Chandra function $\Xi$:
\begin{equation}
\label{eqnvolxig}
{\rm vol}(g\, C\cap C) \leq M_C\,\Xi(g)
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation}

To prove the direct implication 
``$L^2(G/H)$ tempered $\Longrightarrow$ $\rho_{\gs h}\leq \rho_{\gs g/\gs h}$'',
we will just estimate in Proposition \ref{prolghtem} this volume \eqref{eqnvolxig} when $C$ is a small neighborhood
of the base point of the space $G/H$.

As in \cite{BeKoI}, the converse implication  
``$\rho_{\gs h}\leq \rho_{\gs g/\gs h}$ $\Longrightarrow$ $L^2(G/H)$ tempered'',
is much harder to prove. 
Using again Corollary \ref{corgh1h2},  we can also assume that both $G$ and $H$ 
are Zariski connected algebraic groups. 
We proceed by induction on the dimension of $G$.
We introduce in Definition-Lemma \ref{defparsub} 
two intermediate subgroups $F$ and $P$:
\begin{equation}
\label{eqnFP}
          H \subset F \subset P \subset G\, .
\end{equation}
The group $P$ is a parabolic subgroup of minimal dimension that contains $H$.
We write $P=LU$ and $H=SV$ where $U$ is the unipotent radical of $P$,
$L$ is a maximal reductive subgroup of $P$,
$V\subset U$ is the unipotent radical of $H$,
and $S\subset L$ a maximal semisimple subgroup of $H$.
The group $F$ is given by $F:=SU$.

When $P$ is equal to $G$, the group $H$ is semisimple and we apply the main result 
of \cite{BeKoI}. We now assume that $P$ is a proper parabolic subgroup of $G$. 
Let $Z$ be the homogeneous space $Z=F/H\simeq U/V$ endowed with the natural $F$-action.
We denote\\ 
- by $\tau$ the regular representation of $F$ in $L^2(F/H)$,\\
- by  $\pi$ the regular representation of $P$ in $L^2(P/H)$,\\ 
- by $\Pi$ the regular representation of $G$ in $L^2(G/H)$.

Let $Z_0$ be the  same space $Z_0= U/V$ but endowed with another $F$-action, 
where $U$ acts trivially
and where $S$ acts by conjugation. 


We denote\\
- by $\tau_0$ the regular representation of $F$ in $L^2(Z_0)$,\\
- by  $\pi_0$ the regular representation of $P$ in $L^2(P\times_FZ_0)$,\\ 
- by $\Pi_0$ the regular representation of $G$ in $L^2(G\times_FZ_0)$.

Note that both the $G$-manifolds $X:=G/H \simeq G \times_F Z$ 
and $X_0 := G \times_F Z_0$ have 
a $G$-equivariant fiber bundle structure 
over the same space $G/F$ with fiber $Z \simeq Z_0$, 
but that the unitary representations of $G$ 
in $L^2(X)$ and in $L^2(X_0)$ are different. 
We want to study the representation of $G$ in  $L^2(X)$, 
whereas the induction hypothesis will give us information on  $L^2(X_0)$.
This is why we will need to  compare in \eqref{eqnvolvolf} below
the volumes in the fibers $Z$ and $Z_0$.

More precisely, the induction hypothesis combined with a reformulation
of our criterion in Proposition \ref{proequthm}
and a simple computation in Lemma \ref{lemparsub}  tell us that the representation $\pi$ is $L$-tempered.
If we knew that $\pi$ were a $P$-tempered representation 
it would be easy to conclude, using Lemma \ref{lemindtem}, that the representation 
$\Pi=\Ind_P^G\pi$ is $G$-tempered. 
However, what we know is the temperedness of $\pi_0$ and $\Ind_P^G \pi_0$, and
%In fact, since the representation $\pi_0$ is equal to $\pi$ as a 
%representation of $L$ and is trivial as a representation 
%of $U$, Lemma \ref{lemparsub2} tells us that $\pi_0$ is $P$-tempered
%and therefore the representation $\Pi_0=\Ind_P^G\pi_0$ is $G$-tempered.
 Corollary \ref{coralpcol} gives us,
for any  
compact subset $C_0$ of $G\times_FZ_0$, 
a bound:
\begin{equation}
\label{eqnvolxi0}
{\rm vol}(g\, C_0\cap C_0) \leq M_{C_0}\,\Xi(g) 
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation}
In order to deduce \eqref{eqnvolxig} from \eqref{eqnvolxi0},
we first focus on the representations $\tau$ in $L^2(Z)$ and $\tau_0$ in $L^2(Z_0)$. 
We prove in Proposition \ref{prodom},
for every compact subset $D$ of the fiber $Z= F/H$, a uniform
 estimate of ${\rm vol}(f D \cap D)$ 
with respect to translates of the element $f\in F$ by elements of the unipotent subgroup $U$.
Namely,  there exists a compact subset 
$D_0\subset Z_0$ such that 
\begin{equation}
\label{eqnvolvolf}
{\rm vol}(fD\cap D) \leq {\rm vol}(fD_0\cap D_0)
\;\;\;\; \mbox{\rm for all $f$ in $F$.}
\end{equation}
Since $\Pi=\Ind_F^G \tau$ and $\Pi_0=\Ind_F^G \tau_0$, 
we deduce  from \eqref{eqnvolvolf} in Proposition \ref{provolgcc} that 
for every compact subset $C\subset G/H \simeq G\times_F Z$, there exists a compact subset 
$C_0\subset G\times_FZ_0$ such that 
\begin{equation}
\label{eqnvolvolg}
{\rm vol}(g\, C\cap C) \leq {\rm vol}(g\, C_0\cap C_0)
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation}
And \eqref{eqnvolxig} follows 
from \eqref{eqnvolxi0} and \eqref{eqnvolvolg}. This ends the
sketch of the  proof 
of the criterion \eqref{eqnlghtem}. 
The details are explained below. 
\vs

Here is the organization of the paper.

In Section \ref{secdefres} we recall the basic definition
and state precisely our criterion.

In Sections \ref{secprepro} we collect the parts 
of the proof which do not involve the intermediate parabolic 
subgroup $P$. 
It includes the proof for the necessity of the inequality $\rho_{\gs h}\leq \rho_{\gs g/\gs h}$,
 and a formulation of the
% (apparently, more general)
 temperedness criterion for $\Ind_H^G(L^2(V))$ (Theorem~\ref{thmlgvtem}).
 

In Section \ref{secparsub} we introduce the intermediate subgroups $F$ and $P$ in \eqref{eqnFP},
 and detail how they work to conclude the proof for
 the hard part, {\textit{i.e.}}, the sufficiency of  the inequality $\rho_{\gs h}\leq \rho_{\gs g/\gs h}$.
% $\Longrightarrow$ $L^2(G/H)$ tempered''.

In Section \ref{secexalgh} we give a few examples which enlight the efficiency 
of our criterion. 



%2
\section{Definition and main result}
\label{secdefres}

We collect in this chapter a few well-known facts  on tempered representations, on almost $L^2$ represen\-ta\-tions,
and on uniform decay of matrix coefficients. 


%21
\subsection{Regular and induced representations}
\label{secregind}
\bq
We first recall the construction of 
the regular representations and the 
induced representations. 
\eq

\subsubsection{Regular representations}
Let $G$ be a locally compact group,
$X$ be a locally compact space endowed with a continuous action of $G$
and let $\nu_X$ be a Radon measure on $X$.

When the $G$-action preserves the measure $\nu_X$, 
one has a natural unitary representation $\la_X$ of $G$ in the 
Hilbert space $L^2(X):=L^2(X,\nu_X)$  called the 
{\it regular representation} and given by
\begin{equation*}
\label{eqnregrep}
(\la_X(g)\varphi)(x)=\varphi(g^{-1}x)
\;\;\;\mbox{\rm for $g$ in $G$, $\varphi$ in $L^2(X)$ and $x$ in $X$.}
\end{equation*} 

When the class of the measure $\nu_X$ is $G$-invariant, one still has a  
natural unitary representation $\pi$ of $G$ in the 
Hilbert space $L^2(X):=L^2(X,\nu_X)$ also called the 
{\it regular representation}. The formula will involve the 
Radon--Nikodym cocycle $c(g,x)$ which is defined,
for all $g$ in $G$ and $\nu_X$-almost all $x$ in $X$ 
by the equality
\begin{equation}
\label{eqnradnik}
g_*\nu_X=c(g^{-1},x)\,\nu_X\, .
\end{equation} 

The regular representation of $G$ in $L^2(X)$ is then given by  
\begin{equation}
\label{eqnregrec}
(\la_X(g)\varphi)(x)=c(g^{-1},x)^{1/2}\,\varphi(g^{-1}x)
\;\;\;\mbox{\rm for $g$ in $G$, $\varphi$ in $L^2(X)$, $x$ in $X$.}
\end{equation} 

\subsubsection{Induced representations}
Assume now that $X$ is a homogeneous space $G/H$ 
where $H$ is a closed subgroup of $G$.
One can choose a $G$-invariant 
Radon measure on $G/H$
if and only if the modular function of $G$ coincides on $H$ with that of $H$,
\begin{equation*}
\label{eqndegdeh}
\De_G(h) =\De_H(h)
\;\;\;\mbox{\rm for all $h$ in $H$.}
\end{equation*}
In general there  always
exists  a measure $\nu$ on $G/H$ 
whose class is $G$-invariant, and the regular representation 
of $G$ in $L^2(G/H)$ is the induced representation of the trivial representation of $H$
\begin{equation*}
\label{eqnregind}
\la_{G/H}=\Ind_H^G({\bf{1}}).
\end{equation*}
More generally, for any unitary representation $\pi$ of $H$,
one defines the (unitarily) induced representation 
$\Pi:=\Ind_H^G(\pi)$ in the following way.
The projection 
$$
G\longrightarrow G/H
$$ 
is a principal bundle with structure group $H$. 
We fix
a $G$-equivariant Borel measurable trivialization of this principal bundle 
\begin{equation}
\label{eqngsighh}
G\simeq G/H\times H
\end{equation}
which sends relatively compact subsets to relatively compact subsets.
The action of $G$ by left multiplication through this trivialization can be read as 
\begin{equation*}
\label{eqngxhgxs}
g\,(x,h)=(gx,\si(g,x)h)
\;\;\;\;\mbox{\rm 
for all $g\in G$, $x\in G/H$ and $h\in H$,}
\end{equation*}
where $\si\colon G\times G/H\rightarrow H$ 
is a Borel measurable cocycle. 


