Berezin kernels and Analysis on Makarevich Spaces
Jacques Faraut
Michael Pevzner
Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. \bigskip We consider the conformal group ${Conf}(V)$ of a simple real Jordan algebra $V$. The maximal degenerate representations $\pi _s$ ($s\in {\mathbb C}$) we shall study are induced by a character of a maximal parabolic subgroup $\bar P$ of ${Conf}(V)$. These representations $\pi _s$ can be realized on a space $I_s$ of smooth functions on $V$. There is an invariant bilinear form ${\mathfrak B}_s$ on the space $I_s$. The problem we consider is to diagonalize this bilinear form ${\mathfrak B}_s$, with respect to the action of a symmetric subgroup $G$ of the conformal group ${Conf}(V)$. This bilinear form can be written as an integral involving the Berezin kernel $B_{\nu }$, an invariant kernel on the Riemannian symmetric space $G/K$, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of $B_{\nu }$. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity : $$D(\nu )B_{\nu } =b(\nu )B_{\nu +1},$$ where $D(\nu )$ is an invariant differential operator on $G/K$ and $b(\nu )$ is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from $I_{-s}$ to $I_s$. Furthermore we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group $U$ of the conformal group ${Conf}(V)$.
math0411294.tex

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\begin{document}

\title{Berezin kernels and Analysis on Makarevich Spaces}   %
\author{Jacques Faraut,
 Michael Pevzner}
\address{  J.Faraut : Institut de Math\'ematiques de Jussieu, UMR 7586,
Universit\'e de Paris 6, Pierre-et-Marie-Curie, Case 247, 4 place
Jussieu, F-75252 Paris Cedex} \address{ M.Pevzner : Laboratoire de
Math\'ematiques, UMR 6056 CNRS, Universit\'e de Reims, Moulin de la
Housse, BP 1039, F-51687, Reims}
 \email{faraut@math.jussieu.fr, pevzner@univ-reims.fr}

\thanks{First author partially supported by the European Commission (IHP Network HARP)}%
\subjclass[2000]{43A35, 22E46}%
\keywords{Berezin Kernel, Spherical Fourier Transform, Conformal Groups, Makarevich Spaces}%


\dedicatory{Dedicated to Gerrit van Dijk on the occasion of his 65-th birthday}%
%\commby{}%
% ----------------------------------------------------------------
\begin{abstract}
Following ideas of van Dijk and Hille we study the link which
exists between maximal degenerate representations and Berezin
kernels.

\bigskip

We consider the conformal group ${\rm Conf}(V)$ of a simple real
Jordan algebra $V$. The maximal degenerate representations $\pi _s$
($s\in {\mathbb C}$) we shall study are induced by a character of a
maximal parabolic subgroup $\bar P$ of ${\rm Conf}(V)$. These
representations $\pi _s$ can be realized on a space $I_s$ of smooth
functions on $V$. There is an invariant bilinear form ${\mathfrak
B}_s$ on the space $I_s$. The problem we consider is to diagonalize
this bilinear form ${\mathfrak B}_s$, with respect to the action of
a symmetric subgroup $G$ of the conformal group ${\rm Conf}(V)$.
This bilinear form can be written as an integral involving the
Berezin kernel $B_{\nu }$, an invariant kernel on the Riemannian
symmetric space $G/K$, which is a Makarevich symmetric space in the
sense of Bertram. Then we can use results by van Dijk and Pevzner
who computed the spherical Fourier transform of $B_{\nu }$. From
these, one deduces that the Berezin kernel satisfies a remarkable
Bernstein identity :
$$D(\nu )B_{\nu } =b(\nu )B_{\nu +1},$$
where $D(\nu )$ is an invariant differential operator on $G/K$ and
$b(\nu )$ is a polynomial. By using this identity we compute a Hua
type integral which gives the normalizing factor for an
intertwining operator from $I_{-s}$ to $I_s$. Furthermore we
obtain the diagonalization of the invariant bilinear form with
respect to the action of the maximal compact group $U$ of the
conformal group ${\rm Conf}(V)$.

\end{abstract}
\maketitle \thispagestyle{empty}
%\tableofcontents
% ----------------------------------------------------------------
\section{Berezin kernels on Makarevich Spaces}
In this section we shall introduce the notion of a Berezin kernel
on a symmetric space of a particular type by mean of Jordan
algebraic methods.

\subsection{Jordan algebras and their conformal groups}

A finite dimensional vector space $V$ on $\mathbb R$ or $\mathbb C$
is a \emph{Jordan algebra} if it is endowed with a bilinear map
$(x,y)\to xy$ from $V\times V$ into $V$ satisfying the two following
axioms:
\begin{itemize}
\item{(J1)} $xy=yx,\quad\forall x,y\in V$,
\item{(J2)} $x(x^2y)=x^2(xy),\quad x,y\in V$.
\end{itemize}
Let $L(x)\in {\rm End}(V)$ denote for every $x\in V$ the linear map
defined by $L(x)y=xy$ for every $y\in V$. \vskip 10pt

Let $r$ and $n$ denote respectively the rank and the dimension of the
Jordan algebra $V$.
For a regular element $x$, the minimal polynomial $f_x$ is of degree $r$,
$$f_x(\lambda )=\lambda ^r -a_1(x)\lambda ^{r-1}+\cdots +(-1)^ra_r(x).$$
The coefficient $a_j$ is a homogeneous polynomial of degree $j$, $\Delta (x)=a_r(x)$ is the \emph{Jordan polynomial}, and ${\rm tr}\, (x)=a_1(x)$ is the \emph{Jordan trace} of $x$.
%For any regular element $x\in V$ one defines in a
%canonical way a polynomial $\Delta(x)$ homogeneous of degree $r$
%which is the constant term of the minimal polynomial of $x\in V$. We
%call $\Delta(x)$ \emph{Jordan determinant} of an element $x\in V$.
(See \cite{[FK]} p.28 for more details).

Define the so-called \emph{quadratic representation} of $V$ by:
$$
P(x)=2L(x)^2-L(x^2),\qquad x\in V.
$$
One checks that $\Delta(x)^{2}=\det(P(x))^{\frac r{n}}$, where
$P(x)$ is seen as an endomorphism of $V$.\vskip 10pt

A real Jordan algebra $V$ is said to be \emph{semi-simple} if the
bilinear form $\tr L(xy)$ is non degenerate on $V$, such an algebra
is unital. Furthermore it is called \emph{Euclidean} if the above
bilinear form is positive definite.

An involutive automorphism $\alpha$ of $V$ is called
\emph{Euclidean} if the bilinear form $\tr L(\alpha(x)y)$ is
positive definite on $V$. For a semi-simple Jordan algebra such a
Euclidean involution always exists. \vskip 10pt

Assume from now on that $V$ is a simple real Jordan algebra.
According to the general construction of Kantor-K\"ocher-Tits one
associates to $V$ a simple Lie
group which can be understood as a group of conformal
transformations of the corresponding Jordan algebra (in a sense we
shall explain).

Let us remind this classical construction. The \emph{structure
group} ${\rm Str}(V)$ of $V$ can be defined as the subgroup of
$GL(V)$ of elements $g$ such that there exists a real number
$\chi(g)$ for which
\begin{equation}\label{chi}
    \Delta(g.x)=\chi(g)\Delta(x),\quad x\in V.
\end{equation}
The map $g\to\chi(g)$ is a character of ${\rm Str}(V)$ which is a
reductive Lie group. \vskip 10pt
The Jordan algebra $V$ can be identified with the abelian group $N$
of its own translations via the map $y\To n_y$ from $V$ to $N$,
where $n_y(x)=x+y,\:\forall x\in V$.
The \emph{conformal group} $\Conf(V)$ (or the Kantor-K\"ocher-Tits
group) of the Jordan algebra $V$ is the group of rational
transformations of $V$ generated by translations, elements in
$\Str(V)$ and the inversion map $j:x\To -x^{-1}$. It is a simple Lie
group.
A transformation $g\in\Conf(V)$ is conformal in the sense that, at
each point $x$, where $g$ is well defined, its differential $(Dg)_x$
belongs to the structure group $\Str(V)$.

The subgroup of all affine conformal transformations
$P=\Str(V)\ltimes N$ is a maximal parabolic subgroup of $\Conf(V)$.
Let $\sigma$ be the involution of $\Conf(V)$ given by
$$
\sigma(g)=j\circ g\circ j,\qquad g\in \Conf(V),
$$
where $j\in \Conf(V)$ is the inversion map. We define $\overline
N=\sigma(N)$ and $\overline P:=\Str(V)\ltimes\overline N$.

From the geometric point of view the subgroup $\overline P$ can be
characterized in the following way:
$$
\overline P=\{g\in\Conf(V)'\:|\:g(0)=0\},
$$
where $\Conf(V)'$ is the subset of $\Conf(V)$ of all conformal
transformations well defined at $0\in V$. It is open and dense in
$\Conf(V)$. Moreover $\Conf(V)'=N\Str(V)\overline N$. The map
$N\times\Str(V)\times\overline N\To\Conf(V)'$ is a diffeomorphism.
We shall refer to this decomposition as to the \emph{Gelfand-Naimark
decomposition} of the conformal group.
Furthermore, for every transformation $g\in\Conf(V)$ which is well
defined at $x\in V$, the transformation $gn_x$ belongs to
$\Conf(V)'$ and its Gelfand-Naimark decomposition is given by :
\begin{equation}\label{gelf-naimark}
gn_x=n_{g.x}(Dg)_x\bar n',
\end{equation}
where $(Dg)_x\in\Str(V)$ is the differential of the conformal map
$x\to g.x$ at $x$ and $\bar n'\in\overline N$ (see \cite{[P]} Prop.
1.4).

\vskip 10pt

The flag variety ${\mathcal M}=\Conf(V)/\overline
P$, which is compact, is the \emph{conformal compactification} of $V$. In
fact the map $x\To(n_x\circ j)P$ gives rise to an embedding
of $V$ into $\mathcal M$ as an open dense subset, and every
transformation in $\Conf(V)$ extends to $\mathcal{M}$.

\vskip 15pt
 The
Euclidean involution $\alpha$ of $V$ introduced above also defines
an involution $\theta$ of  $\Conf(V)$
by:
\begin{equation}\label{theta}
 \theta(g)=\alpha\circ j\circ g\circ
j\circ\alpha.
\end{equation} It turns out that $\theta$ is a
Cartan involution of $\Conf(V)$ (see \cite{[P]} Prop. 1.1). So the
fix points subgroup of $\theta$ : $ U=\Conf(V)^{\theta}$ is a
maximal compact subgroup of $\Conf(V)$.

Let us remind that a simple real Jordan algebra is either a real
form of a simple complex Jordan algebra or a simple complex Jordan
algebra considered as a real one. We conclude this section with the
classification of simple Jordan algebras, together with their
conformal groups and maximal compact subgroups $U$.
\medskip

{\footnotesize\begin{tabular}{l|c|c|c|c}
    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \hline & Complex   & Euclidean & Non Euclidean & Non Euclidean  \\
   &     Non split      &  Split           &Split & Non split $(m=2\ell)$\\
   \hline
  $V$ & $Sym_m(\mathbb C)$ & $Sym_m(\mathbb R)$ & $\times$ & $Sym_{2\ell}(\mathbb R)\cap M_{\ell}(\mathbb H)$ \\
   $\Conf(V)$& $Sp_m(\mathbb C)$ & $Sp_m(\mathbb R)$ & $\times$ & $Sp(\ell,\ell)$ \\
  $U$ &  $Sp(m)$& $U(m)$ & $\times$ & $Sp(\ell)\times Sp(\ell)$ \\
  &&&&\\
 \hline
$V$ & $M_m(\mathbb C)$ & $Herm_m(\mathbb C)$ & $M_m(\mathbb R)$ & $ M_{\ell}(\mathbb H)$ \\
   $\Conf(V)$& $SL_{2m}(\mathbb C)$ & $SU(m,m)$ & $SL_{2m}(\mathbb R)$ & $SL_{2\ell}(\mathbb H)$ \\
  $U$ & $SU(2m)$ & $S(U_m\times U_m)$ & $SO(2m)$ & $SO_{2\ell}(\mathbb H)$ \\&&&&\\
 \hline
 $V$ & $Skew_{2m}(\mathbb C)$ & $Herm_m(\mathbb H)$ & $Skew_{2m}(\mathbb R)$ & $\times$ \\
   $\Conf(V)$& $SO_{4m}(\mathbb C)$ & $SO^{*}(4m)$ & $SO(2m,2m)$& $\times$ \\
  $U$ & $SO(4m)$ & $U(2m)$ & $SO_{2m}\times SO_{2m}$ & $\times$ \\&&&&\\
 \hline
 $V$ & $\mathbb C\times\mathbb C^{n-1}$ & $\mathbb R\times\mathbb R^{n-1}$ & $\mathbb R^p\times\mathbb R^{q}$
 & $\mathbb R^n$ \\
   $\Conf(V)$ & $SO_{n+2}(\mathbb C)$ & $SO_o(2,n)$ & $SO_o(p+1,q+1)$  & $SO_o(1,n+1)$ \\
  $U$ & $SO_{n+2}$ & $SO_2\times SO_n$ & $SO_{p+1}\times SO_{q+1}$ & $SO(n+1)$  \\&&&&\\
 \hline
 $V$ & $Herm(3,\mathbb O)_{\mathbb C}$ & $Herm(3,\mathbb O)$ & $Herm(3,\mathbb O_s)$ & $\times$ \\
   $\Conf(V)$& $E_7(\mathbb C)$ & $E_{7(-25)}$ & $E_{7(7)}$ & $\times$ \\
  $U$ & $E_7$ & $E_{6(6)}\times SO(2)$ & $SU(8)$ & $\times$\\
  \hline
  Type & IV&I&II&III\\
  \hline
\end{tabular}}\medskip

We shall refer to the Jordan algebras given in the first and forth
columns as to the \emph{non split case}, and to those of the second
and third columns as to the \emph{split case}. \medskip

\subsection{Makarevich Riemannian Symmetric Spaces}

A Makarevich symmetric space is a reductive symmetric space which
can be realized as an open symmetric orbit in the conformal
compactification $\mathcal M$ of a simple real Jordan algebra $V$.
We refer the reader to \cite{[B1],[B2]} and literature cited there
for a detailed description. We shall concentrate our interest on
Makarevich symmetric spaces carrying an invariant Riemannian
metric.

