math0202018.tex

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\begin{document}

\def\D{{\rm D}}





\def\R {{\Bbb R }}
\def\C {{\Bbb C }}
\def\KK{{\Bbb K}}
\def\HH{{\Bbb H}}


\def\U{{\rm U}}

\def\const{{\rm const}}
\def\B{{\rm B}}



\def\OO{{\rm O}}
\def\SO{{\rm SO}}
\def\GL{{\rm GL}}
\def\SL{{\rm SL}}
\def\SU{{\rm SU}}
\def\Sp{{\rm Sp}}
\def\SOS{\SO^*}


\def\Mat{{\rm Mat}}

\def\Subsection{}%Subsection
\def\Subsections{}%Subsections


\newcommand{\rk}{\mathop{\rm rk}\nolimits}



\def\ov{\overline}
\def\phi{\varphi}
\def\epsilon{\varepsilon}
\def\kappa{\varkappa}%don't change to $\kappa$!!!!!!
\def\le{\leqslant}
\def\ge{\geqslant}

\newcommand{\sh}{\mathop{\rm sh}\nolimits}
\newcommand{\ch}{\mathop{\rm ch}\nolimits}


\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\tr}{\mathop{\rm tr}\nolimits}

\def\matritsa{
\begin{pmatrix}a&b\\ \ov b&\ov a\end{pmatrix}}




\begin{center}

{\large\bf
Action of overalgebra in Plancherel
decomposition and shift operators
 in imaginary direction}

 \smallskip

{\sc Yurii A. Neretin }\footnote{Supported by
the grant NWO 047--008--009}

\end{center}

In last 50 years,
there were solved
many problems of
explicit spectral decompositions
for restrictions of unitary representations
to subgroups (see the bibliography
in \cite{MolVINITI}).

This class of problems
 also contains  the following problems,
which are formulated
in other terms.









1. {\it Decomposition of the tensor product
of representations
$\rho_1$, $\rho_2$
of a group $G$}.
Indeed, this is exactly the problem
of restriction of representations of $G\times G$
to the diagonal subgroup $G\subset G\times G$.

\smallskip

2.
{\it Decomposition of $L^2$
on a pseudo-Riemannian symmetric space
 $G/H$}.
As was observed\footnote{This phenomenon
could be easily observed
from the Makarevich paper \cite{Mak}
published in 1973,
in a strange way it remained
 nonformulated
for a long time}
 in \cite{pse},
 for each classical pseudo-Riemannian
symmetric space  $G/H$,
there exists a canonical classical group
 $\widetilde G\supset G$
and a representation
$\rho$ of the group $\widetilde G$
of degenerated principal series
satisfying one of two following properties
(usually the first variant is realized):

--- the restriction of
$\rho$ to $G$ is  $L^2(G/H)$

--- the restriction of
$\rho$ to $G$ is
the  direct sum
of the spaces
$L^2(G/H_j)$, there $G/H_j$
is a finite collection of   symmetric spaces,
and $G/H$ is one of the spaces $G/H_j$.

Hence the decomposition of $L^2$
on a classical Riemannian symmetric space
can be considered as a restriction problem.

Description of the spectral type
(without explicit Plancherel formula)
for all the pseudo-Riemannian
symmetric spaces was recently obtained
in the works of van den Ban,
Schlichtkrull,  Delorme,
and Oshima
(the proof is contained in the union of
a large collection of papers,
for references, see \cite{Ban}, \cite{Del}).
It seems that for classical
symmetric spaces
the problem of evaluation
of the Plancherel measure is near the final solution.

For some cases, the explicit Plancherel
measure is known; in particular,
for
 $L^2$ on semisimple groups (\cite{GN}, \cite{HC}),
on Riemannian symmetric spaces
   (\cite{GN},
\cite{GK1}, \cite{GK2}, see also \cite{Hel}),
  on rank 1 spaces  (\cite{Molhyper}, \cite{Molrank})
and on the spaces
 $G_\C/G_\R$, where $G_\C$ is a complex group,
and $G_\R$
is its real form  (\cite{Har1}, \cite{Har2}).

\smallskip

3. {\it Berezin kernel representations}
 (deformations of $L^2$
on Riemannian noncompact symmetric space $G/K$)
also can be obtained by the restriction
from some overgroup $G^\circ\supset G$,
see \cite{Nerpl}, \cite{Nerhyper}
and references in \cite{Nerhyper}.