The space of the representation $\Pi$ is the space
$\mc H_{\Pi}:=L^2(G/H;\mc H_{\pi})$ of $\mc H_\pi$-valued $L^2$-functions on $G/H$ and the action of $G$ is given,
for $g$ in $G$, $\psi$ in $\mc H_{\Pi}$, $x$ in $G/H$, by
\begin{equation*}
\label{eqnindrep}
(\Pi(g)\psi)(x)=c(g^{-1},x)^{1/2}\,\pi(\si(g,g^{-1}x))\psi(g^{-1}x),
\end{equation*} 
where $c$ is again the Radon--Nikodym cocycle \eqref{eqnradnik}. 

\subsubsection{Induced actions}
When the closed subgroup $H$ of  $G$ is acting continuously 
on a locally compact space $Z$ one can define the induced action
of $G$ on the fibered space 
$$
G\times_H Z:=  (G\times Z)/H
$$
where the quotient is taken for the right $H$-action 
$(g,z)h=(gh,h^{-1}z)$ and where the $G$-action is given by
$g_0\,(g,z)=(g_0g,z)$, for all $g_0$, $g$ in $G$, $z$ in $Z$ and $h$ in $H$.
Using \eqref{eqngsighh}, one gets a $G$-equivariant Borel measurable trivialization of this fibered space
$$
G\times_H Z\simeq G/H\times Z.
$$ 
Through this identification, the $G$-action
is given by 
$$
g(x,z)=(gx,\si(g,x)z)\, 
\;\;\;\mbox{\rm 
for all $g\in G$, $x\in G/H$ and $z\in Z$.}
$$
When the $H$-action preserves the class of a measure $\nu_Z$ on $Z$, 
the $G$-action preserves the class of the measure $\nu_X:=\nu\otimes\nu_Z$.
In this case the regular representation of $G$ in $L^2(G\times_HZ)$ is unitarily equivalent to the 
induced representation of the regular representation of $H$ in $Z$:
\begin{equation}
\label{eqnrexind} 
L^2(G\times_H Z)\simeq \Ind_H^G(L^2(Z))
\;\;\;\mbox{as unitary representations of $G$.}
\end{equation} 

%22
\subsection{Decay of matrix coefficients}
\label{sectemrep2}

\bq
We now recall the control of the matrix coefficients of tempered 
representations of a semisimple Lie group.
\eq


\subsubsection{Tempered representations}
Let $G$ be a locally compact group and $\pi$ be
a unitary representation  of $G$ in a Hilbert space ${\mathcal H}_\pi $.
All  representations $\pi$ of $G$ will be assumed to be continuous 
{\it{i.e.}} the map $G\ra\mc H_\pi ,\, g\mapsto \pi(g)v$ is continuous
for all $v$ in $\mc H_\pi$.
The notion of tempered representation is due to Harish-Chandra.


\begin{Def}
\label{deftem}
The unitary representation $\pi$ is said to be tempered
or $G$-tempered
if $\pi$ is weakly contained 
in the regular representation $\lambda_G$ of $G$ in $L^2(G)$ {\it{i.e.}} if every matrix coefficient of $\pi$ 
is a uniform limit on every compact subset of $G$  of a sequence of 
sums of matrix coefficients of $\lambda_G$. 
\end{Def}
We refer to \cite[Appendix F]{BeHaVa} for more details on weak containments.

\begin{Rem}
\label{remtem}
The notion of temperedness is stable by passage to a finite index subgroup 
$G'$ of $ G$, {\it{i.e.}}
a unitary representation $\pi$ of  $G$ is tempered if and only if $\pi$
is tempered as a representation of $G'$.
\end{Rem}


This notion is also preserved by induction. 

\begin{Lem}
\label{lemindtem}
Let $G$ be a locally compact group, $H$ be a closed subgroup of $G$
and $\pi$ be a unitary representation of $H$.
If $\pi$ is $H$-tempered then the induced representation
$\Ind_H^G(\pi)$ is $G$-tempered.
\end{Lem}

\begin{proof} 
Since the $H$-representation 
$\pi$ is weakly contained in the regular representation $\la_H$
of $H$,
the $G$-representation $\Ind_H^G(\pi)$ 
is weakly contained in the regular representation $\la_G=\Ind_H^G(\la_H)$, and 
hence  is $G$-tempered.  
\end{proof}

\begin{Rem}
\label{remtemame}
$1)$ When $G$ is amenable, according to the Hulanicki--Reiter theorem 
in \cite[Th.~G.3.2]{BeHaVa}, every unitary representation of $G$ is tempered. 

\noindent 
$2)$ When $G$ is a product of two closed subgroups $G=SZ$ with $Z$ central, {\it a unitary representation $\pi$ of $G$ is 
$G$-tempered if and only if it is $S$-tempered.}
Indeed the regular representation of $G$ in $L^2(G)$ is clearly 
$S$-tempered. Conversely, we want to prove that any unitary representation $\pi$ of $G$ which is $S$-tempered 
is also $G$-tempered.
We can assume that $\pi$ is $G$-irreducible. The action of $Z$ in this representation is given by a unitary character $\chi$ and $\pi$ 
is weakly contained in the representation $\Ind_Z^G\chi$. 
Since $\chi$ is $Z$-tempered, this representation is $G$-tempered. 
\end{Rem}

\subsubsection{Matrix coefficients}
Let  now $G$ be a semisimple Lie group (always implicitly assumed to be real Lie groups with finitely many connected components and whose identity component has finite center). 

\begin{Def}\label{defwl2}
A unitary representation $\pi$ of $G$ is said to be almost $L^2$
if there exists  a dense subset $\mc D\subset{\mathcal H}_\pi $ 
for which the matrix coefficients
 $ g\mapsto \langle  \pi(g)v_1,v_2\rangle $ are in
$L^{2+\varepsilon}(G)$ for all $\varepsilon>0$ and all $v_1$, $v_2$ in $\mc D$. 
\end{Def}

We fix a maximal compact subgroup $K$ of $G$.
Let $\Xi$ be the Harish-Chandra spherical function on $G$ 
(see \cite{CHH}).
By definition, $\Xi$ is  the matrix coefficient of a normalized $K$-invariant vector $v_0$
of the spherical unitary principal representation 
$\pi_0={\Ind}_{P_{{\operatorname{min}}}}^G({\bf 1}_{P_{{\operatorname{min}}}})$ 
where $P_{\operatorname{min}}$ is a minimal parabolic subgroup of $G$.
That is 
\begin{equation}
\label{eqnxighar}
\Xi(g)=\langle\pi_0(g)v_0,v_0\rangle
\;\; 
\mbox{\rm for all $g$ in $G$}. 
\end{equation}
Since $P_{\operatorname{min}}$ is amenable, the representation $\pi_0$ is $G$-tempered. Moreover, the function $\Xi$ belongs to $L^{2+\varepsilon}(G)$ for all $\varepsilon>0$ 
(see \cite[Prop.~7.15]{Kn01}).
We will need the following  much more precise version of
this fact. 



\begin{Prop} 
[Cowling, Haagerup and Howe {\cite{CHH}}]
\label{protemal2}
Let $G$ be a connected semisimple Lie 
group with finite center and $\pi$
be a unitary representation  of $G$. The following are equivalent:\\
$(i)$ the representation $\pi$ is tempered,\\
$(ii)$ the representation $\pi$ is almost $L^2$,\\
$(iii)$ for every $K$-finite vectors $v$, $w$ in ${\mathcal H}_\pi $, for every $g$ in $G$,
one has 
$$
|\langle  \pi(g)v,w\rangle |\leq \Xi(g) 
\| v\| \| w\| (\dim \langle  K v\rangle )^\frac12 (\dim \langle  K w\rangle )^\frac12.
$$
\end{Prop}

See \cite[Thms.~1, 2 and Cor.]{CHH}.
See also  \cite{HoTa}, \cite{Nev98} and \cite{Oh1} for other applications
of Proposition \ref{protemal2}.

For  regular representations this proposition becomes:

\begin{Cor}\label{coralpcol}
Let $G$ be a connected semisimple   Lie 
group with finite center
and $X$ a locally compact space endowed with a continuous action of $G$ 
preserving a Radon measure ${\rm vol}$.
The regular representation of $G$ in $L^2(X)$  is tempered if and only if,
for any  compact subset 
$C$ of $X$, the function $g\mapsto {\rm vol} (g\, C\cap C)$ 
belongs to $L^{2+\eps}(G)$
for all $\eps>0$.

In this case, when $C$ is $K$-invariant,
one has
\begin{equation}
\label{eqnvolgcc}
{\rm vol} (g\, C\cap C)\leq {\rm vol}( C)\;\Xi(g) 
\;\;\;\mbox{\rm for all $g$ in $G$}.
\end{equation}
\end{Cor}

The notation  $g\, C$ denotes the set  $g\, C: =\{gx\in X: x\in C\}$.

\begin{proof}
Note  that  a compact subset $C$ of $X$ is always included
in a $K$-invariant compact subset $C_0$, that
the function ${\bf 1}_{C_0}$ is a $K$-invariant vector in $L^2(X)$ and that 
$$
\langle \la_X(g){\bf 1}_{C_{_0}},{\bf 1}_{C_{_0}}\rangle =
{\rm vol} (g\, C_0\cap C_0)
\;\;\;\mbox{\rm for all $g$ in $G$}.
$$
Note also that  the 
functions  ${\bf 1}_{C}$
span a  dense subspace in $L^2(X)$.
\end{proof}




%23
\subsection{The function $\rho_V$}
\label{secfunrov}

\bq
We now define the functions $\rho_\gs h$ and $\rho_{\gs g/\gs h}$ 
occurring in the temperedness criterion, explain how to compute them  
and emphasize
their geometric meaning.
\eq


When $H$ is a  Lie group we denote by the corresponding gothic letter 
$\g h$ the Lie algebra of $H$. Let $V$ be a real 
finite-dimensional representation of $H$.
For an element $Y$ in $\g h$,
we consider the eigenvalues of $Y$ in $V$
(more precisely in the complexification $V_{\m C}$) and
we denote by $V_+$, $V_0$ and $V_-$ 
the largest vector subspaces of $V$ on which 
the real part of all the eigenvalues of  $Y$
are respectively positive, zero and negative.
One has the decomposition 
$V=V_+\oplus V_0\oplus V_-$.
We define the non-negative functions  
$\rho^+_V$ and $\rho_V$ on $\g h$ by
\begin{eqnarray*}
\label{eqnrhopmv}
\rho^+_V(Y)
&: =& {\rm {\operatorname{Tr}}}(Y|_{V_+}),\\
\rho_V(Y)
&: =& \tfrac12\,\rho^+_V(Y)+\tfrac12\,\rho^+_V(-Y),
\end{eqnarray*}
where ${\rm {\operatorname{Tr}}} $ denotes the trace of a matrix. Note that one has the equality
$
{\rm {\operatorname{Tr}}}(Y|_{V_-})=-\rho^+_V(-Y).
$

By definition, one always has the equality
$\rho_V(-Y) = \rho_V(Y).$ 
Moreover, when the action of $H$ on $V$ is volume preserving
one has the equality
\begin{equation*}
\label{eqnrhovpy}
\rho_V(Y) = \rho^+_V(Y).
\end{equation*}
The function called $\rho_V$  in \cite[Sec. 3.1]{BeKoI} is what we call now $\rho_V^+$. It coincides with our
$\rho_V$  since in \cite{BeKoI} we only need to consider volume preserving actions.  