Let $\alpha$ be as previously a Euclidean involution of the Jordan
algebra $V$ and let
$$
V_0:=\{x\in V\,\vert\,\alpha(x)=x\},\quad V_1:=\{x\in
V\,\vert\,\alpha(x)=-x\},
$$
be the corresponding eigenspaces of $\alpha$ on $V$. Notice that the
set $V_0$ is a Euclidean Jordan algebra, whose dimension and rank
will be denoted by $n_0$ and $r_0$. This fact implies that the
interior $\Omega_0$ of the set $\{x^2\,\vert\,x\in V_0\}$ of
"positive" elements in $V_0$ is a symmetric cone in $V_0$. Notice
that $r=r_0$ in the split case and $r=2r_0$ in the non split case.
The Jordan algebra $W=V_0+iV_1$ is a Euclidean real form of the
complexification $V^{\mathbb C}=V+iV$. We will denote by $\Omega $
the symmetric cone of $W$. If $V$ is a simple Jordan algebra of type
I, II, or III, then $W$ is simple, while, if $V$ is of type IV,
$W\simeq V_0\times V_0$.\medskip

 According to \cite{[B1]} one introduces two groups:
 $$
 G:=\{g\in\Conf(V)\,\vert\,(-\alpha)\circ
 g\circ(-\alpha)=g\}_o,\, {\rm and}\, K:=\{g\in
 G\,\vert\,g.e=e\},
 $$
 where the subscript $_o$ stands as usually for the connected
 component of the identity transformation and $e$ denotes the
 identity element of the Jordan algebra $V$. It follows that $K$
 is a maximal compact subgroup of $G$. Moreover, the associated
 Riemannian symmetric space is a real tube domain:
 \begin{equation}\label{makarevic}
    {\mathcal X}:=G/K=\Omega_0+V_1.
\end{equation}

The set $ {\mathcal X}$ is a Riemannian \emph{Makarevich symmetric space}. We shall
refer to the previous description as to the tube realization of
$\mathcal X$.
%The classification table of all Makarevich spaces
%obtained in such a way is listed at the end of this section.

\bigskip

Such a Riemannian Makarevich symmetric space can be obtained as a
real form of a Hermitian symmetric space of tube type. The transform
$$x\mapsto u=(x-e)(x+e)^{-1}.$$
maps the symmetric space $\mathcal X$ onto a bounded domain
$\mathcal D$, which is the unit ball in $V$ with respect to a
so-called spectral norm. Its inverse is the \emph{Cayley transform}
$$
c\,:u\ \mapsto x=c(u)=(e+u)(e-u)^{-1}.
$$

\bigskip
If $V$ is a Euclidean Jordan algebra, then $V_0=\{0\}$, and
$\mathcal X$ is a symmetric cone. If $V$ has a complex structure,
then $\mathcal X$ is a Hermitian symmetric tube.

The following table gives the classification of Riemannian
Makarevich symmetric spaces obtained in this way.
\medskip

{\footnotesize
\begin{tabular}{c|c|c|c|c}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
   & Complex   & Euclidean & Non Euclidean & Non Euclidean  \\
   &           &            &Split & Non split $(m=2\ell)$\\
   \hline & $C_m$ & $A_{m-1}$& $D_m$& $C_{\ell}$\\
  \hline
  $V$ & $Sym_m(\mathbb C)$ & $Sym_m(\mathbb R)$ & $\times$ & $Sym_{2\ell}(\mathbb R)\cap M_{\ell}(\mathbb H)$ \\
  $G$& $Sp(m,{\Real})$ &
$SL(m,{\Real})\times {\Real}_+$ & $\times $ & $Sp(\ell ,{\mathbb C})$  \\
  $K$ &  $U_m$ & $SO_m$ & $\times $ & $Sp(\ell )$ \\&&&&\\
 \hline
$V$ & $M_m(\mathbb C)$ & $Herm_m(\mathbb C)$ & $M_m(\mathbb R)$ & $ M_{\ell}(\mathbb H)$ \\
   $G$& $SU(m,m)$  &  $SL(m,{\mathbb C})\times {\Real}_+$ & $SO(m,m)$ &
$Sp(\ell ,\ell )$ \\
  $K$& $S(U_m\times U_m)$ & $SU_m$ &
$SO_m\times SO_m$ & $Sp(\ell )\times Sp(\ell )$  \\&&&&\\
 \hline
 $V$ & $Skew_{2m}(\mathbb C)$ & $Herm_m(\mathbb H)$ & $Skew_{2m}(\mathbb R)$ & $\times$ \\
   $G$&  $SO^*(4m)$ & $SL(m,{\mathbb H})\times {\Real}_+$ & $SO(2m,{\mathbb
C})$ & $\times $ \\
  $K$ & $U_{2m}$ & $Sp(m)$ & $SO_{2m}$ & $\times $  \\&&&&\\
 \hline
 $V$ & $\mathbb C\times\mathbb C^{n-1}$ & $\mathbb R\times\mathbb R^{n-1}$ & $\mathbb R^p\times\mathbb R^{q}$
 & $\mathbb R^n$ \\
   $G$ &  $SO_0(2,n)$&$SO_0(1,n-1)\times {\Real}_+$&$SO_0(1,p)\times
SO_0(1,q)$& $SO_0(1,n)$\\
  $K$ &   $SO_2\times SO_n$ & $SO(n-1)$ &
$SO(p)\times SO_q$ & $SO_n$ \\&&&&\\
 \hline
 $V$ & $Herm(3,\mathbb O)_{\mathbb C}$ & $Herm(3,\mathbb O)$ & $Herm(3,\mathbb O_s)$ & $\times$ \\
   $G$&$E_{7(-25)}$ & $E_{6(-26)}\times {\Real}_+$ & $SU^*(8)$ &
$\times $  \\
  $K$ &  $E_6\times SO_2$ & $F_4$ & $Sp(4)$ & $\times $ \\
  \hline
\end{tabular}
}
\bigskip

The second row of this table represents the root system of the pair
$(\mathfrak g,\mathfrak a)$, where $\mathfrak g$ is the Lie algebra
of  $G$, and $\mathfrak a$ is a Cartan subspace.
\bigskip

\noindent
\emph{Example 1}.
Let $V=M(m,{\mathbb R})$, the space of $m\times m$ real matrices, be equipped
with the Jordan product $x\circ y=\frac 12 (xy+yx).$ Then the
Jordan determinant  coincides with the usual matrix determinant :
$\Delta (x)=\det x.$
The structure group ${\rm Str}(V)$ is the group
$S\bigl(GL(m,{\Real})\times GL(m,{\Real})\bigr)$, acting on $V$ by
$$x\mapsto g_1xg_2^{-1} \quad \bigl(g_1,g_2\in GL(m,{\Real})\bigr).$$
The conformal group ${\rm Conf}(V)$ is the group
$SL(2m,{\Real})/\{\pm I\}$ acting on $V$ by
$$x\mapsto (ax+b)(cx+d)^{-1} \quad
{\rm if}\ g=\left(%
\begin{array}{cc}
  a & b \\
  c & d \\
\end{array}%
\right).$$

The differential of a conformal transformation $g$ is given by $
\left(Dg\right)_xy=h_1(x)yh_2(x)$, where, under the condition that
$\det c\neq0$,
$$
h_1(x)=(ac^{-1}d-b)(cx+d)^{-1}c,\:h_2(x)=(cx+d)^{-1}.
$$
Since $\det(ac^{-1}d-b)\det c=\det\left(%
\begin{array}{cc}
  a & b \\
  c & d \\
\end{array}%
\right)=1$ we finally get:
$$
\chi\left(\left(Dg\right)_x\right)=\det(cx+d)^{-2}.
$$
The Euclidean involution $\alpha$ on $V=M(m,\Real)$ is given by
the usual matrix transposition : $ \alpha(x)=x^T$, and
$$V_0=Sym _m(\Real ),\ V_1=Skew _m(\Real ).$$
Remind
that an element $g\in\Conf(V)$ belongs to the group $G$ if and
only if
$$
g\circ(-\alpha)=(-\alpha)\circ g.
$$
If $g.x=(ax+b)(cx+d)^{-1}$, the above condition on $g$ to be in $G$
leads to
$$
(xc^T+d^T)(-ax+b)=-(xa^T+b^T)(-cx+d),
$$
or equivalently,
$$
a^Tc+c^Ta=0,\quad b^Td+d^Tb=0,
$$
and, for every $x\in V$,
$$ x(a^T+dc^Tb)=(d^Ta+b^Tc)x.$$
By Schur's
lemma this last condition says that there is $\lambda\in\Real$ such
that $ a^Td+c^Tb=\lambda{\rm Id}_m. $

Define the matrix
$$
\Upsilon=\left(%
\begin{array}{cc}
  0 & {\rm Id}_m \\
  {\rm Id}_m & 0 \\
\end{array}%
\right).
$$
Then the element $g$ belongs to $G$ if and only if $ g^T\Upsilon
g=\lambda \Upsilon.$ Since $\det g=1$, it follows that
$\lambda^{2m}=1,$ and since $G$ is connected, $\lambda=1$.\medskip

Finally we have shown that $G=SO_o(\Upsilon)/\pm{\rm Id}$, where
$SO(\Upsilon)$ stands for the special orthogonal group of the
quadratic form defined by the matrix $\Upsilon$ which has
signature $(m,m)$. Therefore $G\simeq SO_o(m,m)/\pm{\rm Id}$.

\bigskip
 The conformal compactification $\mathcal M$ of $V$ is the
Grassmann manifold $\mathcal M={\rm Gr}(2m,m)$ of $m$-dimensional
vector subspaces in ${\Real}^{2m}$.
\bigskip

 The corresponding Riemannian
Makarevich symmetric space $\mathcal X$ in its tube realization is
the set of matrices $x\in M(m,{\Real })$ with a positive definite
symmetric part:
$$x+x^T\gg 0.$$
%$$V_0={\rm Sym}(m,{\Real}),\quad V_1={\rm Skew}(m,{\Real}).$$

Its bounded realization is given by:
$${\mathcal
D}=\{x\in M(m,{\Real})\mid \|x\|_{\rm op}<1\}.$$

\medskip
\noindent \emph{ Example 2.} Let $V$ be the space ${\Real}\times
{\Real}^{n-1}$. One writes $x=(x_0,x_1),\ x_0\in {\Real},\ x_1\in
{\Real}^{n-1}$. Then the Jordan product is defined as follows :
$z=x\circ y$ if $z_0=x_0y_0-(x_1|y_1),\ z_1=x_0y_1+y_0x_1,$ where
$(x_1|y_1)$ is the usual inner product on $\Real^{n-1}$. The Jordan
determinant is then given by $\Delta (x)=x_0^2+\|x_1\|^2.$ (This
Jordan algebra is actually a Jordan field.)

The corresponding structure group is the group ${\rm
Str}(V)={\Real}_+\times O(n)$, and the conformal group ${\rm
Conf}(V)$ is equal to $O(1,n+1)$. The conformal compactification of
$V$ is the $n$-sphere $S^n$.

Consider the Euclidian involution $\alpha : (x_0,x_1)\mapsto
(x_0,-x_1)$. The corresponding groups $G$ and $K$ are isomorphic
respectively to $SO_0(1,n)$ and $SO(n)$.