\smallskip

Usually the problem of the noncommutative
harmonic analysis is formulated as
the problem about the spectrum of a representation
or as the more complicated problem of an explicit
decomposition of a representation
into a direct integral of irreducible
representations;
the last question
includes
 the explicit evaluation of the spectral
measure (the so-called
Plancherel measure).
Recent works of the author
 \cite{Nerindex}, \cite{Nerhyper}
contain an attempt
of an investigation of "the analysis
after the Plancherel formula".

This work is a continuation of
 \cite{Nerindex}.
 Here we are trying to understand
  the answer to following
 question.

\smallskip

{\sc Question.} Assume that we know the explicit
 Plancherel formula for the restriction
 of a unitary representation $\rho$
 of a group  $G$
to a subgroup $H$.
Is it possible to write the action of Lie algebra
of
 $G$ in the direct integral of  representations
 of
$H$?

\smallskip

We obtain the positive solution
of this problem for one of the simplest possible
examples, precisely, for the tensor product
of a representation of $\SL(2,\R)$
with a highest weight and the conjugate representation
with a lowest weight.
This tensor product and its decomposition
were widely discussed in the literature
on representation theory and special
functions in last 40 years (some references: \cite{Puk},
\cite{dB}, \cite{VGG}, \cite{tensor},
\cite{vD}). Nevertheless,
the formula for the action of $sl_2\oplus sl_2$
in tensor product were not appear.
The reason is the unusual for representation
theory form of these Lie algebra operators.


It turns out  that the operators
(12)--(14)
of Lie algebra $sl_2\oplus sl_2$
 are the second order differential
operators with respect to one variable
and the second order difference operators with respect
to the another variable; moreover,
it turns out that the difference operators are
defined in the terms of a shift
in the imaginary direction, i.e.,
in our formula,
there appear the operators
of the form
$$ T f(x)=f(x+i) \qquad \qquad \text{for $f\in L^2(\R)$}$$
($i^2=-1$).
There is no self-contradiction in this expression,
the shift operators are well defined
on functions admitting the holomorphic continuation
to an appropriate strip.
Operators $f\mapsto xf$ and $f\mapsto \frac d{dx}$
also are not defined on the whole $L^2(\R)$.

First, these Lie algebra  operators were obtained by
author using theorems on the
operational calculus for the index hypergeometric
transform (17) from \cite{Nerindex}.
But the final formulas are elementary
and admit a direct  verification
by elementary tools.

Since  the approximate
 structure of formulas for action
 of overalgebra now became more clear,
 it is natural to formulate the general
 problem given above.


%Shifts in imaginary direction
%can seem unusual
%in the analysis on homogeneous spaces,
%but they appear quite often in the theory of special
%functions. For instance, the Meixner--Pollachek,
%continuous Hahn, continuous dual Hahn
%and Wilson orthogonal
%polynomials are eigenfunctions of difference
%operators in imaginary direction,
%see \cite{AAR}, \cite{KS}.

\medskip

This work was prepaired
during my visit to the Erwin Schr\"odinger
Institute (Vienna).
I thank the administrators of the Institute
for their hospitality.



\smallskip

{\bf 1. Group $\SL(2,\R)$.}
We realize the group $\SL(2,\R)$
as the group of complex
 $2\times 2$ matrices having the form
$$
\begin{pmatrix}a&b\\
\ov b&\ov a\end{pmatrix}, \qquad \text{where}\quad
  |a|^2-|b|^2=1
.$$

By $\D$
we denote the disk $|z|<1$
on the complex plane
$\C$, by $S^1$
we denote the circle $|z|=1$;
we  represent the points
of the circle in the form  $z=e^{i\phi}$.

The group $\SL(2,\R)$
acts on the disk by
the M\"obius
transformations
\begin{equation}
\begin{pmatrix}a&b\\ \ov b&\ov a\end{pmatrix}: \quad
z\mapsto \frac{az+b}{\ov b z+\ov a}
.\end{equation}
If $z\in S^1$, then its image
under (1) also
is contained in $S^1$.