Since this function $\rho_V \colon \g h \to \mathbb{R}_{\ge0}$ 
plays a crucial role in our criterion,
we begin by a few trivial but useful comments, which make  it easy to compute when dealing with examples.
To simplify these comments, we assume that $H$ is an algebraic subgroup of ${\rm GL}(V)$.
Let $\g a=\g a_{\g h}$ be a maximal split abelian Lie subalgebra of $\g h$ 
{\it{i.e.}} the Lie subalgebra of a 
maximal split torus $A$ of $H$.
Any element $Y$ in $\g h$ admits a  unique  Jordan decomposition
$Y=Y_e+Y_h+Y_n$ as a sum of three commuting elements of $\g h$
where $Y_e$ is a semisimple matrix with imaginary eigenvalues,
$Y_h$ is a semisimple matrix with real eigenvalues and 
$Y_n$ is a nilpotent matrix. Moreover there exists an element 
$\la_Y$ in $\g a$ which is $H$ conjugate to $Y_h$.
Then one has the equality
\begin{equation*}
\label{eqnrhovly}
\rho_V(Y) = \rho_V(\la_Y)
\;\;\mbox{for all $Y$ in $\g h$.} 
\end{equation*}
This equality tells us that the function $\rho_V$ is completely determined by its restriction to $\g a$.   


This function $\rho_V\colon \g a \to \mathbb{R}_{\geq 0}$ is 
continuous and is piecewise linear
{\it{i.e.}} there exist finitely many convex polyhedral cones 
which cover $\g a$ and on which $\rho_V$ is linear.
Indeed, let $P_V$ be the set of weights of $\g a$ in $V$ 
and, for all $\al$ in $P_V$, let $m_\al:=\dim V_{\al}$ 
be the dimension of the corresponding 
weight space. Then one has the equality
\begin{equation}\label{eqnrhovys}
\rho_V(Y) = \tfrac12\sum_{\al\in P_V}m_\al|\al(Y)|
\;\;\;\mbox{for all $Y$ in $\g a$.} 
\end{equation}




For example, when $\g h$ is semisimple and $V = \g h$
 via the adjoint action,
our function $\rho_{\gs h}$ is equal on each  positive Weyl chamber $\g a_+$
of $\g a$ to the sum of the corresponding positive roots {\it{i.e.}} to twice the usual ``$\rho$'' 
linear form.
For other representations $V$, the maximal convex polyhedral cones on which 
$\rho_V$ is linear are most often much smaller than the Weyl chambers. 
Explicit computations of the functions $\rho_V$ will be given in Section \ref{secexalgh}.  
\vs

The geometric meaning of this function $\rho_V$ is given by the following elementary Lemma
as in \cite[Prop. 3.6]{BeKoI}.





\begin{Lem}
\label{lemrovvol}
Let $V=\m R^d$. Let 
$\g a$ be an abelian split Lie subalgebra of ${\rm End}(V)$ and $C$ be a compact neighborhood of $0$ in $V$.
Then there exist constants $m_{_C } >0$ , $M_{_C }  > 0$ such that
\[
m_{_C } e^{-\rho_V(Y)}
\le e^{-\Tr(Y)/2}\,{\rm vol}(e^Y C \cap C)
\le M_{_C }  e^{-\rho_V(Y)}
\;\;\mbox{\rm for all $Y \in \g a$. 
}
\]
\end{Lem}
Such a  factor $e^{-\Tr(Y)/2}$ occurs 
in computing the matrix coefficient of the vector 
${\bf 1}_C$ in the 
regular representation  
$L^2(V)$ when the action  on $V$  does not preserve the  volume.
Here ${\rm vol}$ denotes the volume with respect to
the Lebesgue measure on $V$.
The proof of Lemma \ref{lemrovvol} goes similarly to that of  \cite[Prop. 3.6]{BeKoI} which deals with 
the case where the action is volume preserving.

%24
\subsection{Temperedness criterion for $L^2(G/H)$}
\label{sectemcri}

\bq
We can now state precisely our temperedness criterion.
\eq

Let $G$ be a  semisimple Lie group
and $H$ a closed  subgroup of $G$.
Let $\g g$ and $\g h$ be the Lie algebras of $G$ and $H$.
The temperedness criterion for the regular representation of $G$ 
in $L^2(G/H)$ will involve 
the functions
$\rho_\gs h$ and $\rho_{\gs g/\gs h}$
for the $H$-modules $V=\g h$ and $V=\g g/\g h$.

\begin{Thm}
\label{thmlghtem}
Let $G$ be a connected semisimple Lie group with finite center,
$H$ a closed connected subgroup of $G$.
Then, one has the equivalence~:\\
\centerline{$L^2(G/H)$ is $G$-tempered\;\;\;
$\Longleftrightarrow$\;
$\rho_{\gs h}  \le \rho_{\gs g/\gs h} $.}
\end{Thm}

\begin{Rem} 
The assumption that $G$ and $H$ are connected are not very important. As we shall explain in Corollary \ref{corgh1h2}, Theorem \ref{thmlghtem} is still true when $G$ and $H$ have finitely many connected components as soon as 
the identity component $G_e$ has finite center.
\end{Rem}

\begin{Rem} 
When $H$ is algebraic and 
$\g a$ is a maximal abelian split Lie subalgebra of $\g h$,
Inequality
$\rho_{\gs h}  \le \rho_{\gs g/\gs h} $ holds on $\g h$
if and only if it holds
on~$\g a$.
\end{Rem}

\begin{Rem} 
When $H$ is a minimal parabolic subgroup of $G$ the representation of 
$G$ in $L^2(G/H)$ is tempered because the group $H$ is amenable. 
Our criterion is easy to check in this case since the functions  
$\rho_{\gs h}$ and $\rho_{\gs g/\gs h}$ are equal. 
This example explains why, when $H$ is non-unimodular, our temperedness criterion involves 
the functions $\rho_V$ instead of the functions $\rho^+_V$.
\end{Rem}



%3
\section{Preliminary proofs}
\label{secprepro}

In this section we state a  useful reformulation of Theorem \ref{thmlghtem}
and prove the direct  implication in Theorem \ref{thmlghtem}.


%31
\subsection{The Herz majoration principle}
\label{sechermaj}

\bq
We first explain how to reduce the proof of Theorem \ref{thmlghtem}
to the case where both $G$ and $H$ are algebraic and
how to deal with groups having finitely many connected components.
\eq

\begin{Prop} 
\label{progh1h2}
Let $G$ be a semisimple Lie group with finitely many compo\-nents
such that the identity component $G_e$ has finite center
and
$H'\subset H $ two closed subgroups of $G$.\\
$1)$ If $L^2(G/H)$ is $G$-tempered then $L^2(G/H')$ is $G$-tempered.\\
$2)$ The converse is true when $H'$ is normal in $H$ and $H/H'$ is amenable
(for instance finite,  compact, or abelian).
\end{Prop}

\begin{Lem} 
\label{lemgh1h2}
Let $G$ be a semisimple Lie group with finitely many connected components
such that $G_e$ has finite center,
and $H$ be a closed subgroup of $G$.
If the regular representation in $L^2(G/H)$ is $G$-tempered then the induced representation $\Pi=\Ind_H^G(\pi)$ is also $G$-tempered
for any  unitary representation $\pi$ of $H$.
\end{Lem}

\begin{proof}[Proof of Lemma \ref{lemgh1h2}] This  classical lemma is called ``Herz majoration principle''
(see \cite[Chap.~6]{BeGu}). 
We recall the short argument  since it will be very useful in Proposition \ref{provolgcc}. 
We use freely the notation of Section \ref{secdefres}.
For a function $\ph$ in the space $L^2(G/H,\mc H_\pi)$ 
of the induced representation $\Pi=\Ind_H^G(\pi)$, we denote by   $|\ph|$
the function in the space $L^2(G/H)$ of the regular representation 
$\Pi_0=\Ind_H^G({\bf{1}})$ given by $|\ph|(x):=\|\ph(x)\|$ for   $x$ in $G/H$.
The space $\mc D$ of bounded functions with compact support 
is dense in $L^2(G/H,\mc H_\pi)$. 
For $\ph$ and $\psi$ in $\mc D$, one can compute the matrix coefficients
\begin{eqnarray*}
\langle\Pi(g)\ph,\psi\rangle
&=&\int_{G/H}c(g^{-1},x)^{1/2}\langle\pi(\si(g,g^{-1}x)) \ph(g^{-1} x),\psi(x )\rangle \rmd\nu(x), \\
%\end{eqnarray*}
%Therefore, one gets the bounds
%\begin{eqnarray*}
|\langle\Pi(g)\ph,\psi\rangle|
& \leq&\int_{G/H}c(g^{-1},x)^{1/2} \|\ph (g^{-1} x)\|\,\|\psi(x )\| \rmd\nu(x)\\
& \leq& \langle\Pi_0(g)|\ph|,|\psi|\rangle  .
\end{eqnarray*}
Since $\Pi_0$ is tempered, these matrix coefficients belong to $L^{2+\eps}(G)$ 
for all $\eps>0$. 
Therefore the representation $\Pi$ is almost $L^2$ and hence is  $G$-tempered by Proposition \ref{protemal2}.
\end{proof}


\begin{proof}[Proof of Proposition \ref{progh1h2}] 
$1)$ This follows from Lemma \ref{lemgh1h2} applied to the regular representation $\pi$ of $H$ in $L^2(H/H')$.

$2)$ Since $H/H'$ is amenable, the trivial representation 
of $H$ is weakly contained in the regular representation 
of $H$ in $L^2(H/H')$.
Therefore, inducing to $G$, 
the regular representation of $G$ in $L^2(G/H)$ is 
weakly contained in the 
regular representation of $G$ in $L^2(G/H')$
and hence is $G$-tempered.
\end{proof}



The following corollary tells us that the temperedness of $L^2(G/H)$
depends only on the Lie algebras $\g g$, $\g h$ 
and does not change if we replace $\g h$ by its derived Lie algebra 
$[\g h,\g h]$. 


\begin{Cor} 
\label{corgh1h2}
Let $G$ be a semisimple Lie group with finitely many connected components
such that $G_e$ has a finite center $Z_G$
and
$H$ be a closed subgroup with finitely many connected components.
Then the following are equivalent.\\
\centerline{$
(i) \mbox{ $L^2(G/H)$ is $G$-tempered}
\Longleftrightarrow
(ii) \mbox{ $L^2(G_e/H_e)$ is $G_e$-tempered}
\Longleftrightarrow
$}
\centerline{$
(iii) \mbox{ $L^2(G/HZ_G)$ is $G/Z_G$-tempered}
\Longleftrightarrow
(iv)  \mbox{ $L^2(G/[H,H])$ is $G$-tempered.}
$}
\end{Cor}

\begin{proof} 
This follows from Proposition \ref{progh1h2} 
and Remark \ref{remtem} since 
the  quotients $H/H_e$, $HZ_G/H$ and $H/[H,H]$ 
are amenable groups.
\end{proof}

\begin{Rem}
\label{remredalg}
Corollary \ref{corgh1h2} is useful to reduce the proof 
of 
Theorem \ref{thmlghtem} to the case where both $G$ and $H$ are algebraic groups.