The Riemannian symmetric space $\mathcal X$ is the real hyperbolic
space of dimension $n$, realized as a half-space
$${\mathcal X}=\{(x_0,x_1)\in {\Real}\times {\Real}^{n-1}
\mid x_0>0\}.$$ The bounded realization of this symmetric space is
isomorphic to the Euclidean ball:
$${\mathcal
D}=\{x\in {\Real}^n \mid \|x\|<1\}.$$ \medskip

\subsection{Iwasawa decomposition and spherical Fourier Transform}
According to (\ref{theta}) the Cartan involution  of the group $G$
is given by
$$
\theta(g)=(-j)\circ g\circ(-j),\quad g\in G.
$$
The Lie algebra $\mathfrak g$, which is an algebra of quadratic
vector fields on $V$, decomposes under $d\theta$ into a direct sum
of two eigenspaces:
$$
\mathfrak g=\mathfrak k+\mathfrak p,
$$
where the Lie algebra $\mathfrak k$ is the Lie algebra of the group
$K$ introduced above. Let us fix a Jordan frame
$\{c_1,...,c_{r_0}\}$ in $V_0$. Then the space $\mathfrak a$ of the
linear vector fields
$$\xi (x)=\sum _{j=1}^{r_0}t_jL(c_j)x\quad (t_j\in {\Real}),$$
is a Cartan subspace in $\mathfrak p$. The root system of the pair
$({\mathfrak g},{\mathfrak a})$ is of type $A$, $C$, or $D$. We
choose the Weyl chamber ${\mathfrak a}^+$ defined by $t_1<\cdots
<t_{r_0}$ in case of type $A$, or $0<t_1<\cdots <t_{r_0}$ in case
of type $C$ or $D$.

For type $A$ the positive roots are :
$\left\{\frac{t_j-t_i}2,i<j\right\}$, for type $D$ :
$\left\{\frac{t_j-t_i}2,i<j,\:\frac{t_j+t_i}2,i\neq j\right\}$ and
for type $C$: $\left\{\frac{t_j-t_i}2,i<j,\:\frac{t_j+t_i}2,i\neq
j,\:t_i\right\}$.

We define $A=\exp {\mathfrak a}$, and, as usual the nilpotent
subgroup $N$. It can be written $N=N_0\ltimes N_1$, where $N_0$ is
a triangular subgroup in ${\rm Str}(V_0)$, and $N_1$ is the group
of translations
$$x\mapsto x+v\quad {\rm with}\ v\in V_1.$$

The \emph{Iwasawa decomposition} can be described as follows: every
$x\in {\mathcal X}$ can be uniquely written
$$x=na\cdot e=n_0a\cdot e+v,$$
with $a\in A$, $n_0\in N_0$, $v\in V_1$. One writes
$$a=\exp {\mathcal A}(x)\quad {\rm with}\ {\mathcal A}(x)\in {\mathfrak a}.$$
If $\lambda \in {\mathfrak a}^*_{\mathbb C}$, we write, for $\xi
(x)=\sum _jt_j L(c_j)x$,
$$\langle \lambda ,\xi \rangle=\sum _{j=1}^{r_0} \lambda _jt_j.$$
Then
$$e^{\langle \lambda ,{\mathcal A}(x) \rangle}=\Delta _{\lambda }(x_0),$$
where $\Delta _{\lambda }$ is the power function of the symmetric
cone $\Omega _0$ in the Euclidean Jordan algebra $V_0$, and $x_0$
denotes the $V_0$-component of $x$:
$$x=x_0+x_1,\quad x_0\in V_0,\ x_1\in
V_1.$$

\medskip

The \emph{spherical functions} $\varphi _{\lambda }$ for
${\mathcal X}=G/K$ are given by
\begin{equation}\label{sphfunction}
   \varphi _{\lambda }(x)
=\int _K \Delta _{\rho -\lambda }\bigl((k\cdot x)_0\bigr) dk,
\end{equation}
where $dk$ is the normalized Haar measure of $K$, and $\rho $ is
the half-sum of the positive roots:
$$
\langle\rho,\xi\rangle=\frac12{\rm Tr}({\rm ad}\:\xi)_{\vert
Lie(N)}=\frac12{\rm Tr}({\rm ad}\:\xi)_{\vert Lie(N_0)}+\frac12{\rm
Tr}({\rm ad}\:\xi)_{\vert Lie(N_1)}.
$$
One obtains $\rho_j=\frac {d_0}4(2j-r_0-1)+\frac{n_1}{2r_0},$ where
the integer $d_0$ is defined by $n_0=r_0+\frac{d_0}2r_0(r_0-1).$

\medskip

The \emph{spherical Fourier transform} of a $K$-invariant
integrable function $f$ on $\mathcal X$ is defined on $i{\mathfrak
a}^*$ by
$${\mathcal F}f(\lambda )=\int _{\mathcal X} \varphi _{-\lambda }(x)f(x)\mu(dx)
=\int _{\mathcal X}\Delta _{\rho +\lambda }(x_0)f(x)\mu(dx),$$ where
$\mu(dx)=\Delta(x_0)^{-\frac nr}dx$ is a $G$-invariant measure on
$\mathcal X$. Here and elsewhere further the measure $dx$ denotes
the Euclidean measure associated to the Euclidean structure defined
on $V$ by $(x\:\vert\:y)={\rm tr}(x\alpha(y))$.

\subsection{Berezin kernels and the Kantor cross-ratio}

The {\it Kantor cross ratio} of four points of a simple Jordan
algebra $V$ is the rational function
\begin{equation}\label{cross-ratio}
\{x_1,x_2,x_3,x_4\}={\Delta (x_1-x_3)\over \Delta (x_2-x_3)}
:{\Delta (x_1-x_4)\over \Delta (x_2-x_4)}.
\end{equation}
 It is invariant under
conformal transformations and extends to the conformal
compactificaion $\mathcal M$ of $V$.

The invariance by translations is obvious. The invariance under
${\rm Str}(V)$ comes from the fact that $\Delta $ is semi-invariant
under ${\rm Str}(V)$. The invariance under the inverse $j:x\mapsto
-x^{-1}$ follows from the formula (\cite{[FK]}, Lemma X.4.4):
$$
\Delta \bigl(j(x)-j(y)\bigr)={\Delta (y-x)\over \Delta (x)\Delta
(y)}.
$$

Furthermore it is proved that a local transformation which preserves
the cross ratio is the restriction of an element of ${\rm Conf(V)}$
( \emph{cf}. \cite{[Kantor]}, Theorem 6).
\medskip
Being given the Kantor cross-ratio (\ref{cross-ratio}) we shall
introduce the kernel $F$ on $\mathcal X$ by
\begin{eqnarray}
F(x,y)&=&\{x,y,-\alpha (x),-\alpha (y)\}\\
& =&{\Delta \bigl(x+\alpha (x)\bigr)\Delta \bigl(y+\alpha
(y)\bigr) \over \Delta \bigl(x+\alpha (y)\bigr)\Delta
\bigl(y+\alpha (x)\bigr)}.\nonumber
\end{eqnarray}
 This function is invariant under $G$ and
positive. The {\it Berezin kernel} $B_\nu$ can be defined for every
$\nu\in\mathbb C$ as
\begin{equation}\label{berezin}
    B_{\nu }(x,y)=F(x,y)^{\frac{r_0}{r}\nu }=\left\{
\begin{array}{l}
F(x,y)^\nu\quad{\rm in}\:{\rm
the}\:{\rm split}\:{\rm case}\\
F(x,y)^{\frac\nu2}\quad{\rm in}\:{\rm the}\,{\rm non}\,{\rm
split}\,{\rm case} \end{array}\right.
\end{equation}
In virtue of the $G$-invariance of the kernel $B_{\nu }$ the
\emph{Berezin function}
$$
\psi _{\nu }(x) =B_{\nu }(x,e) =\left({\Delta (x_0)\over \Delta
\Bigl({x+e\over 2}\Bigr)^{2}}\right)^{\frac{r_0}{r}\nu }
$$
 is $K$-invariant. In this formula $x_0$ denotes as before the $V_0$-component of
$x$ in the decomposition $V=V_0+V_1,\ x=x_0+x_1.$ \medskip

If $\{c_1,\ldots ,c_{r_0}\}$ is a Jordan frame of $V_0$, and if
$x=\sum _{j=1}^{r_0} e^{t_j}c_j,$ then
$$\psi _{\nu }(x)=\prod _{j=1}^{r_0} \Bigl(\cosh {t_j\over 2}\Bigr)^{-2\nu }.$$

\medskip
\begin{thm}\label{thm1} Assume $\Re\nu>\frac n{r_0}-1$. Then the
function $\psi_{\nu}$ is integrable and its spherical Fourier
transform is given by
\begin{equation}\label{fourierspherik}
 {\mathcal F}\psi _{\nu}(\lambda ) =\frac{P(\lambda,\nu)P(-\lambda,\nu)}{Q(\nu)},
\end{equation}
with
\begin{eqnarray*}
  P(\lambda,\nu) &=& \prod_{j=1}^{r_o}\Gamma\left(\frac 12+\nu-\delta+\lambda_j\right),\quad \delta=\frac{n}{2r_0}\\
  Q(\nu) &=& c\prod_{j=1}^{2r_o}\Gamma\left(\nu-\beta_j\right).
\end{eqnarray*}
where the constant $c$\footnote{We denote by $c$ throughout this
paper different constants depending only on $V$.} and the real
numbes $\beta_j$'s depend on the Jordan algebra $V$. In particular,
for $\nu$ real and $\lambda\in i\mathfrak{a}^*$,
$\bar\lambda=-\lambda$ and
$$
 {\mathcal F}\psi _{\nu}(\lambda ) =\frac{|P(\lambda,\nu)|^2}{Q(\nu)}\geq0,
$$
\end{thm}

Remind that the Gindikin $\Gamma$-function of a symmetric cone
$\Omega$ is defined by
$$
\Gamma_{\Omega}(\nu)=\int_{\Omega}e^{-{\rm
tr}(x)}\Delta(x)^{\nu-\frac nr}dx.
$$
If the cone $\Omega$ is irreducible (the corresponding Euclidean
Jordan algebra $W$ is simple), then
$$
\Gamma_{\Omega}(\nu)=(2\pi)^{\frac{n-r}2}\prod_{j=1}^r\Gamma\left(\nu-(j-1)\frac
d2\right),
$$
where the integer $d$ is defined by $n=r+\frac d2r(r-1)$.

Therefore, for types I and II:
$$
Q(\nu)=c2^{-2r\nu}\Gamma_{\Omega}(2\nu).
$$
For type III:
$$
Q(\nu)=c \Gamma_{\Omega}(\nu).
$$
For type IV:
$$
\Gamma_{\Omega}(\nu)=(\Gamma_{\Omega_0}(\nu))^2,
$$
notice that in this case the Jordan algebra $W$ is not simple
anymore : $W\simeq V_0\times V_0$.

 The above theorem was proved by different methods  in
\cite{[DP],[Z01],[UU]}.


\bigskip

 Let $h(z,w)$ stand for the so-called \emph{canonical polynomial}
defined on $V^{\mathbb C}\times V^{\mathbb C}$ by the following
conditions (see \cite{[FK]} p.262):

-It is holomorphic in the first variable and anti-holomorphic in the
second one.

-For every $g\in\Str(V^{\mathbb C})$ one has $h(gz,w)=h(z,g^*w)$.

-For every $x\in W$ one has $h(x,x)=\Delta(e-x^2)$.\medskip

Recall that the Bergman kernel of the Hermitian symmetric space,
whose $\mathcal{X}$ is a real form, is given by
$h(z,w)^{-\frac{2n}r}$.
\bigskip

\emph{Example 1.} For $V=M(m,\mathbb R)$ the canonical polynomial is
given by $h(x,x)=\det(I-xx^T)$.\medskip

\emph{Example 2.} For $V=\Real\times \Real^{n-1}$ the canonical
polynomial is given by $h(x,x)=(1-\Vert x\Vert^2)^2.$\bigskip


Remind that the Cayley transform
 $$
 c: u\mapsto c(u)=(e+u)(e-u)^{-1},
 $$
 maps the bounded domain $\mathcal{D}$ onto the tube
 $\mathcal{X}=\Omega_0+V_1$.

 Define
$$\tilde F(u,v)=F(c(u),c(v)).$$ Since
$-\alpha(c(u))=c(\alpha(u^{-1}))$ we obtain
$$
\tilde F(u,v):=\{u,v,\alpha(u^{-1}),\alpha(v^{-1})\}={\Delta
\bigl(u-\alpha (u^{-1})\bigr)\Delta \bigl(v-\alpha (v^{-1})\bigr)
\over \Delta \bigl(u-\alpha (v^{-1})\bigr)\Delta \bigl(v-\alpha
(u^{-1})\bigr)}.
$$
Furthermore $\tilde F(u,0)=\lim_{v\to 0}\tilde
F(u,v)=\Delta(\alpha(u^{-1})-u)\Delta(\alpha(u))$, and one can
show that $\tilde F(u,0)=h(u,u)$.
\bigskip

Define similarly
\begin{eqnarray*}
  \tilde B_{\nu}(u,v) &=& B_{\nu}(c(u),c(v)), \\
  \tilde\psi_{\nu}(u) &=& \psi_{\nu}(c(u)).
\end{eqnarray*}
Then
$$
\tilde\psi_{\nu}=\left\{
\begin{array}{l}
h(u,u)^\nu\quad{\rm in}\:{\rm
the}\:{\rm split}\:{\rm case},\\
h(u,u)^{\frac\nu2}\quad{\rm in}\:{\rm the}\:{\rm non}\,{\rm
split}\:{\rm case}.
\end{array}\right.
$$

Consider the integral
$$
I(\nu)=\int_{\mathcal D}\tilde\psi_{\nu}(u)h(u,u)^{-\frac nr}du.
$$
It can be written as
$$
\int_{\mathcal D}h(u,u)^{\frac{r_0}r\nu-\frac nr}du.
$$
Since $h(u,u)^{-\frac nr}du$ is an invariant measure on $\mathcal D$
the last integral equals
$$
 I(\nu)=c\int_{\mathcal X}\psi_{\nu}(x)\mu(dx)=c{\mathcal
F}\psi_{\nu}(\rho).
$$
Therefore we obtain
\begin{cor}
$$
I(\nu)={\rm vol}\:{\mathcal D}\frac{Q(\frac nr)}{P(\rho,\frac
nr)P(-\rho,\frac nr)}\cdot\frac{P(\rho,\nu)P(-\rho,\nu)}{Q(\nu)}.
$$
%where ${\rm vol}\:{\mathcal D}=I(\frac nr)=c\frac{P(\rho,\frac
%nr)}{Q\left(\frac nr\right)}.$
\end{cor}

Eventually we may notice that according to the Jordan algebra's type
some simplifications in the above formula, due to explicit
expressions of $\Gamma$-factors, are possible, see examples below.
\bigskip

\emph{Example 1.} Let $V=M(m,\Real)$. Then $V_0=Sym_m(\Real),
V_1=Skew_m(\Real)$, the invariant measure on $\mathcal X$ is given
by $\det(x_0)^{-m}dx_0dx_1$, and the Berezin function is
$$
\psi_{\nu}(x)=4^{m\nu}\left[\frac{\det
x_0}{\det(e+x)^2}\right]^{\nu},\:{\rm with}\: \phi_\nu(e)=1.
$$
Its push-forward to $\mathcal D=\{u\in V\:|\:\Vert u\Vert_{\rm
op}<1\}$ equipped with the invariant measure $\det(e-uu^T)^{-m}du$
is given by
$$
\widetilde\psi_\nu(u)=\det(e-uu^T)^\nu.
$$

The spherical Fourier transform of $ \psi_{\nu}$ is the given by
\begin{eqnarray*}
  {\mathcal F}\psi_{\nu}(\lambda) &=& \int_{\omega_0+V_1}\left[\frac{\det
x_0}{\det(e+x)^2}\right]^{\nu}\Delta^o_{\rho-\lambda}(x_0)\Delta^o(x_0)^{-m}dx_0dx_1 \\
   &=& c\frac{\prod_{j=1}^m\Gamma(\nu-\frac{m-1}2+\lambda_j)\Gamma(\nu-\frac{m-1}2-\lambda_j)}
   {4^{-m\nu}\prod_{j=1}^m\Gamma(2\nu-j+1)}.
\end{eqnarray*}
Notice that in this case
\begin{eqnarray*}
Q(\nu)&=&c4^{-m\nu}\Gamma_\Omega(2\nu)=c'\Gamma_{\Omega_0}(\nu)\Gamma_{\Omega_0}(\nu+\frac
12)\\&=&c''\prod_{j=1}^m\Gamma(\nu-\frac{j-1}2)\prod_{j=1}^m\Gamma(\nu+\frac
12-\frac{j-1}2).
\end{eqnarray*}
 Moreover
$$
I(\nu)=\frac{\Gamma_{\Omega_0}(\nu-\frac{m-1}2)}{\Gamma_{\Omega_0}(\nu+\frac
12)}.
$$

\emph{Example 2.} In the case when $V=\Real^n=\Real+\Real^{n-1}$ the
Jordan determinant is given by $\Delta(x)=x_0^2+\Vert x_1\Vert^2$
and the Berezin function is given by
$$
\psi_{\nu}(x)=\left[\frac{4x_0}{(1+x_0)^2+\Vert
x_1\Vert^2}\right]^\nu.
$$
Its spherical Fourier transform is equal to
$$
{\mathcal
F}\psi_{\nu}(\lambda)=c\frac{\Gamma(\nu+\lambda-\frac{n-1}2)\Gamma(\nu-\lambda-\frac{n-1}2)}
{\Gamma(\nu)\Gamma(\nu+\frac 12-\frac{n-1}2)}.
$$
And finally,
$$
I(\nu)=c\frac{\Gamma(\nu-n+1)}{\Gamma\left(\nu+\frac12-\frac{n-1}2\right)}.
$$
%\bigskip

\subsection{A Bernstein identity}

\bigskip

An interesting consequence of Theorem (\ref{thm1}) is a Bernstein
identity. Let $D(\nu )$ be the invariant differential operator on
${\mathcal X}=G/K$ whose symbol, {\it i.e.} its image through the
Harish-Chandra isomorphism
$$\gamma : {\mathbb D}({\mathcal X})\rightarrow S({\mathbb C}^{r_0}), \quad
D\mapsto \gamma _D(\lambda ),$$ is
$$\gamma _{D(\nu )}(\lambda )
=\gamma _{\nu }(\lambda ) =\prod _{j=1}^{r_0}(\nu +\frac
12-\delta+\lambda _j)(\nu+\frac 12-\delta-\lambda_j).$$

\begin{cor}\label{bernstein} The Berezin kernel (\ref{berezin}) satisfies the
following identity:
$$
D(\nu)B_{\nu }=b(\nu )B_{\nu +1},
$$
where $b(\nu )$ is the polynomial of degree $2r_0$ given by:
$$
b(\nu)=\frac{Q(\nu+1)}{Q(\nu)}=\prod_{j=1}^{2r_o}(\nu-\beta_j).$$
\end{cor}

In the split case (types I and II):
$$
b(\nu)=\prod_{j=1}^{r_0}\left(\nu-\frac
d4(j-1)\right)\left(\nu+\frac12-\frac d4(j-1)\right).
$$
In case when $V$ is of type III $(r=2r_0)$,
$$
b(\nu)=\prod_{j=1}^{2r_0}\left(\nu-\frac{d}2(j-1)\right),
$$
where the integer $d$ is defined through $n=r+\frac d2r(r-1)$.

In case when $V$ is complex (type IV):
$$
b(\nu)=\prod_{j=1}^{r_0}\left(\nu-\frac{d_0}2(j-1)\right)^2.
$$

 This identity has been established for $V$ complex
by Engli$\check{\rm s}$ \cite{[Englis]}, also in a slightly
different form by Unterberger and Upmeier \cite{[UU]}, and has been
generalized in \cite{[Fa]}.

In general we do not know any explicit expression for the
differential operator $D(\nu )$. However an explicit formula has
been obtained in the special case of a simple complex Jordan
algebra, and for $\nu =\delta $ \cite{[Khlif]}:
$$
   D\left(\delta\right )=\Delta (y)^{1+{n\over
r}} \Delta \Bigl({\partial \over \partial z}\Bigr) \Delta
\Bigl({\partial \over
\partial \bar z}\Bigr) \Delta (y)^{1-{n\over r}},$$ where $z=x+iy$.


Observe that, for $V={\mathbb C}$ we get:
$$D\left(\delta\right)=y^2{\partial ^2\over \partial z\partial \bar
z}$$ is nothing but the Laplace-Beltrami operator of the upper
hyperbolic half-plane.

\bigskip
\emph{Example 1.} When $V=M(m,{\Real})$ the Bernstein polynomial and
the Harish-Chandra symbol are respectively given by
\begin{eqnarray*}
b(\nu)&=&\prod_{j=1}^m\left(\nu-\frac{j-1}2\right)\left(\nu+\frac
12-\frac{j-1}2\right),\\
\gamma_\nu(\lambda)&=&\prod_{j=1}^m\left(\nu-\frac12({m-1})+\lambda_j\right)\left(\nu-\frac12({m-1})-\lambda_j\right).
\end{eqnarray*}

\emph{Example 2.} When $V={\Real}\times {\Real}^{n-1}$ the Bernstein
polynomial and the Harish-Chandra symbol are respectively given by
$$
b(\nu)=\nu\left(\nu-\frac
n2+1\right),\quad\gamma_\nu(\lambda)=\left(\nu+\lambda-\frac{n-1}2\right)\left(\nu-\lambda-\frac{n-1}2\right).$$
\bigskip
\subsection{ Hua Integral}
Let us introduce the compact dual $U/K$ of the symmetric space
$\mathcal X=G/K$ in the complexification $\mathcal X^{\mathbb
C}=G^{\mathbb C}/K^{\mathbb C}$ of the symmetric space $\mathcal X$.

Define $\mathfrak u=\mathfrak k+i\mathfrak p$ and let $U$ be the
analytic subgroup of $G^{\mathbb C}$ with the Lie algebra $\mathfrak
u$. The compact symmetric space $\mathcal Y=U/K$ is isomorphic to
the conformal compactification $\mathcal M$ of the Jordan algebra
$V$.
 %The real linear space $iV$ can be seen as an open and dense
%subset of $\mathcal Y$. Indeed:
%$$
%\mathcal Y\cap V^{\mathbb C}=iV.
%$$

The set $\mathcal{Y}'$ of invertible elements $y$ in $V^{\mathbb C}$
such that $\bar y=y^{-1}$ is open and dense in $\mathcal{Y}$.

For $\nu=-\kappa,\:(\kappa\in\mathbb N)$ the function
$\psi_{-\kappa}$ extends as a meromorphic function on $\mathcal
X^{\mathbb C}$. For $y=\sum\limits_{j=1}^{r_0}e^{i\theta_j}c_j$ we
have
$$
\psi_{-\kappa}(y)=\prod_{j=1}^{r_0}\left(\cos^2\frac{\theta_j}2\right)^{\kappa}.
$$
This shows that this function is well defined on $\mathcal Y$ for
$\kappa\in \mathbb C$, and bounded for $\Re\kappa\geq0$.

Denote by $\mu_o$ the normalized invariant measure on $\mathcal{Y}$
and define
\begin{equation}\label{hua}
    J(\kappa)=\int_{\mathcal{Y}}\psi_{-\kappa}(y)\mu_o(dy).
\end{equation}

\begin{thm}\label{thmhua}
For $\Re\kappa\geq0$ the following identity holds:
\begin{eqnarray*}
  J(\kappa) &=& c%\frac{\prod_{j=1}^{r_o}\Gamma(\frac
%12+\delta-\rho_j)\Gamma(\frac 12+\delta+\rho_j)}
%{\prod_{j=1}^{2r_o}\Gamma(1+\beta_j)} \\
%   &\cdot&
   \frac{\prod_{j=1}^{2r_o}\Gamma(\kappa+1+\beta_j)}{\prod_{j=1}^{r_o}\Gamma(\kappa+\frac
   12+\delta-\rho_j)\Gamma(\kappa+\frac 12+\delta+\rho_j)}.
\end{eqnarray*}
\end{thm}
\proof Assume first that $\kappa\in\mathbb N$. The Bernstein
identity (\ref{bernstein}) implies that
$$
 D_{-\kappa}\psi_{-\kappa}=b(-\kappa)\psi_{-\kappa+1},
$$

By integrating this identity on $\mathcal{Y}$ we obtain
\begin{eqnarray*}
b(-\kappa)J(\kappa+1)&=&\int_{\mathcal{Y}}D_{-\kappa}\psi_{-\kappa}\mu_o(dy)\\
&=&\int_{\mathcal{Y}}\psi_{-\kappa}(y)D_{-\kappa}1\mu_o(dy),
\end{eqnarray*}
since the operator $D_{-\kappa}$ is self adjoint. The constant term
of the differential operator $D_{-\kappa}$ is given by
$$
D_{-\kappa}1=\gamma_{-\kappa}(\rho).
$$

We finally get
\begin{equation}\label{recurHua}
    \gamma_{-\kappa}(\rho)J(\kappa)=b(-\kappa)J(\kappa-1).
\end{equation}

On the other hand the sequence
$$
\phi(\kappa)=\frac{\prod_{j=1}^{2r_o}\Gamma(\kappa+1+\beta_j)}
{\prod_{j=1}^{r_o}\Gamma(\kappa+\frac12+\delta-\rho_j)\Gamma(\kappa+\frac12+\delta+\rho_j)}
$$
satisfies the same recursion relation. Indeed,
$$
\phi(\kappa)=\frac{\prod_{j=1}^{2r_o}(\kappa+\beta_j)}{\prod_{j=1}^{r_o}(\kappa+\frac12+\delta-\rho_j)
(\kappa+\frac12+\delta+\rho_j)}\phi(\kappa-1).
$$
Therefore $J(\kappa)=c\phi(\kappa)$ and because of the ``initial"
condition $J(0)=1$, we finally get
$$
J(\kappa)=\frac{\phi(\kappa)}{\phi(0)}.
$$
The functions $\phi(\kappa)$ and $J(\kappa)$ are holomorphic for
$\Re\kappa>0$ and grow polynomially :
\begin{eqnarray*}
  |\phi(\kappa)| &\leq& c_1(1+|\kappa|)^{N_1} \\
  |J(\kappa)|&\leq& J(0)=1.
\end{eqnarray*}
Indeed, $0\leq \psi_{-\kappa}(y)\leq 1$. Therefore, the Carlson's
theorem implies that
$$
J(\kappa)=\frac{\phi(\kappa)}{\phi(0)},
$$
for every $\kappa\in\mathbb C$ such that
$\Re\kappa>0.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box$
\bigskip

By writing $y'=c(iv)$, $J(\kappa)$ becomes an integral over $iV$.
One shows that
$$
\tilde\psi_{-\kappa}(iv):=\psi_{-\kappa}(c(iv))=h(v,-v)^{-\frac{r_0}r\kappa}.
$$
Therefore, $$J(\kappa)=\int_Vh(v,-v)^{-\frac{r_0}r\kappa-\frac
nr}.$$ In fact, in this realization an invariant measure $\mu_o$ on
$\mathcal Y$ is given by $h(v,-v)^{-\frac nr}dv$.