Indeed, every semisimple Lie algebra $\g g$ is the Lie algebra of an algebraic group: the group ${\rm Aut}(\g g)$. Therefore, 
using $(i)\Leftrightarrow (ii)\Leftrightarrow (iii)$,
we can assume that $G$ is algebraic.

Moreover, by Chevalley's ``th\'eorie des repliques'' in \cite{Che48}, for any closed subgroup $H$ of an algebraic group $G$,
there exists two algebraic subgroups $H_1$ and $H_2$ of $G$
whose Lie algebras satisfy 
$$\g h_1\subset \g h \subset \g h_2
\;\;{\rm and}\;\;\;
\g h_1 = [\g h,\g h]= [\g h_2,\g h_2].
$$
Therefore, 
using $(i)\Leftrightarrow (iv)$, we can assume that $H$ is an algebraic subgroup.
\end{Rem}

\begin{Rem}
\label{remreduni}
Since the  group  $[H,H]$ is   unimodular, Corollary \ref{corgh1h2} is also useful to reduce the proof 
of 
Theorem \ref{thmlghtem} to the case where  $H$ is unimodular.
\end{Rem}


%32
\subsection{A strengthening of the main theorem}
\label{secstrthm}

\bq
Theorem \ref{thmlghtem} will be proven by induction on the dimension of $G$.
This induction process forces us to prove simultaneously an apparently 
stronger theorem which involves $L^2(V)$-valued sections over $G/H$ associated to
a finite-dimensional $H$-module $V$.
\eq

\begin{Thm}
\label{thmlgvtem}
Let $G$ be an algebraic semisimple Lie group,
$H$ an algebraic subgroup of $G$ and 
$V$ a real finite-dimensional algebraic representation of $H$.
Then, one has the equivalence~:\\
\centerline{$\Ind_H^G(L^2(V))$ is $G$-tempered\;\;\;
$\Longleftrightarrow$\;
$\rho_{\gs h}  \le \rho_{\gs g/\gs h} +2\,\rho_V $.}
\end{Thm}

Again, we only need to check this 
inequality on a maximal split abelian Lie subalgebra $\g a$ of $\g h$.
Note also that, by Remark \ref{remredalg}, 
Theorem \ref{thmlghtem} is the special case of 
Theorem \ref{thmlgvtem} where $V=\{ 0\}$.


%33
\subsection{The direct implication}
\label{secdirimp}

\bq
We  first prove the direct implication in Theorems \ref{thmlghtem} and \ref{thmlgvtem}.
\eq

From now on, we will set $\g q:=\g g/\g h$.

\begin{Prop}
\label{prolghtem}
Let $G$ be an algebraic semisimple Lie group,
$H$ an algebraic subgroup of $G$ and $V$ an algebraic representation of $H$.
If the representation $\Pi=\Ind_H^G(L^2(V))$ is $G$-tempered then one has 
$\rho_{\gs h}  \le \rho_{\gs q} +2\rho_V$.
\end{Prop}

\begin{proof}
By \eqref{eqnrexind}  this representation $\Pi$ is also the regular representation of the $G$-space
$X:=G\times_HV$. 
Let $A$ be a maximal split torus of $H$ and $\g a$ be the Lie algebra of $A$.
We choose an $A$-invariant decomposition 
$\g g=\g h\oplus \g q_0$ and small closed balls $B_0\subset \g q_0$ and
$B_V\subset V$ centered at $0$. We can see $B_V$ as a subset of $X$ and the map
\begin{eqnarray}
\label{eqnbobvgv}
B_0\times B_V \longrightarrow G\times_H V, 
\qquad
(u,v)\mapsto \exp (u)v 
\nonumber
\end{eqnarray}
is a homeomorphism onto its image $C$. 
Since $\Pi$ is tempered one has a bound
as in   \eqref{eqnvolgcc}
\begin{equation}
\label{eqnpigxig}
\langle\Pi(g )1_C,1_C\rangle\leq M_C\; \Xi(g)
\;\;\;\mbox{\rm for all $g$ in $G$}.
\end{equation}
We will exploit this bound for elements $g=e^Y$ with $Y$ in $\g a$.
In our coordinate system \eqref{eqnbobvgv} we can choose 
the measure $\nu_X$ to coincide with the Lebesgue measure on 
$\g q_0\oplus V$. Taking into account the Radon--Nykodim derivative and the $A$-invariance of $\g q_0$,
one computes
\begin{eqnarray*}
\label{eqnpigvol}
\langle\Pi(e^Y)1_C,1_C\rangle
\!\!&\geq &\!\! 
e^{-{\Tr}_{\gs q_0}(Y)/2} e^{-{\Tr}_{V}(Y)/2}\;
{\rm vol}_{\gs q_0}(e^YB_0\cap B_0)\;{\rm vol}_V(e^YB_V\cap B_V),
\end{eqnarray*}
and therefore, using Lemma \ref{lemrovvol}, one deduces
\begin{eqnarray}
\label{eqnpigrho}
\langle\Pi(e^Y )1_C,1_C\rangle
&\geq &m_{_C}\,e^{-\rho_{\gs q}(Y) } e^{-\rho_V(Y)} 
\;\;\;\mbox{\rm for all $Y$ in $\g a$.} 
\end{eqnarray}
Combining  \eqref{eqnpigxig} and \eqref{eqnpigrho} 
with known bounds for the 
spherical function $\Xi$ as in \cite[Prop 7.15]{Kn01},
one gets, for suitable positive constants $d$, $C$,
\begin{equation*}
\label{eqnroxiro}
\frac{m_C}{M_C}e^{-\rho_{\gs q}(Y)  -\rho_V(Y) }\leq \Xi(e^Y)\leq M_0\;(1+\|Y\|)^{d}e^{-\rho_{\gs g}(Y)/2 }  
\;\;\;\mbox{\rm for all $Y$ in $\g a$.} 
\end{equation*}
Therefore one has $\rho_{\gs g}\leq 2\,\rho_{\gs q}+2\,\rho_V$, 
and hence  $\rho_{\gs h}\leq  \rho_{\gs q}+2\,\rho_V$ as required.
\end{proof}

%34
\subsection{Equivalence of the main theorems}
\label{secequthm}
\bq
We have already noticed that 
Theorem \ref{thmlghtem} is a special case of Theorem \ref{thmlgvtem}.
We explain now why Theorem \ref{thmlgvtem} is a consequence of 
Theorem \ref{thmlghtem}.
\eq

\begin{Prop}
\label{proequthm}
Let $G$ be an algebraic semisimple Lie group.
If the conclusion of Theorem \ref{thmlghtem} is true for all  algebraic subgroups $H$ of $G$, then 
the conclusion of Theorem \ref{thmlgvtem} is also true for all  algebraic subgroups $H$ of $G$.
\end{Prop}

The proof relies on the following lemma.

\begin{Lem}
\label{lemtratri}
Let $H$ be a Lie group and $V$ a finite-dimensional representation of $H$.
Let $v\in V$ be a point  whose orbit $Hv$ has maximal dimension and $H_v$ be the stabilizer of $v$ in $H$. 
Then the action of $H_v$ on  $V/\g h v$ is trivial.  
\end{Lem}

\begin{proof}[Proof of Lemma \ref{lemtratri}]
Assume by contradiction that there exist $Y$ in $\g h$ 
and $w$ in $V$ such that the vector 
$Yw$ does not belong to $\g h v$.

Choose a complementary subspace $\g m$ of $\g h_v$ in $\g h$ so that $\g h= \g h_v\oplus \g m$.
Choose also a point $v_\eps= v+\eps w$ 
near $v$.  For $\eps$ small,
the tangent space $\g h\, v_\eps$ to the orbit
$H\, v_\eps$ contains both the subspace $\g m\, v_\eps$
which is near $\g m \,v=\g h\, v$
and the vector $\eps^{-1}Yv_\eps=Y w$.
Therefore, for $\eps$ small, one has the inequality
$\dim \g h v_\eps>\dim \g h v$ which gives us a contradiction.
\end{proof}

\begin{proof}[Proof of Proposition \ref{proequthm}] We assume that $\rho_{\gs h}\leq  \rho_{\gs q}+2\,\rho_V$ and we want to prove, using Theorem \ref{thmlghtem}, that the regular representation of $G$
in $L^2(G\times_H V)$ is tempered.
%Let $V'\subset V$ be the union of the $H$-orbits of 
%maximal dimension. This set $V'$ is open and has full measure.
Since the action is algebraic, there exists a Borel measurable subset 
$T\subset V$ which meets each of these $H$-orbits in exactly one point.
Let $\nu_V$ be a probability measure on $V$ with positive density and $\nu_T$ be the probability measure on $T\simeq H\backslash V$ given as the image of $\nu_V$.
One has a direct integral decomposition 
of the regular representation
\begin{equation*}
\label{eqnregint}
L^2(G\times_H V) =\int_T^{\oplus}L^2(G/H_v)\rmd\nu_T(v)
\end{equation*}
where $H_v$ is the stabilizer of $v$ in $H$.  
Since the direct integral of tempered represen\-tations is tempered,
we only need to prove that, for $\nu_T$-almost all $v$ in $T$,
\begin{equation}
\label{eqnghvtem}
L^2(G/H_v) 
\;\;\mbox{is $G$-tempered.}
\end{equation}
Our assumption implies that 
\begin{equation}
\label{eqnrhohqv}
\rho_{\gs h}(Y)\leq  \rho_{\gs q}(Y)+2\,\rho_V(Y)
\;\;\mbox{for all $Y$ in $\g h_v$}
\end{equation}
For $\nu_T$-almost all $v$ in $T$, the orbit $Hv$ has maximal dimension,
hence, by Lemma \ref{lemtratri},
the action of $\g h$ on the quotient $V/\g h v$ is trivial,
and therefore one has the equality 
\begin{equation}
\label{eqnrhovhv}
\rho_{V}(Y)=  \rho_{\gs h}(Y)-\rho_{\gs h_v}(Y)
\;\;\mbox{for all $Y$ in $\g h_v$}
\end{equation}
Combining \eqref{eqnrhohqv} and \eqref{eqnrhovhv}, one gets,
for $\nu_T$-almost all $v$ in $T$,
\begin{equation*}
\label{eqnrhohvg}
2\,\rho_{\gs h_v}(Y)\leq
\rho_{\gs q}(Y)+\rho_{\gs h}(Y)=\rho_{\gs g}(Y)
\;\;\mbox{for all $Y$ in $\g h_v$}
\end{equation*}
which can be rewritten as 
the temperedness criterion 
$
\rho_{\gs h_v}(Y)\leq
\rho_{\gs q_v}(Y)
$ 
for 
$L^2(G/H_v)$ in Theorem \ref{thmlghtem} and hence proves \eqref{eqnghvtem}.
\end{proof}



%4
\section{Using parabolic subgroups}
\label{secparsub}

The aim of this section is to prove the converse implication in 
Theorem \ref{thmlghtem}. As we have seen in Remarks
\ref{remredalg} and \ref{remreduni}, 
we can assume that  $G$ is a Zariski connected algebraic group and that 
$H$ is a Zariski connected algebraic subgroup such that $\g h=[\g h,\g h]$.