 The above integral has been considered and computed by Hua
in \cite{[Hua]}, Chapter II in several special cases.
\section{Berezin kernels and representations}
Following ideas developed in \cite{[vdh],[vdm],[Hille]} we shall now
give an alternative approach to the theory of Berezin kernels
considering intertwining operators for the maximal degenerate series
representations of a conformal group.

\subsection{Maximal degenerate series representations}

The character $\chi$ of the structure group $\Str(V)$ (\emph{cf}.
(\ref{chi})) can be trivially extended to the whole parabolic
subgroup $\overline P$ by $\chi(h\bar n):=\chi(h)$ for every
$h\in\Str(V),\:\bar n\in\overline N$.

For every $s\in\mathbb C$ we define a character $\chi_{s}$ of
$\overline P$ by
$$
\chi_{s}(\bar p):=|\chi_(\bar p)|^{s+\frac{n}{2r}}.
$$

The induced representation $\widetilde{\pi_{s}}=\Ind_{\overline
P}^{\Conf(V)}\left(\chi_{s}\right)$ of the conformal group acts on
the space
$$
\widetilde I_{s}:=\{f\in C^{\infty}(\Conf(V))\:|\:f(h\bar
p)=\chi_{s}(\bar p)f(h),\forall h\in\Str(V),\bar p\in\overline P\},
$$
by left translations. A pre-Hilbert structure on $\widetilde I_{s}$
is given by
$$\norm f^2=\int_U\abs{ f(u)}^2du,$$ where $U$ is
the maximal compact subgroup of $\Conf(V)$ associated with the
Cartan involution $\theta$, and $du$ is the normalized Haar measure
of $U$.

According to the Gelfand-Naimark decomposition a function
$f\in\widetilde I_{s}$ is determined by its restriction
$f_V(x)=f(n_x)$ on $N\simeq V$. Let $ I_{s}$ be the subspace of
$C^{\infty}(V)$ of functions $f_V$ with $f\in \widetilde I_{s}$. The
conformal group acts on $I_{s}$ by:
\begin{equation}\label{action}
    \pi_{s}(g)f(x)=|A(g,x)|^{s+{n\over 2r}}f(g^{-1}.x),\:g\in\Conf(V),\:x\in V,
\end{equation}
where
$$A(g,x):=\chi_{s}\bigl((Dg^{-1})_x\bigr).$$
This action is usually called the \emph{even maximal degenerated
series} representation of $\Conf(V)$.

One shows that the norm of a function $f(n_x)=f_V(x)\in I_{s}$ is
given by:
\begin{equation}\label{norm}
    \norm f^2=\int_V\vert f_V(x)\vert^2h(x,-x)^{2\Re s}dx,
\end{equation}

The formula (\ref{norm}) implies that for $\Re s=0$ the space
$I_{s}$ is contained in $L^2(V)$ and the representation $\pi_{s}$
extends as a unitary representation on $L^2(V)$.

%Notice that
%\begin{equation}\label{h(x,x)}
%\delta_o(n_x)=h(x,-x),
%\end{equation}
%where $h(z,w)$ is the canonical polynomial introduced in the first
%section.
\bigskip

In order to address the question of irreducibility and unitarity of
these representations we refer the reader to \cite {[P]},
\cite{[S2],[S1]} and to \cite{[Z02]}. \vskip 15pt

 Following the standard procedure
we introduce an intertwining operator between $\pi_{s}$ and
$\pi_{-s}$.

Consider the map $\widetilde A _{s}$ defined on $\widetilde I_{s}$
by
\begin{equation}\label{intertwiner}
    f\To(\widetilde A_sf)(g):=\int_Nf(gjn)dn,\quad\forall g\in\Conf(V),
\end{equation}
where $dn$ is a left invariant Haar measure on $N$. We will see that
this integral converges for $\Re s>{n\over 2r_0}$.


%We postpone
%the description of the set $\mathcal W$ of values of the parameter
%$s$ for which the above integral is meaningful, i.e. its kernel
%function is locally integrable, till the end of this section.

\begin{prop}
For every $f\in\widetilde I_{s}$ the function $\widetilde A_sf$
belongs to $\widetilde I_{-s}$ and the map $\widetilde A_s$ given by
(\ref{intertwiner}) intertwines the corresponding representations of
the conformal group:
\begin{equation}\label{intertw-repr}
    \widetilde{\pi}_{-s}(g)(\widetilde A_sf)=\widetilde A_s(\widetilde{\pi}_{s}(g)f),\:\forall
    f\in\widetilde I_{s},\:g\in\Conf(V).
\end{equation}
\end{prop}

\emph{Proof.} The map $\widetilde A_s$ obviously commutes with the
left action of $\Conf(V)$. We have to check that $\widetilde A_sf$
transforms in an appropriate way under the right action of
$\overline P$. Notice first that $\widetilde A_sf$ is $\overline N$
right invariant. Indeed
\begin{eqnarray*}
(\widetilde A_sf)(g\bar n)&=&\int_Nf(g\bar
njn)dn=\int_Nf(gjn'jjn)dn\\&=&\int_Vf(gjn'')dn''=(\widetilde
A_sf)(g).
\end{eqnarray*}
For what concerns the action of $\Str(V)$ we have for every
$f\in\widetilde I_{s}$:
\begin{equation}\label{AfH}
    (\widetilde A_sf)(g\ell)=\int_Nf(g\ell jn)dn=\int_Nf(gj(\ell^t)^{-1}n)dn,
\end{equation}
where $h^t$ denotes the transformation adjoint to $h\in\Str(V)$ with
respect to the bilinear form $\tr L(xy)$ on $V$.

Indeed, one shows, (see \cite{[FK]}Prop.VIII.2.5) that $j\ell
j=(\ell^t)^{-1},\:\forall \ell\in\Str(V)$. Thus the equality
(\ref{AfH}) implies that
\begin{eqnarray*}
  (\widetilde A_sf)(g\ell) &=& \int_Nf(gj(\ell^t)^{-1}n\ell^t(\ell^t)^{-1})dn \\
  &=& \abs{\det \ell}^{\frac nr}\int_Nf(gjn'(\ell^t)^{-1})dn' \\
   &=& |\det \ell|^{\frac nr-s\frac{n}{2r}}(\widetilde A_sf)(g)=\chi_{-s}(\ell)(\widetilde A_sf)(g)\qquad\qquad\qquad\Box
\end{eqnarray*}

\vskip 10pt

Define the map $A_s: I_s\rightarrow I_{-s}$ by
$$
A_s(f_V)=(\widetilde A_s f)_V\quad (f\in \widetilde I_s).
$$

\begin{prop}
For every $f\in I_s$ we have,
$$(A_sf)(x)=\int _V |\Delta (x-u)|^{2s-{n\over r}}f(u)du.$$
\end{prop}

\proof By definition of $\widetilde A_s$,
$$
(\widetilde A_sf)(n_x)=\int _V f(n_xjn_v)dv\quad (f\in \widetilde I_s).
$$
According to the Gelfand-Naimark decomposition (\ref{gelf-naimark})
for $g=n_xj$ we have
$$
  n_xjn_v=n_{(n_x\circ j).v}(D(n_x\circ j))_v\bar
  n')=n_{(x-v^{-1})}P(v^{-1})\bar n'.
$$

Thus
$$
\left(A_sf\right)(x) =\int_Vf(x-v^{-1})\chi_{s}(P(v)^{-1})dv.
$$
Let $u=x-v^{-1}$. The Jacobian of this transformation being equal to
$|\Delta (v)|^{\frac{2n}r},$ we finally get:
\begin{equation}\label{intertwiner_V}
 \left(A_sf\right)(x)=\int_Vf(u)|\Delta(x-u)|^{2s-\frac{n}r}du.\qquad\quad\qquad\Box
\end{equation}
\medskip
The bilinear form on $ \widetilde I_{-s}\times \widetilde I_{s}$
defined by
$$
\langle f_1,f_2\rangle=\int_Uf_1(u)f_2(u)du
$$
is $\Conf(V)$-invariant. By using the fact that
$$
\langle f_1,f_2\rangle =c\int _V (f_1)_V(x)(f_2)_V(x)dx.
$$
it gives rise to a $\Conf(V)$-invariant bilinear form on $I_s\times
I_s$:

\begin{prop}\label{Bs}
For $\Re s>{n\over 2r_0}$, the bilinear form $\mathfrak{B}_s$,
$$
 \mathfrak{B}_s(f_1,f_2)
=\int \int _{V\times V}|\Delta (x-y)|^{2s-{n\over
r}}f_1(x)f_2(y)dxdy,
 $$
is well defined on $I_s\times I_s$ and is ${\rm Conf}(V)$-invariant.
\end{prop}
\proof The invariance follows from the intertwining property of
$A_s$. It can also be shown by a straightforward verification.
Indeed, let us recall following property of conformal
transformations (\emph{cf}. \cite{[FG]}, Lemma 6.6) :
\begin{equation}\label{FG}
    \Delta(g.x-g.y)^2=A(g,x)A(g,y)\Delta(x-y)^2,\quad g\in\Conf(V),\:x,y\in V.
\end{equation}
Therefore,
\begin{eqnarray*}
  &&\mathfrak{B}_s(\pi_s(g)f_1,\pi_s(g)f_2)= \\
  &=& \iint\limits_{V\times V}|\Delta(x-y)|^{2s-\frac nr}|A(g^{-1},x)|^{s+\frac n{2r}}
| A(g^{-1},y)|^{s+\frac n{2r}} f_1(g^{-1}.x)f_2(g^{-1}.y)dxdy
\end{eqnarray*}
Denote $x=g.x',\:y=g.y'$. Thus $dx=A(g,x')^{\frac nr}dx'$ and
$dy=A(g,y')^{\frac nr}dy'$. Notice that $g\cdot g^{-1}={\rm Id}$ and
therefore $\left(Dg^{-1}\right)_{g.x}\circ\left(Dg\right)_x={\rm
Id}$. It implies that $A(g^{-1},g.x)A(g,x)=1$.

Taking into account the identity (\ref{FG}) we finally get
\begin{eqnarray*}
 &&\mathfrak{B}_s(\pi_s(g)f_1,\pi_s(g)f_2) = \\
  &=& \iint\limits_{V\times V}|\Delta(x'-y')|^{2s-\frac nr}|A(g^{-1},x')|^{s-\frac n{2r}}
  |A(g^{-1},y')|^{s-\frac n{2r}}\\
  &\cdot&A(g,x')^{-s-\frac n{2r}} A(g,y')^{-s-\frac n{2r}}
  |A(g,x')|^{\frac nr}|A(g,y')|^{\frac nr}f_1(x')f_2(y')dx'dy'\\&=&\mathfrak{B}_s(f_1,f_2).
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box
\end{eqnarray*}

\bigskip

 According to the general theory (see for instance \cite{[K]})
these maximal degenerate series representations $\pi_{s}$ are
spherical. We shall determine the corresponding $U$-fixed vector of
the representation $\pi_{s}$.

Using the Iwasawa decomposition of $\Conf(V)$ we write
$g=u\ell(g)\bar n$, where $\ell(g)\in\Str(V)$, which is defined up
to a multiplication by an element of $U\cap\Str(V)$ on the left.
Notice by the way that with the above notation we have
$\theta(g^{-1})g=n^{-1}\ell(g)^2\bar n$.

Consider $n_x\in N$ admitting the decomposition $n_x=u\ell(n_x)\bar
n$. Therefore
$$
\ell(n_x)^2=D(\theta(n_{-x})n_x)(0)=P(x^{-1}+\alpha(x))^{-1}P(x)^{-1}.
$$
Let $f^{o}\in\widetilde I_{s}$ be a $U$-invariant function, with
$f^{o}(e)=1$. Then
$$
f^{o}(g)=f^{o}(u\ell(g)\bar
n)=\chi_{s}(\ell(g))f^{o}(u)=\chi_{s}(\ell(g)).
$$
Eventually we get that the $U$-fixed vector $f^{o}_V$ in $I_{s}$ is
given by
$$
f^{o}_V(x)=|\Delta(x^{-1}+\alpha(x))\Delta(x)|^{-(s+\frac{n}{2r})}=h(x,-x)^{-(s+\frac
n{2r})}.
$$

\bigskip

In order to conclude this section we shall discuss conditions under
which the intertwining operator $A_s$ is well defined.

Let us evaluate $A_sf^o$ at the identity of the conformal group.
$$
\widetilde
A_sf^o(e)=\int_Nf^o(jn)dn=\int_Nf^o(n)dn=\int_Vh(x,-x)^{-(s+\frac
n{2r})}dx.$$ Notice that the value $C(s)$ of this integral equals
$$C(s)=J\left(\frac r{r_0}(s-\frac{n}{2r})\right),$$
where $J(\nu)$ is given by (\ref{hua}). For $u\in U$,
$f^o(u)\not=0$. therefore, for $f\in \widetilde I_s$,
$$|f(g)|\leq M|f^o(g)|,$$
with
$$M= \sup _{u\in U}{|f(u)|\over {|f^0(u)|}}.$$
Now it is clear that the integral defining $A_s$ is well defined for
$\Re s>{n\over 2r_0}$. One shows that the map $s\to A_s$ can be
actually extended to whole complex plane as a meromorphic function.
Because of the intertwining property we eventually get:
$$
A_s\circ A_{-s}=C(s)C(-s){\rm Id}_{ I_s}.
$$

As it has been noticed by several authors \cite{[BLIN],[KS],[Z02]}
for some particular values of the parameter $s$ the intertwining
operator $A_s$ turns out to be a differential operator, which,
furthermore, satisfies a generalized Capelli identity. Indeed, the
map $A_s$ may be seen as a convolution operator with the kernel
$|\Delta(x-y)|^{2s-\frac nr}$ which has a meromorphic continuation
with respect to $s$. One shows that, for $2s-\frac
nr=-2\ell,\:\ell\in\mathbb N$, this is a differential operator
proportional to $\Delta\left(\frac{\partial}{\partial
x}\right)^{2\ell}.$
\bigskip

\subsection{Restriction of $\pi_s$ and Canonical representations}
\bigskip

We shall study the representations $\pi_{s}$  when restricted to the
subgroup $G$. The representation space $\widetilde I_s$ can be seen
as a line bundle over the conformal compactification $\mathcal{M}.$

The Makarevich space $\mathcal{X}$ is one of the open orbits of $G$
acting on $\mathcal{M}$. The space $I_{s}({\mathcal X})$ of
functions in $I(s)=I_{s}(V)$ supported in the closure
$\overline{\mathcal X}\subset V$ is then invariant. We shall
consider the corresponding representation $T_s$ of $G$ on
$I_{s}({\mathcal X})$.

The problem is to determine for which values of the parameter $s$
the representation $T_s$ of the group $G$ is unitarizable, and then
to decompose it into irreducible ones. According to an established
terminology taking its roots in \cite{[GVG]} one calls such
representations \emph{canonical representations} of the group $G$.

A key observation is the connection between the canonical
representation $T_s$ and the Berezin kernel. We shall make this link
clear.\medskip

Consider the bilinear form on $I_{s}({\mathcal X})\times
I_{s}({\mathcal X})$ given by
\begin{equation}\label{bformmakar}
    \widetilde{\mathfrak B}_s^{\alpha}(f_1,f_2):=\mathfrak{B}_s(f_1,f_2\circ(-\alpha)),
\end{equation}
where $\mathfrak{B}_s(f_1,f_2)$ is the bilinear form on
$I_{s}(V)\times I_{s}(V)$ introduced in Proposition (\ref{Bs}).
\begin{prop}
The bilinear form $\widetilde{\mathfrak B}_s^{\alpha}$ is invariant
under the action of the group $G$.
\end{prop}
The proof uses the invariance property of the cross-ratio, and the
two following lemm\ae.
\begin{lem}\label{lemAalpha}
For every $g\in G$ the following identity holds:
$$
A(g,-\alpha(x))=A(g,x).
$$
\end{lem}
\proof By definition the transformation $g$ commutes with $-\alpha$
for every $g\in G$: $ g\circ(-\alpha)=(-\alpha)\circ g. $
Differentiating this equality we have
$$
\left(Dg\right)_{-\alpha(x)}\circ(-\alpha)=(-\alpha)\circ
\left(Dg\right)_x,
$$
or equivalently $\left(Dg\right)_{-\alpha(x)}=(-\alpha)\circ
\left(Dg\right)_x\circ(-\alpha).$
 Notice that for every $h\in\Str(V)$  one has
 $$
 (-\alpha)\circ h\circ(-\alpha)=\alpha\circ h\circ\alpha\in\Str(V),
 $$
 and on the other hand $\chi(\alpha\circ h\circ\alpha)=\chi(h)$. Since
$\Delta(\alpha(x))=\Delta(x)$ we finally get
$$
\Delta(\alpha\circ h\circ\alpha(x))=\chi(h)\Delta(x),
$$
which implies the statement for
$h=\left(Dg\right)_x.\qquad\qquad\qquad\Box$
\begin{lem}\label{lemgx0}
Let $x$ be an element in $V$, then for every $g\in G$ the following
identity holds
$$
\Delta((gx)_0)=A(g,x)\Delta(x_0).
$$
(Recall that the subscript $_0$ means the $V_0$-component.)
\end{lem}
Notice that
$\Delta((gx)_0)=\Delta(g.x-g.(-\alpha(x))=\Delta(g.x+\alpha(g.x))$
for every $g\in G$. Let us substitute $y$ by $-\alpha(x)$ in
(\ref{FG}), then
$$
\Delta(g.x-g.(-\alpha(x)))^2=A(g,x)A(g,-\alpha(x))\Delta(x+\alpha(x))^2.
$$
Therefore, according to the lemma (\ref{lemAalpha}), we have
$$
\Delta((gx)_0)=\pm A(g,x)\Delta(x_0).
$$
Since for $g=Id$ the sign in the above equation is positive, and the
group $G$ being connected it remains positive for every $g\in
G.\quad\quad\Box$
\bigskip

Let us introduce the multiplication operator on the space
$I_s(\mathcal X)$ by
$$
M_{s }f(x)=\Delta (x_0)^{s+\frac n{2r} } f(x).
$$
These operators intertwine the canonical representation $T_s$ and
the left regular action $L$ of the group $G$,
$$
M_s\circ T_s(g)=L(g)\circ M_s,\quad g\in G,
$$
where $ \bigl(L(g)f\bigr)(x)=f(g^{-1}\cdot x).$

\begin{prop}
Let us define the bilinear form ${\mathfrak B}^{\alpha}_s$ on
$\mathcal C_c(\mathcal X)\times \mathcal C_c(\mathcal X)$ by
$$
\mathfrak B_s^{\alpha }(F_1,F_1) =\mathfrak B_s^{\alpha
}(M_sf_1,M_sf_2) =\widetilde{ \mathfrak B}_s^{\alpha }(f_1,f_2).
$$
Then
\begin{equation}\label{berezinmakarintegral}
    {\mathfrak B}^{\alpha}_s(F_1,F_2)
=\iint\limits_{\mathcal X\times\mathcal X}{B}_{\nu
}(x,y)F_1(x)F_2(y)\mu(dx)\mu(dy),
\end{equation}
where $B_{\nu}(x,y)$ is the Berezin kernel introduced in
(\ref{berezin}) and \newline $\nu=-\frac{r}{r_0}\left(s-\frac
n{2r}\right)$.
\end{prop}

Therefore the decomposition of the canonical representations reduces
now to a classical problem in spherical harmonic analysis.


 Let $\Lambda $ be the set of parameters $\lambda \in \mathfrak a _{\mathbb C}^*$
for which the spherical function $\varphi _{\lambda }$  given by
(\ref{sphfunction}) is positive definite. Let $\psi $ be a
continuous function of positive type on $G$ which is
$K$-biinvariant. By the Bochner-Godement theorem there is a unique
bounded positive measure $m $ on $\Lambda $ such that
$$\psi (x)=\int _{\Lambda } \varphi _{\lambda }(x)m (d\lambda ).$$
If $\psi $ is integrable, then the measure $m $ is absolutely
continuous with respect to the Plancherel measure, with a density
given by the Fourier transform of $\psi $:
$$m(d\lambda )=\mathcal F \psi (\lambda ){d\lambda \over |c(\lambda )|},$$
where $c(\lambda)$ is the Harish-Chandra $c$-function of the
symmetric space $\mathcal{X}$.


\bigskip

Therefore the problem is :

(1) Determine the set ${\mathcal W}\subset {\Real}$ of values $\nu $
for which the Berezin kernel ${\mathfrak B}^\alpha_\nu$ (or the
corresponding $K$-biinvariant function $\psi _\nu$) is of positive
type.

(2) For $\nu \in {\mathcal W}$, determine the positive measure $m
_\nu$ on $\Lambda $ such that
$$\psi _{\nu }(x)=\int _{\Lambda } \varphi _{\lambda }(x)m_{\nu}(d\lambda ).$$

These problems have been solved for hyperbolic spaces by van Dijk,
Hille, Pasquale \cite{[vdh],[vdpas]}. Notice however that only real
hyperbolic spaces are Makarevich spaces as introduced above.

The case $G=U(p,q)$ was studied by Hille, and  Neretin
\cite{[Hille],[Ner]}, and the case when $G=SO(p,q)$  by Neretin in
\cite{[Ner]}.

\bigskip

For $\nu>\frac n{r_0}-1$ the function $\psi _\nu$ is integrable and,
for $\lambda\in i\mathfrak a^*$:

$$\mathcal{F}\psi _\nu(\lambda)=\frac{|P(\lambda,\nu)|^2}{Q(\nu)}\geq 0.
$$

It follows that $\psi _\nu$ is of positive type, and
$$
\psi _\nu(x)=\int_{i\mathfrak
a^*}\phi_{\lambda}(x)\frac{|P(\lambda,\nu)|^2}{Q(\nu)}\frac{d\lambda}{|c(\lambda)|^2}.
$$

According to Helgason we define the Fourier transform of an
integrable function $f$ on $\mathcal{X}$. The Iwasawa decomposition
of the group $G$ can be written as $G=NAK$. If $k^{-1}g\in
N\exp(H)K$ with $H\in\mathfrak a$, one writes
$$
H=\mathcal A(x,b),\quad x\in gK,\:b\in kM\in B=K/M,
$$
where $M$ is the centralizer of $A$ in $K$.

The Fourier transform of $f$ is the function defined on $i\mathfrak
a^*\times B$ by
$$
\hat f(\lambda,b)=\int_{\mathcal
X}f(x)e^{\langle-\lambda+\rho,\mathcal{A}(x,b)}\rangle dx.
$$

We consider $\hat f(\lambda):=\hat f(\lambda,\cdot)$ as an element
of the space $\mathcal{H}_{\lambda}\simeq L^2(B)$ which carries a
unitary spherical principal series $\Pi _\lambda$ of $G$. Then the
map $f\mapsto\hat f(\lambda)$ intertwines the left regular
representation $L$ and $\Pi _\lambda$.
\begin{thm} %Plancherel formula for the bilinear form
%$\mathfrak{B}_s^{\alpha}$.

For $\nu>\frac n{r_0}-1$, the Berezin form is positive definite.
Therefore the representation $T_s$ is unitarizable, decomposes
multiplicity free as a direct integral of spherical principal series
$\Pi _\lambda$ of $G$, which corresponds to the following Plancherel
formula :
$$
\mathfrak{B}_s^{\alpha}(f,\bar f)=\int_{i\mathfrak a^*}\Vert\hat
f(\lambda)\Vert^2\frac{|P(\lambda,\nu)|^2}{Q(\nu)}\frac{d\lambda}{|c(\lambda)|^2},
$$
where $\nu=-\frac r{r_0}(s-\frac n{2r})$ and $f\in C_c(\mathcal X)$.
\end{thm}
\bigskip

\subsection{Berezin kernel on the Riemannian Compact Dual}
In order to study the "deformation" of the regular representation of
the compact group $U$ we shall investigate the spherical Fourier
transform of the Berezin kernels on the compact dual symmetric space
$\mathcal Y$.