The  proof relies on the presence of two nice intermediate 
subgroups 
$$
H\subset F \subset P\subset G.
$$

%41
\subsection{The intermediate subgroups}
\label{secintpar}

\bq
We first explain the construction of these intermediate subgroups
$F$ and $P$.
\eq

Let $G$ be an algebraic semisimple Lie group
and $H$ a Zariski connected  algebraic subgroup of $G$
such that $\g h=[\g h,\g h]$.


\begin{LemDef}
\label{defparsub}
We fix a parabolic subgroup $P$ of $G$ of minimal dimension
that contains $H$ and denote by $U$ the unipotent radical of $P$. 
There exists a reductive subgroup $L\subset P$ such that 
$P=LU$ and $H=(L\cap H)(U\cap H)$. Moreover the group
$S:=L\cap H$ is semisimple and the group $V:=U\cap H$ 
is the unipotent radical of $H$.
We denote by $F$ the group  $F=SU$.
\end{LemDef}


\begin{proof}
The group $V:=U\cap H$ is a unipotent normal subgroup of $H$. 
The quotient $S':=H/V$ is a Zariski connected  subgroup of
the reductive group $P/U$ which is not contained in any proper
parabolic subgroup of $P/U$. 
Therefore, by \cite[Sec.~VIII.10]{Bou7a9}
this group $S'$ is reductive. 
Since $\g h=[\g h,\g h]$, this group $S'$ is semisimple
and there exists a semisimple subgroup $S\subset H$ such that 
$H=SV$. Since $S$ is semisimple, the group $V$ is the unipotent radical of $H$.
Since maximal reductive subgroups $L$ of $P$ are $U$-conjugate,
one can choose $L$ containing $S$ and therefore one has $S=L\cap H$.
\end{proof}

The following two lemmas will be useful 
in our induction process.


\begin{Lem}
\label{lemparsub}
With the notation of Definition \ref{defparsub},
the following two functions on $\g s$ are equal~:
\begin{equation}
\label{eqnparsub}
\rho_{\gs g/\gs h}-\rho_{\gs h}=
\rho_{\gs l/\gs s}+2\, \rho_{\gs u/\gs v}
-\rho_{\gs s}. 
\end{equation}
\end{Lem}

\begin{proof} Since $\rho_{\gs g/\gs p}=\rho_{\gs u}$,
one has the equalities of functions on $\g s$,\\
\mbox{ }\hfill
$
\rho_{\gs g/\gs h}=
\rho_{\gs u}+\rho_{\gs l/\gs s}+ \rho_{\gs u/\gs v}
\;\;\; {\rm and}\; \;\;\;
\rho_{\gs h}=
\rho_{\gs s}+ \rho_{\gs v}.
$\hfill
\end{proof}

\begin{Lem}
\label{lemparsub2}
Let $P=LU$ be a real algebraic group which is
a semidirect product of a reductive subgroup $L$ and its unipotent radical $U$. Let $\pi_0$ be a unitary representation
of $P$ which is $L$-tempered and trivial on $U$.
Then the representation $\pi_0$ is also $P$-tempered.
\end{Lem}

\begin{proof}
The weak containment $\pi_0 \prec L^2(L)$ as unitary representations of $L$
implies 
the weak containment $\pi_0 \prec L^2(P/U)$ as unitary representations of $P$
because $U$ acts trivially on the both sides.
Since $U$ is amenable, the
trivial representation of $U$ is $U$-tempered,
therefore  by Lemma \ref{lemindtem} the
regular representation of $P$ in $L^2(P/U)$ is 
$P$-tempered, and $\pi_0$ is also $P$-tempered.
\end{proof}


%42
\subsection{Bounding volume of compact sets}
\label{secvolcom}
\bq
The proof of Theorem \ref{thmlghtem} relies on a control of the volume of the intersection of translates of compact sets in $X=G/H$.
We first explain how to bound such  volumes in $Z=F/H$.    
This bound is  quite general. 
\eq

\begin{Prop}
\label{prodom}
Let $F=SU$ be a real algebraic  group which is a semidirect product of a reductive subgroup $S$ and 
its unipotent radical $U$. 
Let $H=SV$ be an algebraic subgroup of $F$ containing
$S$  where $V=U\cap H$. 
Let $Z$ be the $F$-space $Z=F/H=U/V$ endowed with a $U$-invariant Radon measure.
Then for every compact subset $D\subset Z$, there exists a compact subset 
$D_0\subset Z$ such that for all $s\in S$ and $u\in U$, one has
\begin{equation}
\label{eqnvolsus}
{\rm vol}(su D\cap D)\leq {\rm vol}(sD_0\cap D_0).
\end{equation}
\end{Prop}

Here is the reformulation 
of Proposition \ref{prodom} that  will be used later on.
\begin{Def}
\label{defzousuh}
Let $Z_0$ be the  same space $Z_0= U/V$ as $Z$ 
but endowed with another $F$-action 
where $U$ acts trivially
and where $S$ acts by conjugation. 
\end{Def}

\begin{Cor}
\label{cordom}
Same notation as in Proposition \ref{prodom}.
Then for every compact subset $D\subset Z$, there exists a compact subset 
$D_0\subset Z_0$ such that for every $f\in F$, one has
\begin{equation}
\label{eqnvolfdd}
{\rm vol}(f D\cap D)\leq {\rm vol}(fD_0\cap D_0).
\end{equation}
\end{Cor}
The proof of Proposition \ref{prodom} is by induction on the dimension of $Z$. It relies only on geometric arguments and 
uses no representation theory.
\vs

Before studying the proof of Proposition \ref{prodom} 
the reader could as an exercise focus on the following very simple example where 
$Z=\m R^2$ is the affine $2$-plane and 
$F$ is the group of affine bijections
$\left(\begin{matrix}a&r\\ 0&b\end{matrix}\right)
\left(\begin{matrix}t\\ s\end{matrix}\right)\; $
that preserve
the horizontal foliation. In this case $S$ is the 
$2$-dimensional group 
of diagonal matrices and $U$ is the $3$-dimensional
Heisenberg group.
The proof for this example relies on the same ideas
while being very concrete. 





\begin{proof}[Proof of Proposition \ref{prodom}]
{\bf First case: }$S$ is a split torus. 

We denote by $C$ the center of $U$ and $C_V=V\cap C$.
%The projection 
%$$U\longrightarrow U/C_U$$ 
%is a principal bundle with group $C_U$. Similarly, 
Let $W$ be the closed subgroup $W:=VC\subset U$. The projection
$$
Z=U/V\longrightarrow Z':=U/W
$$
is a principal bundle of group $C_W:=C/C_V=W/V$.
According to Lemma \ref{lemdom} below, there exists a continuous 
trivialization of this principal bundle
\begin{equation}
\label{eqnuvuwzz}
Z\simeq Z'\times C_W
\end{equation}
such that the action of $U$ and $S$ through this trivialization can be read as 
\begin{eqnarray}
\label{eqnuyzcuy}
su\,(z',c)&=& (su\, z'\, ,\, s\,c+s\,c_0(u,z'))
\end{eqnarray}
for all $s\in S$, $u\in U$, $z'\in Z'$, 
$c\in C_W$, 
where $c_0$ is a continuous cocycle $c_0:U\times Z'\rightarrow C_W$. We fix three compatible invariant measures 
${\rm vol}_Z$, ${\rm vol}_{Z'}$ and ${\rm vol}$ on $Z$, $Z'$, and $C_W$.

We start with a compact set $D\subset Z$.
Through the trivialization \eqref{eqnuvuwzz}, 
this set $D$ is included in a product 
of two compact sets $D'\subset Z'$ and $B\subset C_W$
\begin{equation*}
D\subset D'\times B\, ,
\end{equation*}
where $B$ is a symmetric convex set in the group $C_W$
seen as a real vector space. 
By the induction hypothesis,
there exists a compact set $D'_0\subset Z'$
which satisfies 
the bound \eqref{eqnvolsus} for $D'$, {\it{i.e.}} such that
\begin{equation}
\label{eqnvolfd2}
{\rm vol}_{Z'}(su D'\cap D')\leq {\rm vol}_{Z'}(sD'_0\cap D'_0)
\;\;\;\mbox{for all $s\in S$, $u\in U$}.
\end{equation}
We  compute using \eqref{eqnuyzcuy} 
and Lemma \ref{lemconset} below, 
for all $s\in S$ and $u\in U$,

\begin{eqnarray*}
{\rm vol}_Z(su D\cap D)
&\leq&
\int_{suD'\cap D'}{\rm vol}((sB+s\,c_0(u,(su)^{-1}z'))\cap B)\rmd z'\\
&\leq &
\int_{suD'\cap D'}{\rm vol}(sB\cap B)\rmd z'
\end{eqnarray*} 
where $\rmd z'$ also denotes the $U$-invariant measure on $Z'$.
Hence, using  \eqref{eqnvolfd2}, we go on
\begin{eqnarray*}
{\rm vol}_Z(su D\cap D)
&\leq&{\rm vol}_{Z'}(suD'\cap D')\;{\rm vol}(sB\cap B)\\
&\leq &
{\rm vol}_{Z'}(sD_0'\cap D_0')\; {\rm vol}(sB\cap B)\\
&= &
{\rm vol}_{Z}(sD_0\cap D_0),
\end{eqnarray*} 
where $D_0$ is the compact subset of $Z$
given by  $D_0:=D'_0\times B$.
\vs

{\bf Second case: }$S$ is a reductive group.