 The spherical functions of the symmetric space $\mathcal Y=U/K$ are
given by
\begin{equation}\label{spherY}
    \Phi_{{\bf m}}(x)=\int_K\Delta_{{\bf m}}((k.x)_0)dk=\varphi_{{\bf m}-\rho}(x),
\end{equation}
where the weights ${\bf m}=(m_1,\dots,m_n)\in\mathbb Z^n$ are given
according to different cases by following conditions :
\begin{itemize}
\item If $V$ is a Euclidean Jordan algebra (root system of type A),
then $m_1\geq m_2\geq\dots\geq m_n$.
\item If $V$ is a non Euclidean algebra of split type (root system
of type D), them $-|m|_1\geq m_2\geq\dots\geq m_n$.
\item If $V$ is either a complex or  a non Euclidean Jordan algebra of non split
type (root system of type C), then $0\geq m_1\geq m_2\geq\dots\geq
m_n$.
\end{itemize}
\medskip

The spherical Fourier coefficients of a $K$-invariant integrable
function $f$ on $\mathcal Y$ are given by
$$a({\bf m})=\int _{\mathcal Y} f(y)\Phi _{\bf m}(y)\mu _0(dy).$$
For $\Re \kappa \geq 0$, the Berezin function $\psi _{-\kappa}$ is
bounded on $\mathcal Y$. We will determine its spherical Fourier
coefficients
$$
a_{\kappa}({{\bf m}})=\int_{\mathcal Y}\psi_{-\kappa}(y)\Phi_{{\bf
m}}(y)d\mu_0(dy).
$$
Notice that $a_{\kappa}(0)=J(\kappa)$.
\begin{thm}\label{thmcoefffourier}
The Fourier coefficients of the Berezin kernel function
$\psi_{-\kappa}$ are given by
$$
a_{\kappa}({{\bf
m}})=J(\kappa)\frac{\prod_{j=1}^{n}\Gamma(\kappa+\frac12+\delta-\rho_j)\Gamma(\kappa+\frac12+\delta+\rho_j)}
{\prod_{j=1}^{n}\Gamma(\kappa+\frac12+\delta-\rho_j+m_j)\Gamma(\kappa+\frac12+\delta+\rho_j-m_j)}.
$$
\end{thm}

\emph{Proof} We use the same method as in \cite{[Z03]}.

(a) We show first that the measure on $\mathcal Y$ given by
$${1\over J(\kappa )}\psi _{-\kappa }(y)\mu _0(dy)$$
converges to the Dirac measure $\delta $ at the base point $e$ of
$\mathcal Y$ in the sense of tight convergence of measures. The
proof of this fact is based on the following lemma.

\begin{lem}\label{suitedelta}
Let $\mathcal K$ be a compact topological space, and $\mu $ a
positive measure on it such that every non empty open set has a
positive measure. Let $q\geq 0$ be a continuous function on
$\mathcal K$ which attains its maximum at only one point $x_0$.
Define, for $n\in \mathbb N$,
$$a_n=\int _{\mathcal K} q(x)^n\mu (dx),$$
and, for a continuous function $\varphi $ on $\mathcal K$,
$$L_n(\varphi )={1\over a_n}\int _{\mathcal K} \varphi (x)q(x)^n\mu (dx).$$
Then
$$\lim _{n\to \infty }L_n(\varphi )=\varphi (x_0).$$
\end{lem}

\emph{Proof} For $0<\alpha <M=\max q$, there exits a constant
$C_{\alpha }$ such that
$$a_n\geq C_{\alpha }\alpha ^n.$$
In fact there is a neighborhood $\mathcal V$ of $x_0$ such that
$q(x)\geq \alpha $ for $x\in \mathcal V$, and
$$a_n\geq \mu (\mathcal V)\alpha ^n.$$
Let $\mathcal W$ be a neighborhood of $x_0$. For $x\in {\mathcal
K}\setminus \mathcal W$, $q(x)\geq \beta <M$. Choose $\alpha $ such
that $\beta <\alpha <M$. then
$${1\over a_n}\int _{\mathcal K\setminus \mathcal W}q(x)^n\mu (dx)
\leq {1\over C_{\alpha }} \mu (\mathcal K)\Bigl({\beta \over \alpha
}\Bigr)^n,$$ and
$$\lim _{n\to \infty }{1\over a_n}\int _{\mathcal K\setminus \mathcal W}q(x)^n\mu
(dx)=0. \qquad\qquad\qquad \Box $$

\bigskip

The function $\psi _{-\kappa }$ attains its maximum $M=1$ only at
$y=e$. Therefore Lemma (\ref{suitedelta}) applies. It follows thet,
for every $\bf m$,
$$\lim _{\kappa \to \infty }{1\over J(\kappa )}a_{\kappa }({\bf m})=1.$$

\medskip

(b) Assume that $\kappa \in \mathbb N$. From the Bernstein identity
(\ref{bernstein}) it follows that the spherical Fourier coefficients
$a_{\kappa}({{\bf m}})$ satisfy the following recursion relation :
$$
\gamma_{-\kappa}(\rho-{{\bf m}})a_{\kappa}({{\bf
m}})=b(-\kappa)a_{\kappa-1}({{\bf m}}).
$$
Since, for ${\bf m}= 0$,
$\gamma_{-\kappa}(\rho)a_{\kappa}(0)=b(-\kappa)a_{\kappa-1}({\bf
0})$ we obtain
$$
\frac{\gamma_{-\kappa}(\rho-{{\bf m}})}{\gamma_{-\kappa}(\rho)}
\frac{a_{\kappa}({{\bf m}})}{a_{\kappa}(0)}
=\frac{a_{\kappa-1}({{\bf m}})}{a_{\kappa-1}(0)}.
$$
Furthermore, since $a_{\kappa }(0)=J(\kappa )$, by (a)
$$
\lim_{\kappa\to\infty}\frac{a_{\kappa}({{\bf m}})}{a_{\kappa}(0)}=1.
$$
The sequence given by
$$
\phi(\kappa)=\frac{\prod_{j=1}^{n}\Gamma(\kappa+\frac12+\delta-\rho_j)\Gamma(\kappa+\frac12+\delta+\rho_j)}
{\prod_{j=1}^{n}\Gamma(\kappa+\frac12+\delta-\rho_j+m_j)\Gamma(\kappa+\frac12+\delta+\rho_j-m_j)},
$$
satisfies the same recursion relation as $\frac{a_{\kappa}({{\bf
m}})}{a_{\kappa}(0)}$ does. Moreover, from the asymptotic
equivalence
$$\frac{\Gamma(z+a)}{\Gamma(z+b)}\sim
z^{a-b}\left(1+\frac1{2z}(b-a)(b+a-1)+O(z^{-2})\right),\quad{\rm
as}\:{z\to\infty},
$$
 it follows that
 $$
 \lim_{\kappa\to\infty}\phi(\kappa)=1.
 $$
 Therefore we have proved the theorem for $\kappa\in \mathbb N$.

\medskip

\noindent(c) By using Carlson's theorem, as we did in the proof of
Theorem (\ref{thmhua}), we conclude that the statement is
 still valid for $\Re\kappa\geq0.\qquad\Box$
\medskip

\subsection{Restriction of the representation $\pi_s$ to the compact
group $U$} The compact symmetric space $\mathcal Y=U/K$ is also of
Makarevich type. Indeed,
$$
U=\{g\in\Conf(V)\:|\:\alpha\circ j\circ g\circ j\circ\alpha=g\}_o.
$$
From the generalized cross-ratio (\ref{cross-ratio}) we define the
kernel $F^c(u,v)$ by
\begin{eqnarray*}
  F^c(u,v) &=& \{u,v,-\alpha(u^{-1}),-\alpha(v^{-1})\} \\
  &=&{\Delta \bigl(u+\alpha (u^{-1})\bigr)\Delta \bigl(v+\alpha
(v^{-1})\bigr) \over \Delta \bigl(u+\alpha (v^{-1})\bigr)\Delta
\bigl(v+\alpha (u^{-1})\bigr)}
\end{eqnarray*}
and $F^c(u,0)=h(u,-u)$.

Similarly to what we did in Section 2.2 we twist the bilinear form
$\mathfrak B_s$ by the involution $j\circ\alpha$, in other words we
replace $y$ by $\alpha(y^{-1})$, and by introducing multiplication
operators
$$
M^c_sf(x)=h(x,-x)^{s+\frac n{2r}}f(x)
$$
we eventually obtain a $U$-invariant bilinear form on
$C^{\infty}(\mathcal Y)$:
$$
\mathfrak{B}_s^c(F_1,F_2)=\iint_{\mathcal{Y}\times\mathcal{Y}}B_s^c(x,y)F_1(x)F_2(y)\mu_0(dx)\mu_0(dy),
$$
where $B_s^c(x,y)=F^c(x,y)^{\frac {r_0}r \nu}$.
\medskip

The spherical dual $\hat U_K$ is parameterized by the set of weights
${\bf m}=(m_1,\ldots,m_n)$ described in the previous section. To
such a weight $\bf m$ corresponds a (class of) unitary spherical
representation $\Pi _{\bf m}$ on a finite dimensional vector space
$\mathcal H_{\bf m}$ of dimension $d_{\bf m}$. The highest weight
$\mu $ of $\Pi _{\bf m}$ is given by
$$
\langle \mu,\xi \rangle =-\sum _{j=1}^{r_0} m_jt_j,$$ if $\xi
(x)=\sum t_jL(c_j)x$. In the space $\mathcal H_{\bf m}$ there is a
normalized $K$-fixed vector $v_{\bf m}$. The Fourier coefficient
$\hat f(\bf m)$ of an integrable function $f$ on $\mathcal{Y}$ is
the vector in $\mathcal H_{\bf m}$ defined by
$$
\hat f({\bf m})=\int_{U/K}\Pi _{\bf m}(g)v_{\bf m}f(g)dg.
$$
The map $f\mapsto\hat f(\bf m)$ intertwines the left regular
representation of $U$ and $\Pi _{\bf m}$.

\begin{thm}
For  a $\kappa\in\mathbb N$ the Berezin form $\mathfrak B_s^c$ is
positive definite. Therefore the restriction of $\pi_s$ to the
compact group $U$ decomposes into a direct sum of spherical
principal series representations $\Pi_{\bf m}$ of $U$ according to
the following Plancherel formula
$$
\mathfrak B_s^c(f,\bar f)=\sum_{{\bf m}\in\hat U_K}d_{\bf
m}a_{\kappa}({\bf m}) \Vert\hat f({\bf m})\Vert^2,
$$
where $\kappa=\frac r{r_0}(s-\frac{n}{2r})$, $f\in C(U/K)$.
\end{thm}

By the first part of the proof of Theorem 2.8, in some sense, the
canonical representations of the compact dual of $G$ tend to the
left regular representation when the parameter $\kappa$ goes through
the negative integer points. In fact, for $f\in \mathcal C_c
(\mathcal Y)$,
$$
\lim_{\kappa\to\infty} \frac1{J(\kappa )}\int _{\mathcal Y\times
\mathcal Y} \mathfrak B_{\kappa }^c(x,y)f(x)f(y)\mu _0(dx)\mu _0(dy)
=\int _{\mathcal Y}|f(y)|^2\mu _0(dy).
$$
 Similarly, in the non compact case, as
$\nu\to \infty $, the probability measure $ I(\nu)^{-1}\psi _{\nu
}(x)\mu(dx)$ converges to the Dirac measure $\delta _e$ at the
identity element $e$. It follows that, for $f\in \mathcal C
_c(\mathcal X)$,
$$\lim _{\nu\to \infty }
{1\over I({\nu })}\int _{{\mathcal X}\times {\mathcal X}} {\mathcal
B}^{\alpha}_{\nu }(x,y)f(x)\overline{f(y)}\mu(dx)\mu(dy) =\int
_{\mathcal X} |f(x)|^2\mu(dx).$$ In some sense, as $\nu \to \infty
$, the canonical representation $T_\nu$ tends to the regular
representation of $G$ on $L^2({\mathcal X})$.