This general case will be deduced from the first case.
Indeed any reductive group admits a Cartan decomposition
$S=K_{_S}A_{_S}K_{_S}$ where $K_{_S}$ is a maximal compact subgroup 
of $S$ and where $A_{_S}$ is a maximal split torus of $S$.
We start with a compact set $D$ of $Z$.
According to the first case, 
there exists a $K_{_S}$-invariant compact set $D_0\subset Z$
such that, for all $a\in A_{_S}$ and  $u\in U$, one has
\begin{equation*}
\label{eqnvolfd3}
{\rm vol}(au K_{_S}D\cap K_{_S}D)\leq {\rm vol}(aD_0\cap D_0).
\end{equation*}
Therefore, for all $s$ in $S$ and $u$ in $U$, writing
$s=k_1ak_2$ with 
$k_1$, $k_2$ in $K_{_S}$ and $a$ in $A_{_S}$, one has 
\begin{eqnarray*}
\label{eqnvolfd4}
{\rm vol}(su D\cap D)
&\leq& 
{\rm vol}(a(k_2uk_2^{-1})k_2D\cap k_1^{-1}D)\\
&\leq &
{\rm vol}(aD_0\cap D_0)\\
&= &
{\rm vol}(sD_0\cap D_0),
\end{eqnarray*}
as required.
\end{proof}
In the proof of Proposition \ref{prodom}, we have used the following
two lemmas.


\begin{Lem}
\label{lemdom}
Let $U$ be a unipotent group, $V\subset U$ a unipotent subgroup, $C$ be the center of $U$, $W:=VC$ and  
$C_V:=C\cap V$.
Let $S\subset {\rm Aut}(U)$ be a split torus 
which preserves $V$.
Then 
there exists a continuous 
trivialization of the $U$-equivariant principal bundle $U/V\rightarrow U/W$ with structure group $C/C_V$
\begin{equation*}
\label{eqnuvuwzz2}
U/V\simeq U/W\times C/C_V
\end{equation*}
such that the action of $U$ and $S$ through this trivialization can be read as 
\begin{eqnarray*}
\label{eqnuyzcuy2}
su\,(y,c)&=& (su\, y\, ,\, s\,c+s\,c_0(u,y))
\end{eqnarray*}
for all $u\in U$, $s\in S$, $y\in U/W$, 
$c\in C/C_V$, 
where $c_0$ is a continuous cocycle $c_0\colon U\times U/W\rightarrow C/C_V$.
\end{Lem}

\begin{proof}[Proof of Lemma \ref{lemdom}]
These claims are a variation of a classical result of 
Chevalley--Rosenlicht (See for instance \cite[Thm.~3.1.4]{CoGr}).
The proof relies on the existence of ``an adapted basis in a nilpotent
Lie algebra''. 
Here is a sketch of proof of these claims.

As usual, let $\g u$, $\g v$, $\g c$ and $\g w$
be the Lie algebras of the groups $U$, $V$, $C$ and $W$.
Let $I$ be the ordered set  $I=\{1,\ldots,n\}$, where $n=\dim \g u$. We fix a basis $(e_i)_{i\in I}$ of
$\g u$, such that\\
- for every $i\geq 1$, the 
vector space spanned by the $e_j$ for $j\geq i$ is an ideal;\\
- for every $i\geq 1$, the line $\m R e_i$ is invariant by $S$;\\
- there exists a subset $I_V\subset I$ such that 
$\g v$ is spanned by  $e_i$ for $i\in I_V$; \\
- there exists a subset $I_C\subset I$ such that 
$\g c$ is spanned by  $e_i$ for $i\in I_C$; \\
- the Lie algebra
$\g w$ is spanned by  $e_i$ for $i\in I_W:= I_C\cup I_V$. 

Then, the map
\begin{eqnarray*}
\Psi\;\colon\;\;\;\m R^I
\longrightarrow
U
\quad
(t_i)_{i\in I}\mapsto 
\textstyle\prod _{i\in I}\exp(t_ie_i),
\end{eqnarray*} 
where the product is performed using the order on $I$,
is a diffeomorphism
and one has 
$$
\Psi(\m R^{I_V})=V
\;\; ,
\;\;
\Psi(\m R^{I_C})=C
\;\;{\rm and}\;\; 
\Psi(\m R^{I_W})=W.
$$
Setting $J_V:= I\smallsetminus I_V$
and  $J_W:= I\smallsetminus I_W$,
the map 
\begin{eqnarray*}
\Psi_V\;\colon\;\;\;\m R^{J_V}
\longrightarrow
U/V, 
\quad
(t_i)_{i\in J_V} \mapsto 
\textstyle\prod _{i\in J_V}\exp(t_ie_i)V
\end{eqnarray*} 
is also a diffeomorphism and the restriction of this map to the subset
$\m R^{J_W}$ gives an $S$-equivariant section 
of the bundle $U/V\rightarrow U/W$.
\end{proof}

Here is the second basic lemma  used in the proof
of Proposition \ref{prodom}.


\begin{Lem}
\label{lemconset}
Let $B$, $B'$ be two symmetric convex sets of $\m R^d$,
then one has
$$
{\rm vol}((B+v)\cap B'))\leq {\rm vol} (B\cap B')
\;\;\;\mbox{\rm for all $v\in \m R^d$}.
$$
\end{Lem}
\begin{proof}
By the Brunn--Minkowski inequality (see \cite[Sec.~11]{BoFe}),
the map 
$v\mapsto {\rm vol}((B+v)\cap B')^{1/d}$ is concave on the convex set $B'-B$ and hence achieves its maximum value 
at $v=0$.
\end{proof}

%43
\subsection{Matrix coefficients of induced representations}
\label{seccoeind}
\bq
We now explain how to control 
the volume of the intersection of 
translates of compact sets in the $G$-space $X=G/H$
with those in $X_0:=G\times_F Z_0$.    
\eq


\begin{Prop}
\label{provolgcc}
Let $G$ be an algebraic semisimple Lie group
and $H$ a Zariski connected  algebraic subgroup
such that $\g h=[\g h,\g h]$.
Let $P=LU$, $F=SU$ and $H=SV$ 
be the groups introduced in Definition $\ref{defparsub}$.
Let $Z_0$ be the $F$-space  introduced in Definition 
$\ref{defzousuh}$ and $X_0$ the $G$-space $X_0:=G\times_FZ_0$. 
Then, for every compact subset $C\subset G/H$, there exists a compact subset 
$C_0\subset X_0$ such that 
\begin{equation}
\label{eqnvolcc02}
{\rm vol}(g\, C\cap C) \leq {\rm vol}(g\, C_0\cap C_0)
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation}
\end{Prop}

In Proposition \ref{provolgcc} the assumption $\g h=[\g h,\g h]$ can be removed but the conclusion \eqref{eqnvolcc02}
becomes slightly 
more technical when there is no $G$-invariant measure on $G/H$. Indeed, when $\g h\neq [\g h,\g h]$, one has to replace 
the bound \eqref{eqnvolcc02} by a  
bound of $K$-finite matrix coefficients of the induced representation $\Pi=\Ind_F^G(L^2(F/H))$ thanks to $K$-finite matrix  coefficients of the induced representation $\Pi_0=\Ind_F^G(L^2(Z_0))$.

\begin{proof}[Proof of Proposition \ref{provolgcc}]
The projection 
$$
G\rightarrow X':=G/F
$$
is a $G$-equivariant principal bundle with structure group $F$. 
As in Section \ref{secregind}, we fix
a Borel measurable trivialization of this principal bundle 
\begin{equation}
\label{eqngsigh2}
G\simeq X'\times F
\end{equation}
which sends relatively compact subsets to relatively compact subsets.
The action of $G$ by left multiplication through this trivialization can be read as 
\begin{equation*}
\label{eqngxhgx2}
g\,(x',f)=(gx',\si_F(g,x')f)
\;\;\;\;\mbox{\rm 
for all $g\in G$, $x'\in X'$ and $f\in F$,}
\end{equation*}
where $\si_F\colon G\times X'\rightarrow F$ 
is a Borel  measurable cocycle. 
This trivialization \eqref{eqngsigh2}
induces a trivialization of the associated bundles
\begin{eqnarray*}
\label{eqngsigh3}
X= G\times _F Z    
&\simeq&  X' \times Z\, , \\    
X_0 = G \times_F Z_0  
&\simeq &                 
X' \times Z_0\, .
\end{eqnarray*}
We start with a compact set $C$ of $X$.
Through the first trivialization, 
this compact set is included in a product 
of two compact sets $C'\subset X'$ and $D\subset Z$
\begin{equation}
\label{eqncsumxd}
C\subset C'\times D\; .
\end{equation}
We denote by $D_0\subset Z_0$ the compact set given
by Corollary \ref{cordom} 
and we compute using \eqref{eqnvolfdd}, 
for $g$ in $G$,
\begin{eqnarray*}
{\rm vol}_X(g\, C\cap C)
&\leq&
\int_{gC'\cap C'}{\rm vol}_Z(\si_F(g,g^{-1} x')D\cap D)\rmd x'\\
&\leq &
\int_{gC'\cap C'}{\rm vol}_{Z_0}(\si_F(g,g^{-1} x')D_0\cap D_0)\rmd x'\\
&\leq &
{\rm vol}_{X_0}(g\, C_0\cap C_0),
\end{eqnarray*} 
where $\rmd x'$ is a $G$-invariant measure on $X'$
and $C_0$ is a compact subset of $X_0\simeq X'\times Z_0$
which contains $C'\times D_0$.
\end{proof}


%44
\subsection{Proof of the temperedness criterion}
\label{secsemgro}

\bq
We conclude the proof of Theorem \ref{thmlghtem}.
\eq




\begin{proof}[Proof of the converse implication in Theorem \ref{thmlghtem}] We prove it by induction on the dimension of $G$.
By Remarks
\ref{remredalg} and \ref{remreduni}, 
we can assume that  $G$ is a Zariski connected semisimple algebraic group and that 
$H$ is a Zariski connected algebraic subgroup such that $\g h=[\g h,\g h]$.
Let 
$$
H=SV\subset F=SU\subset P=LU \subset G
$$
be the groups introduced in Definition $\ref{defparsub}$.
Let $Z_0=U/V$ be the $F$-space introduced in Definition 
$\ref{defzousuh}$ and $X_0$ be the $G$-space $X_0=G\times_FZ_0$. 

When $P$ is equal to $G$, the group $H$ is semisimple and we apply \cite[Thm 3.1]{BeKoI}. We now assume that $P$ is a proper parabolic subgroup of $G$. 