In conclusion we should mention that originally the one parameter
family of Berezin kernels was introduced in the context of
quantization of K\"ahlerian symmetric spaces in \cite{[Ber]}. The
last remark says that the Berezin symbolic calculus "deforms" the
usual point-wise multiplication of functions. We refer the reader to
\cite{[AO],[UU]} for more details on asymptotics of the Berezin
transform.
% ----------------------------------------------------------------
\bibliographystyle{amsplain}
\begin{thebibliography}{CDRV}
\bibitem{[AO]} J.Arazy, B.\O rsted,
Asymptotic expansions of Berezin transforms. \emph{Indiana Univ.
Math. J.} \textbf{49} (2000), no. 1, pp.7--30.
\bibitem{[Ber]} F.A.Berezin, Quantization, \emph{Math. USSR Izvestija},
\textbf{8}, (1974), pp.1109-1165.
\bibitem{[B1]}W.Bertram, On some causal and conformal groups, {\it J. Lie Theory}
\textbf{6}.2, (1996), pp.215-247.
\bibitem{[B2]} W.Bertram, Algebraic Structures of
Makarevi\v{c} spaces. I. \emph{Transf. Groups}, Vol \textbf{3}, No1,
1998, pp.3-32.
\bibitem{[BLIN]} B.Blind, Distributions homog\`enes sur une
alg\`ebre de Jordan. \emph{Bull. Soc. Math. France}, \textbf{125},
(1997), pp. 493--528.
\bibitem{[vdh]} G. van Dijk, S.Hille, Maximal degenerate
representations, Berezin kernels and canonical representations, in
"Lie Groups and Lie algebras, their representations, generalizations
and applications" (B.Komrakov, G.Litvinov, A.Sossinsky Edt.), Kluwer
Academic, Dordrecht, 1997.
\bibitem{[vdm]} G. van Dijk, V.Molchanov, Tensor products of maximal
degenerate series representations of the group $SL(n,\Real)$, {\it
J. Math. Pures Appl.}, \textbf{78}, (1999), pp. 99--119.
\bibitem{[vdpas]} G. van Dijk, A.Pasquale, Harmonic analysis on vector bundles over
${\rm Sp}(1,n)/{\rm Sp}(1)\times{\rm Sp}(n)$.  \emph{Enseign. Math.}
(2) \textbf{45} (1999), no. 3-4, pp.219--252.
\bibitem{[DP]} G. van Dijk, M.Pevzner, Berezin kernels of tube domains.  \emph{J. Funct. Anal.},
\textbf{181}, (2001), no.2, pp.189--208.
\bibitem{[Englis]} M.Engli\v{s}, A mean value theorem on bounded
symmetric domains, \emph{Proc. Amer. Math. Soc.} \textbf{12}7, pp.
3259--3268.
\bibitem{[Fa]} J.Faraut, Int\'egrales de Riesz sur un espace sym\'etrique ordonn\'e.
Geometry and analysis on finite- and infinite-dimensional Lie groups
(B\c{e}dlewo, 2000), pp.289--308, Banach Center Publ., \textbf{55},
Polish Acad. Sci., Warsaw, 2002.
\bibitem{[FG]} J.Faraut, S.Gindikin, Pseudo-Hermitian symmetric spaces
of tube type. {\it Topics in geometry}, Progr.Nonlinear Diff. Equat.
Appl., \textbf{20} Birkh\"{a}user Boston, 1996, pp. 123--154.
\bibitem{[FK]} J. Faraut and A Kor\'anyi: Analysis on Symmetric Cones.
Oxford Science Publications, 1994.
%\bibitem{[H]} S.Helgason, Differential geometry, Lie groups and
%symmetric spaces, Academic Press, New York, 1978.
\bibitem{[Hille]} S.Hille, Canonical representations, Ph.D Thesis,
Leiden Univeristy, 1999.
\bibitem{[Hua]} L.K.Hua, \emph{Harmonic analysis of functions of several
complex variables in the classical domains}. Translations of
Mathematical monographs, 6, AMS, Providence, 1963.
\bibitem{[Kantor]} I.L.Kantor, Non-linear groups of transformations defined by general
norms of Jordan algebras.  \emph{Soviet Math Dokl.}, \textbf{8},
(1967), pp.176-180.
\bibitem{[Khlif]} Y.Khlif, In t\'egrales de Riesz sur un espace sym\'etrique de type Cayley,
Th\`ese de doctorat, Universit\'e de
 Paris VI (2000).
\bibitem{[K]} B.Kostant, On the existence and irreducibility of
certain series representations, Proceedings of the summer school on
group representations, Bolyai Jonos Math. Soc. Budapest, 1971.
\bibitem{[KS]} B.Kostant, S.Sahi,  Jordan algebras and Capelli
identities. Invent. Math. 112, (1993), pp. 657--664.
\bibitem{[Ner]} Yu.Neretin, Matrix analogues of the $B$-function, and the Plancherel formula for Berezin kernel representations.
\emph{Sb. Math}.  \textbf{191} (2000), no. 5-6, pp.683--715
\bibitem{[P]} M.Pevzner, Analyse conforme sur les alg\`ebres de
Jordan, {\it J. Austral. Math. Soc.} \textbf{73},(2002), 1--21.
\bibitem{[S2]} S.Sahi, Explicit Hilbert spaces for certain
unipotent representations, {\it Invent. math.} {\bf 110} (1992), pp.
409-418.
\bibitem{[S3]} S.Sahi, The Capelli identity and unitary
representations, \emph{Compositio Math}. \textbf{81} (1992),
pp.247-260.
\bibitem{[S1]} S.Sahi, Jordan algebras and degenerate principal
series {\it J. reine angev. Math} {\bf 462} (1995), pp. 1--18.
%\bibitem{[Sat]} I.Satake, Algebraic structures of symmetric domains,
%Iwanami Shoten, Publishers and Princeton Univ.Press, 1980.
\bibitem{[UU]} A.Unterberger, H.Upmeier, The Berezin transform and invariant differential operators.
\emph{Comm. Math. Phys.} \textbf{164} (1994), no.3, pp.563--597.
\bibitem{[GVG]} A.M.Vershik, I.M.Gelfand, M.I.Graev, Representations of the group
${\rm SL}(2,\,R)$, where $R$ is a ring of functions. (Russian)
 \emph{Uspehi Mat. Nauk}, \textbf{28} (1973), no.5(173), pp.83--128.
 \bibitem{[Z02]} G.Zhang, Jordan algebras and generalized principal
 series representations. \emph{Math. Ann}. \textbf{302}, (1995), pp. 773--786.
 \bibitem{[Z03]} G.Zhang, Berezin transform on compact Hermitian
 symmetric spaces, \emph{Manuscripta Math}. \textbf{97} (1998), pp.371-388.
\bibitem{[Z01]} G.Zhang, Berezin transform on real bounded symmetric domains.
\emph{Trans. Amer. Math. Soc.} \textbf{353} (2001), no.9, pp.
3769--3787.
\end{thebibliography}
\end{document}
% ----------------------------------------------------------------

math0411294.extracted_bib
\begin{thebibliography}{CDRV}
\bibitem{[AO]} J.Arazy, B.\O rsted,
Asymptotic expansions of Berezin transforms. \emph{Indiana Univ.
Math. J.} \textbf{49} (2000), no. 1, pp.7--30.
\bibitem{[Ber]} F.A.Berezin, Quantization, \emph{Math. USSR Izvestija},
\textbf{8}, (1974), pp.1109-1165.
\bibitem{[B1]}W.Bertram, On some causal and conformal groups, {\it J. Lie Theory}
\textbf{6}.2, (1996), pp.215-247.
\bibitem{[B2]} W.Bertram, Algebraic Structures of
Makarevi\v{c} spaces. I. \emph{Transf. Groups}, Vol \textbf{3}, No1,
1998, pp.3-32.
\bibitem{[BLIN]} B.Blind, Distributions homog\`enes sur une
alg\`ebre de Jordan. \emph{Bull. Soc. Math. France}, \textbf{125},
(1997), pp. 493--528.
\bibitem{[vdh]} G. van Dijk, S.Hille, Maximal degenerate
representations, Berezin kernels and canonical representations, in
"Lie Groups and Lie algebras, their representations, generalizations
and applications" (B.Komrakov, G.Litvinov, A.Sossinsky Edt.), Kluwer
Academic, Dordrecht, 1997.
\bibitem{[vdm]} G. van Dijk, V.Molchanov, Tensor products of maximal
degenerate series representations of the group $SL(n,\Real)$, {\it
J. Math. Pures Appl.}, \textbf{78}, (1999), pp. 99--119.
\bibitem{[vdpas]} G. van Dijk, A.Pasquale, Harmonic analysis on vector bundles over
${\rm Sp}(1,n)/{\rm Sp}(1)\times{\rm Sp}(n)$.  \emph{Enseign. Math.}
(2) \textbf{45} (1999), no. 3-4, pp.219--252.
\bibitem{[DP]} G. van Dijk, M.Pevzner, Berezin kernels of tube domains.  \emph{J. Funct. Anal.},
\textbf{181}, (2001), no.2, pp.189--208.
\bibitem{[Englis]} M.Engli\v{s}, A mean value theorem on bounded
symmetric domains, \emph{Proc. Amer. Math. Soc.} \textbf{12}7, pp.
3259--3268.
\bibitem{[Fa]} J.Faraut, Int\'egrales de Riesz sur un espace sym\'etrique ordonn\'e.
Geometry and analysis on finite- and infinite-dimensional Lie groups
(B\c{e}dlewo, 2000), pp.289--308, Banach Center Publ., \textbf{55},
Polish Acad. Sci., Warsaw, 2002.
\bibitem{[FG]} J.Faraut, S.Gindikin, Pseudo-Hermitian symmetric spaces
of tube type. {\it Topics in geometry}, Progr.Nonlinear Diff. Equat.
Appl., \textbf{20} Birkh\"{a}user Boston, 1996, pp. 123--154.
\bibitem{[FK]} J. Faraut and A Kor\'anyi: Analysis on Symmetric Cones.
Oxford Science Publications, 1994.
%\bibitem{[H]} S.Helgason, Differential geometry, Lie groups and
%symmetric spaces, Academic Press, New York, 1978.
\bibitem{[Hille]} S.Hille, Canonical representations, Ph.D Thesis,
Leiden Univeristy, 1999.
\bibitem{[Hua]} L.K.Hua, \emph{Harmonic analysis of functions of several
complex variables in the classical domains}. Translations of
Mathematical monographs, 6, AMS, Providence, 1963.
\bibitem{[Kantor]} I.L.Kantor, Non-linear groups of transformations defined by general
norms of Jordan algebras.  \emph{Soviet Math Dokl.}, \textbf{8},
(1967), pp.176-180.
\bibitem{[Khlif]} Y.Khlif, In t\'egrales de Riesz sur un espace sym\'etrique de type Cayley,
Th\`ese de doctorat, Universit\'e de
 Paris VI (2000).
\bibitem{[K]} B.Kostant, On the existence and irreducibility of
certain series representations, Proceedings of the summer school on
group representations, Bolyai Jonos Math. Soc. Budapest, 1971.
\bibitem{[KS]} B.Kostant, S.Sahi,  Jordan algebras and Capelli
identities. Invent. Math. 112, (1993), pp. 657--664.
\bibitem{[Ner]} Yu.Neretin, Matrix analogues of the $B$-function, and the Plancherel formula for Berezin kernel representations.
\emph{Sb. Math}.  \textbf{191} (2000), no. 5-6, pp.683--715
\bibitem{[P]} M.Pevzner, Analyse conforme sur les alg\`ebres de
Jordan, {\it J. Austral. Math. Soc.} \textbf{73},(2002), 1--21.
\bibitem{[S2]} S.Sahi, Explicit Hilbert spaces for certain
unipotent representations, {\it Invent. math.} {\bf 110} (1992), pp.
409-418.
\bibitem{[S3]} S.Sahi, The Capelli identity and unitary
representations, \emph{Compositio Math}. \textbf{81} (1992),
pp.247-260.
\bibitem{[S1]} S.Sahi, Jordan algebras and degenerate principal
series {\it J. reine angev. Math} {\bf 462} (1995), pp. 1--18.
%\bibitem{[Sat]} I.Satake, Algebraic structures of symmetric domains,
%Iwanami Shoten, Publishers and Princeton Univ.Press, 1980.
\bibitem{[UU]} A.Unterberger, H.Upmeier, The Berezin transform and invariant differential operators.
\emph{Comm. Math. Phys.} \textbf{164} (1994), no.3, pp.563--597.
\bibitem{[GVG]} A.M.Vershik, I.M.Gelfand, M.I.Graev, Representations of the group
${\rm SL}(2,\,R)$, where $R$ is a ring of functions. (Russian)
 \emph{Uspehi Mat. Nauk}, \textbf{28} (1973), no.5(173), pp.83--128.
 \bibitem{[Z02]} G.Zhang, Jordan algebras and generalized principal
 series representations. \emph{Math. Ann}. \textbf{302}, (1995), pp. 773--786.
 \bibitem{[Z03]} G.Zhang, Berezin transform on compact Hermitian
 symmetric spaces, \emph{Manuscripta Math}. \textbf{97} (1998), pp.371-388.
\bibitem{[Z01]} G.Zhang, Berezin transform on real bounded symmetric domains.
\emph{Trans. Amer. Math. Soc.} \textbf{353} (2001), no.9, pp.
3769--3787.
\end{thebibliography}