By assumption one has $\rho_\gs h\leq \rho_{\gs g/\gs h}$ on $\g h$. 
Therefore, by Lemma \ref{lemparsub}, 
one has the inequality on $\g s$
\begin{equation}
\label{eqnparsub2}
\rho_{\gs s}\leq \rho_{\gs l/\gs s}+2\, \rho_{\gs u/\gs v}. 
\end{equation}

We introduce the regular representation $\pi_0$ of $P$ in 
$L^2(P\times_FZ_0)$
 which is unitarily equivalent 
 to $\Ind_F^P(L^2({\mathfrak{u}}/{\mathfrak{v}}))$
 by the isomorphism \eqref{eqnrexind}.
As a representation of $L$, one has 
$$
\pi_0|_L=\Ind_S^L(L^2(\g u/\g v)).
$$
Using our induction hypothesis
on the dimension of G
to the derived subgroup of $L$,
Proposition \ref{proequthm} and Remark \ref{remtemame} 
tell us 
that the representation $\pi_0$ is $L$-tempered
 by \eqref{eqnrexind}.
Therefore, by Lemma \ref{lemparsub2},  
the representation $\pi_0$ is $P$-tempered. 
The regular representation  $\Pi_0$ of $G$ in $L^2(G\times_FZ_0)$ 
is unitarily equivalent to $\Pi_0 =\Ind_P^G(\pi_0)$ 
because
$
\Pi_0 \simeq \Ind_P^G (\Ind_F^P (L^2(\g u/\g v))).
$
Now, Lemma \ref{lemindtem} implies that 
this representation $\Pi_0$ is $G$-tempered.
Therefore, by Corollary \ref{coralpcol},
for any $K$-invariant
compact subset $C_0$ of $G\times_FZ_0$, 
one has a bound:
\begin{equation*}
{\rm vol}(g\, C_0\cap C_0) \leq {\rm vol}( C_0)\; \Xi(g) 
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation*}
Hence, by Proposition \ref{provolgcc}, 
for any 
compact subset $C$ of $G/H$, 
one also has a bound
\begin{equation*}
{\rm vol}(g\, C\cap C) \leq M_C\; \Xi(g)
\;\;\;\; \mbox{\rm for all $g$ in $G$.}
\end{equation*}
Again by Corollary \ref{coralpcol}, this tells us 
that the representation of $G$ in $L^2(G/H)$ 
is $G$-tempered.
\end{proof}






%5
\section{Examples}
\label{secexalgh}


The criterion given in Theorem \ref{thmlghtem}  allows to easily detect for a given homogeneous space $G/H$ whether 
the unitary representation of a semisimple Lie group $G$ in $L^2(G/H)$ is tempered or not. 
We collect in this chapter a few examples,
omitting the details of the computational verifications. 



%51
\subsection{Examples of tempered homogeneous spaces}
\label{secredhom}

\bq
We first recall a few examples  
extracted from \cite{BeKoI} where $H$ is reductive.
\eq






\begin{Exa}
\label{exagslopq}
$L^2(SL(p+q,\mathbb{R})/SO(p,q))$ is always tempered.\\
$L^2(SL(2m,\mathbb{R})/Sp(m,\m R))$ is never tempered.\\
$L^2(SL(m+n,\mathbb{C})/SL(m,\mathbb{C})\times SL(n,\mathbb{C}))$ 
is tempered iff $|m-n|\leq 1$.\\
$L^2(SO(m+n,\mathbb{C})/SO(m,\m C))\times SO(n,\m C))$ is tempered iff
$|m-n|\leq 2$.\\
$L^2(Sp(m+n,\mathbb{C})/Sp(m,\mathbb{C})\times Sp(n,\mathbb{C}))$ is tempered iff $m=n$.
\end{Exa} 

\begin{Exa}
\label{exaslprod}
Let $n=n_1+\cdots + n_r$ with $n_1\geq\cdots\geq n_r\geq 1$, $r\geq 2$.\\
$L^2(SL(n,\m R)/\prod SL(n_i,\m R))$ is tempered iff 
$2n_1\leq n+1$.\\
$L^2(Sp(n,\m R)/\prod Sp(n_i,\m R))$ is tempered iff
$2n_1\leq n$.\\
Let $p=p_1+p_2$, $q=q_1+q_2$ with $p_1$, $p_2$, $q_1$, $q_2$ $\geq 1$.\\
$L^2(SO(p,q)/ SO(p_1,q_1)\times SO(p_2,q_2))$ is tempered iff
$|p_1+q_1-p_2-q_2|\leq 2$.
\end{Exa}

\begin{Exa}
\label{exagxg}
Let $G $ be an algebraic semisimple Lie group and 
$K $ a maximal compact subgroup.
$L^2(G_{\m C}/K_{\m C})$ is $G_{\m C}$-tempered iff $G $ is quasisplit.
\end{Exa}

\begin{Rem}
A  way to justify this last example is to
notice that our criterion 
$2\, \rho_{\g k_\m C}\leq \rho_{\g g_\m C}$
means that 
{\it the trivial $K$-type is a small $K$-type of $G$}
in the terminology of Vogan's paper 
\cite[Def. 6.1]{Vog79}, see also in Knapp's book
\cite[Chap.~XV]{Kn01}, and to use 
the following equivalences due to Vogan in the same paper
\cite[Thm.~6.4]{Vog79}:
$$
\mbox{
$G$ has a small $K$-type 
$\Longleftrightarrow$
the trivial $K$-type is small
$\Longleftrightarrow$
$G$ is quasisplit.}
$$
\end{Rem}

Here is a delicate example for semisimple symmetric spaces.
\begin{Exa}
\label{exSp} Let $G/H:=Sp(2,1)/Sp(1) \times Sp(1,1)$.
The Plancherel formula \cite{Del98, Osh88} tells that 
both the continuous part and  a ``generic portion" of the discrete part
of $L^2(G/H)$ are tempered, however, 
 our criterion \eqref{eqnlghtem} tells that $L^2(G/H)$ is nontempered
because $\rho_{\mathfrak h}(Y) = \frac{3}{2} \rho_{\mathfrak q}(Y) >\rho_{\mathfrak q}(Y)$
 if $Y$ is a nonzero hyperbolic element of $\mathfrak h$.
In fact, the discrete part of $L^2(G/H)$ 
consist of Harish-Chandra's discrete series representations, say $\pi_n$ ($n=1,2,...$),
and two more non-vanishing representations $\pi_{0}$ and $\pi_{-1}$
in the coherent family, 
%as a coherent continuation to singular infinitesimal characters of $\{\pi_n\}$,
 where $\pi_0$ is still tempered but $\pi_{-1}$ is nontempered (\cite[Thm.~1]{kob92}).
\end{Exa}

Here is another direct application of our criterion \eqref{eqnlghtem} where $H$ is not anymore assumed to be reductive.

\begin{Cor} 
\label{corgh}
Let $G$ be an algebraic  semisimple Lie group,
and
$H$ an algebraic  subgroup.\\
$1)$ If the representation of $G_{\mathbb{C}}$ in
$L^2(G_{\mathbb{C}}/H_{\mathbb{C}})$ is tempered,
then the representation of $G$ in $L^2(G/H)$ is tempered.\\
$2)$ The converse is true under if $H$ contains a maximal torus 
which is split.
\end{Cor}

%The converse statement of Corollary \ref{corgh}~1) may fail even for
%a semisimple symmetric space $G/H$ if we drop the assumption of 2).
%\begin{Exa}
%\label{exSLH}
%Let $G/H:=SL(2m+1, \mathbb H)/S(GL(m+1,\mathbb H)\times GL(m, \mathbb H))$.
%Then the complexification $G_{\mathbb C}/H_{\mathbb C}$
%is isomorphic to the complex symmetric space
%$SL(4m+2, \mathbb C)/S(GL(2m+2,\mathbb C)\times GL(2m, \mathbb C))$.
%The criterion \eqref{eqnlghtem} tells that $L^2(G/H)$ is tempered,
%whereas $L^2(G_{\mathbb C}/H_{\mathbb C})$ is non-tempered as is seen
%in Example~\ref{exagslopq}.
%It should be noted in this example that $L^2(G/H)$ is tempered although the centralizer of $\mathfrak j$ in $\mathfrak g$ is not
%abelian where $j$ is a Cartan subalgebra for the symmetric pair $(\mathfrak g, \mathfrak h)$,
%see also Example~\ref{exagxg}.
%\end{Exa}

%52
\subsection{Subgroups of $SL(n,\m R)$}
\label{secnonred}
\bq
We now explain how to check our criterion \eqref{eqnlghtem}
on a very concrete example.
\eq

In Table 1, we specify our criterion \eqref{eqnlghtem} when
$
G= {\rm SL}(\m R^p\oplus\m R^q) 
$
and  $H$ is a  subgroup of $G$  
normalized by the 
group ${\rm SL}(\m R^p)\times {\rm SL}(\m R^q) $.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
%&&&\\
$H_1\colon\left(\begin{matrix}*&0\\ 0&I\end{matrix}\right)$&
$H_2\colon\left(\begin{matrix}*&*\\ 0&I\end{matrix}\right)$&
$H_3\colon\left(\begin{matrix}*&*\\ 0&*\end{matrix}\right)$&
$H_4\colon\left(\begin{matrix}*&0\\ 0&*\end{matrix}\right)$\\
&&&\\
$p\leq q+1$&
$p= 1$&
$p=q=1$&
$p\leq q+1$\\
&&&
$q\leq p+1$\\
\hline
\end{tabular}
\end{center}
\vspace{-1em}
{\caption{Table 1:\;
The criterion $\rho_{\gs h}\leq\rho_{\gs g/\gs h}$
when $G= {\rm SL}(p+q,\m R)$}}
\label{figslpq}
\end{figure}

In Table 2, we specify our criterion \eqref{eqnlghtem} when
$
G= {\rm SL}(\m R^p\oplus\m R^q\oplus\m R^r) 
$
and  $H$ is a  subgroup of $G$  
normalized by the 
group ${\rm SL}(\m R^p)\times {\rm SL}(\m R^q) \times {\rm SL}(\m R^r) $.
Note that in these two tables, the center of the diagonal blocks is not important by Corollary \ref{corgh1h2}. 
\begin{figure}[htb]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$H_1:\left(\begin{matrix}*&0&*\\ 0&I&0\\ 0&0&I\end{matrix}\right)$&
$H_2:\left(\begin{matrix}I&0&*\\ 0&*&0\\ 0&0&I\end{matrix}\right)$&
$H_3:\left(\begin{matrix}I&*&*\\ 0&*&0\\ 0&0&I\end{matrix}\right)$&
$H_4:\left(\begin{matrix}*&*&*\\ 0&*&*\\ 0&0&*\end{matrix}\right)$\\
&&&\\
$p\leq q+1$&
$q\leq p+r+1$&
$q\leq r+1$ &
$p= q=r=1$\\
\hline
\hline 
$H_5:\left(\begin{matrix}*&0&0\\ 0&*&0\\ 0&0&I\end{matrix}\right)$&
$H_6:\left(\begin{matrix}*&0&*\\ 0&*&0\\ 0&0&I\end{matrix}\right)$&
$H_7:\left(\begin{matrix}*&*&*\\ 0&*&0\\ 0&0&I\end{matrix}\right)$&
$H_8:\left(\begin{matrix}*&0&*\\ 0&I&0\\ 0&0&*\end{matrix}\right)$\\
&&&\\
$p\leq q+r+1$ &
$p\leq q+1$&
$p=1$&
$p\leq q+1$\\
$q\leq p+r+1$&
$q\leq p+r+1$&
$q\leq r+1$&
$r\leq q+1$\\
\hline
\hline
$H_9\!:\left(\begin{matrix}I&*&*\\ 0&*&0\\ 0&0&*\end{matrix}\right)$&
$H_{10}\!:\left(\begin{matrix}*&0&0\\ 0&*&0\\ 0&0&*\end{matrix}\right)$&
$H_{11}\!:\left(\begin{matrix}*&0&*\\ 0&*&0\\ 0&0&*\end{matrix}\right)$&
$H_{12}\!:\left(\begin{matrix}*&*&*\\ 0&*&0\\ 0&0&*\end{matrix}\right)$\\
&&&\\
$q\leq r+1$ &
$p\leq q+r+1$&
$p\leq q+1$&
$p=1$\\
$r\leq q+1$&
$q\leq p+r+1$&
$q\leq p+r+1$&
$q\leq r+1$\\
&
$r\leq p+q+1$&
$r\leq q+1$&
$r\leq q+1$\\
\hline
\end{tabular}
\end{center}
\vspace{-1em}
\caption{Table 2:\;
The criterion $\rho_{\gs h}\leq\rho_{\gs g/\gs h}$
when $G= {\rm SL}(p+q+r,\m R)$}
\label{figslpqr}
\end{figure}
\begin{Rem}
It is rather easy to guess the inequalities
in Table \ref{figslpqr}.
Here is the heuristic recipe: there is one inequality
for each non-identity diagonal block. 
The left-hand side of this inequality is given by the size
of this diagonal block, while
the right-hand side can be guessed by looking at 
the size of the zero blocks on the right and on the top of it. 
\end{Rem}

We will just explain the proof for the group $H=H_{11}$
in Table \ref{figslpqr}.
The other cases are  similar. 

\begin{Cor}
\label{corslpqr}
Let $G= {\rm SL}(p\!+\!q\!+\!r,\m R)$ and 
$ H$  the subgroup of matrices 
$\left(\begin{matrix}\al &0&z\\ 0&\be&0\\ 0&0&\ga\end{matrix}\right)$ 
with $\al\in {\rm GL}(p,\m R)$, 
$\be\in {\rm GL}(q,\m R)$, $\ga\in {\rm GL}(r,\m R)$,
$z\in {\rm M}(p,r;\m R)$.
Then $L^2(G/H)$ is $G$-tempered if and only if
$\;
p\leq q\!+\!1,\;
q\leq p\!+\!r\!+\!1,\;
r\leq q\!+\!1.
$
\end{Cor}



\begin{proof}[Proof of Corollary \ref{corslpqr}]
We denote by $\g a$ the Lie algebra of diagonal matrices
$$
\g a=\{Y=(x,y,z)\in \m R^p\oplus\m R^q\oplus\m R^r\mid {\Tr}(Y)=0\}.
$$
We only need to check the criterion 
\eqref{eqnlghtem} on the chamber
$$
\g a_+=\{Y=(x,y,z)\in \g a
\mid \mbox{ $x$, $y$ and $z$ have non-decreasing coordinates}\}.
$$
We recall that $\g q=\g g/\g h$ and 
we compute for $Y\in \g a_+$,
$$
\rho_\gs h(Y)=\textstyle
\sum_{i=1}^pa_ix_i+
\sum_{j=1}^qb_jy_j+
\sum_{k=1}^rc_kz_k
\;,
$$
where $\;\;a_i:= 2i-p-1,\;\;
b_j:= 2j-q-1,\;\;
c_k:=2k-r-1$, \; and
$$\textstyle
\rho_\gs q(Y)=
\sum_{i,j}|x_i-y_j| +\sum_{j,k}|y_j-z_k|\; .
$$

Assume first that the criterion $\rho_\gs h \leq \rho_\gs q $ is satisfied on $\g a$. It is then also satisfied on $\m R^{p+q+r}$. 
Applying it successively to the three vectors $Y=e_p$, $Y=e_{p+q}$ and 
$Y=e_{p+q+r}$ of the standard basis $e_1,\ldots ,e_{p+q+r}$ of $\m R^{p+q+r}$, 
one gets successively the three inequalities
$\;
p\leq q\!+\!1,\;
q\leq p\!+\!r\!+\!1,\;
r\leq q\!+\!1.
$
\vs

Assume now that these three inequalities are satisfied.
Note that
\begin{eqnarray*}
\rho_\gs q(Y)\!\!&\geq&\!\! \textstyle
\sum_{a_i>b_j}\! (x_i\!-\! y_j) +
\sum_{b_j>a_i}\! (y_j\!-\! x_i)  +
\sum_{b_j>c_k}\! (y_j\!-\! z_k)  +
\sum_{c_k>b_j}\! (z_k\!-\! y_j)   \\
&=&\textstyle
\sum_{i=1}^p \ell_i x_i +
\sum_{j=1}^q m_j  y_j +
\sum_{k=1}^r n_k z_k, 
\;\; {\rm where} 
\end{eqnarray*}
\begin{eqnarray*}
\ell_i&=&
|\{j\mid b_j<a_i\}|-|\{j\mid b_j>a_i\}|\\
m_j&=&
|\{i\mid a_i<b_j\}|-|\{i\mid a_i>b_j\}| 
+|\{k\mid c_k<b_j\}|-|\{k\mid c_k>b_j\}|\\ 
n_k&=&
|\{j\mid b_j<c_k\}|-|\{j\mid b_j>c_k\}|.
\end{eqnarray*}

Since $p\leq q+1$, one has $\ell_i=a_i$ for all $1\leq i\leq p$.

Since $r\leq q+1$, one has $n_k=c_k$ for all $1\leq k\leq r$.

For $1\leq j\leq q$, one has
$m_{q+1-j}=-m_j$ and, when $j> q/2$,
$$
m_j= \operatorname{min}(b_j,p)+\operatorname{min}(b_j,r).
$$
Since $q\leq p+r+1$, one has 
$m_j\geq b_j$ for all $j> q/2$. 
Then using the fact that 
the $y_j$s are non-decreasing functions of $j$, one gets,
for $Y$ in $\g a_+$,
$$\textstyle
\rho_\gs q(Y)-\rho_{\gs h}(Y)=\sum_{j=1}^q(m_j-b_j)y_j\geq 0.
$$
This proves that the criterion $\rho_\gs h \leq \rho_\gs q $ is satisfied.
\end{proof}

Some of the subgroups in Table \ref{figslpqr} appear naturally
 in analyzing the tensor product representations
 of $SL(n,{\mathbb{R}})$ as below.  

\subsection{Tensor product of nontempered representations}
\label{sectensor}

Suppose $\Pi$ and $\Pi'$ are unitary representations of $G$.  
The tensor product representation $\Pi \otimes \Pi'$ is tempered
 if $\Pi$ or $\Pi'$ is tempered.  
In contrast, $\Pi \otimes \Pi'$ may be and may not be tempered 
 when both $\Pi$ and $\Pi'$ are nontempered.  

For instance, let $n=n_1 + \cdots + n_k$ be a partition,
 and we consider the (degenerate) principal series representation
 $\Pi_{n_1,\cdots,n_k}:=\Ind_{P_{n_1,\cdots,n_k}}^{G}({\bf{1}})$
 of $G=SL(n,{\mathbb{R}})$, 
 where $P_{n_1,\cdots,n_k}$ is the standard parabolic subgroup 
 with Levi subgroup $S(GL(n_1,{\mathbb{R}}) \times \cdots \times GL(n_k,{\mathbb{R}}))$.  
Then $\Pi_{n_1,\cdots,n_k}$ is tempered 
 iff $k=n$ and $n_1= \cdots = n_k=1$.  
Here are some examples of the temperedness criterion \eqref{eqnlghtem}
 applied to the tensor product of two such representations.  
\begin{Prop}
Let $0 \le k, l \le n$ and $a+b+c =n$.  
\begin{enumerate}
\item[{\rm{(1)}}]
$\Pi_{k,n-k} \otimes \Pi_{n-l,l}$ is tempered
 iff $|k-l| \le 1$ and $|k+l-n| \le 1$.  
\item[{\rm{(2)}}]
$\Pi_{a,b,c} \otimes \Pi_{b+c,a}$ is tempered
 iff $\max(b,c) -1 \le a \le b+c+1$.  
\item[{\rm{(3)}}]
$\Pi_{a,b,c} \otimes \Pi_{c,b,a}$ is tempered
 iff $2 \max(a,b,c) \le n+1$. 
\end{enumerate}
\end{Prop}

\begin{proof}
For any parabolic subgroups $P$ and $P'$ of $G$, there
exists an element $w \in G$ such that $P w P'$ is open dense in $G$,
and thus the tensor product $\Ind_{P}^G ({\bf 1}) \otimes \Ind_{P'}^G ({\bf 1})$
is unitarily equivalent to the regular representation in $L^2(G/H)$ 
by the Mackey theory, where $H=w^{-1}P w \cap P'$.
In the above cases, we have the following unitary equivalences:
\begin{alignat*}{2}
\Pi_{k,n-k} \otimes \Pi_{n-l,l} &\simeq L^2(G/H_{12})
\quad
&& 
\text{ with } (p,q,r)=(|k-l|, \min(k,l), n-\max(k,l)), 
\\
\Pi_{a,b,c} \otimes \Pi_{b+c,a} &\simeq L^2(G/H_{11})
\quad
&& 
\text{ with } (p,q,r)=(b,a,c), 
\\
\Pi_{a,b,c} \otimes \Pi_{c,b,a} &\simeq L^2(G/H_{10})
\quad
&& 
\text{ with } (p,q,r)=(a,b,c),
\end{alignat*}
whence Proposition follows from Table \ref{figslpqr} in Section \ref{secnonred}.  
\end{proof}

\subsection*{Acknowledgments}

The authors are grateful to the IHES and to the University of Tokyo for its support through the GCOE program.
The second author was partially supported by Grant-in-Aid for Scientific Research %(B) (22340026) 
(A) (25247006) JSPS.






%9
{\small
\bibliography{temperedII}
}


{\small
\noindent
Y. \textsc{Benoist} :
CNRS-Universit\'e Paris-Sud, Orsay, France\newline
e-mail : \texttt{yves.benoist@math.u-psud.fr}

\medskip
\noindent
T. \textsc{Kobayashi} :
Graduate School of Mathematical Sciences and 
Kavli IPMU-University of Tokyo, Komaba,  Japan\newline
e-mail : \texttt{toshi@ms.u-tokyo.ac.jp}
\end{document}




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