\smallskip


{\bf 2. Highest weight representations
of $\SL(2,\R)$.} Fix
$\alpha>1$.
Consider the space $H_\alpha$
of holomorphic functions  $f$
 on the circle $\D$
 satisfying the condition
\begin{equation}
\int_\D|f(z)|^2 (1-|z|^2)^{\alpha-2}\{dz\}<\infty
,\end{equation}
where $\{dz\}$ denotes the Lebesgue measure
on $\D$. Define
the inner product in the space $H_\alpha$
by
\begin{equation}
\langle f, g\rangle_\alpha=\frac{\alpha-1}\pi
 \int_\D f(z) \ov{g(z)} (1-|z|^2)^{\alpha-2}\{dz\}
;\end{equation}
 the pre-integral factor
 provides the identity
 $\langle 1,1\rangle_\alpha=1$.
 In terms of the Taylor coefficients,
 the inner product is given by
$$
\langle \sum c_k z^k, \sum c_k'z^k\rangle_\alpha=
\sum_{k\ge 0} c_k \ov c_k'\frac{k!}
{\alpha(\alpha+1)\dots (\alpha+k-1)}
,$$
and condition (2) for $f(z)=\sum c_k z^k$
has the form
\begin{equation}
\sum |c_k|^2 k^{-(\alpha-1)}<\infty
.\end{equation}

It can easily be checked that the space $H_\alpha$
 is complete with respect to the  norm
 defined by this inner product,
 i.e., $H_\alpha$ is a Hilbert space.

The group $\SL(2,\R)$
acts in the space $H_\alpha$
by the unitary operators
\begin{equation}
T_\alpha\matritsa(z)=f\Bigl(\frac{az+b}
{\ov b z+\ov a}\Bigr)
  (\ov b z+\ov a)^{-\alpha}
.\end{equation}

First, consider the case, when $\alpha$
is an integer.
The factor $(\ov b z+\ov a)^{-\alpha}$
 is a degree of the derivative of the function (1),
this implies that $T_\alpha$
is a representation:
\begin{equation}
T_\alpha(g_1)T_\alpha(g_2)=T_\alpha(g_1g_2)
.\end{equation}
The operators $T_\alpha(g)$ are unitary, i.e.,
$$
\langle T_\alpha(g) f_1, T_\alpha(g) f_2\rangle_\alpha
 = \langle f_1, f_2\rangle_\alpha,
$$
 this can be easily checked by a change of
the variable.


For a noninteger   $\alpha$,
$T_\alpha(g)$ also is an unitary representation;
we only must explain the
meaning of the expression
\begin{equation}
(\ov b z+\ov a)^{-\alpha}=
(1+b\ov a^{\,-1} z)^{-\alpha} a^{-\alpha}=
(1+b\ov a^{\,-1} z)^{-\alpha}
 e^{-\alpha(\ln a+ 2\pi i k)}
.\end{equation}
Obviously, $|b\ov a^{\,-1}|<1$.
Hence the function
$$(1+b\ov a^{-1} z)^{-\alpha}:=
1+\frac{\alpha}{1!}b\ov a^{-1} z+
\frac{\alpha(\alpha-1)}{2!}
(b\ov a^{-1})^2 z^2+\dots$$
is well defined.

Therefore the operator (5)
is defined up to the factor $e^{-2\pi i k\alpha}$,
 the absolute value of this factor is 1.
Hence  equality (6)
is replaced by
$$
T_\alpha(g_1)T_\alpha(g_2)=
\theta\cdot T_\alpha(g_1g_2)
,$$
where $|\theta|=1$.
Thus,    $T_\alpha(g)$
is an unitary projective
representation of the group
$\SL(2,\R)$.

\smallskip

{\sc Remark.}
Clearly, $T_\alpha(g)$
can be also considered as a linear representation
of the universal covering
group
 of $\SL(2,\R)$.
 %Below this is nonessential.


\smallskip

{\bf 3. Tensor product.}
By $\ov  T_\alpha(g)  $
we denote the representation
complex conjugate to $ T_\alpha(g)$,
it acts in the space
 $\ov H_\alpha$ of
 antiholomorphic functions
 in the circle $|u|<1$
 by the formula
$$
T_\alpha\matritsa f(\ov u)=
f\Bigl(\frac{\ov  a\,\ov u+\ov b}{ b \ov u+ a}\Bigr)
  ( b \ov u+ a)^{-\alpha}.
$$
The inner product in $\ov H_\alpha$
is given by (3).

Consider the space $ H_\alpha\otimes\ov  H_\alpha$.
It consists of functions
 $f(z,\ov u)$ on the bidisk $\D\times\D$,
holomorphic with respect to
the variable $z$ and antiholomorphic
in
$u$; the inner product in $ H_\alpha\otimes\ov  H_\alpha$
is given by
$$
\langle f_1,f_2\rangle=\frac{(\alpha-1)^2}{\pi^2}
\iint_{\D \times \D} f_1(z,\ov u)\ov{ f_2(z,\ov u)}
(1-z\ov z)^{\alpha-2} (1-u\ov u)^{\alpha-2} \{dz\}\, \{du\}
.$$
The group $\SL_2(\R)\times\SL_2(\R)$
acts in $H_\alpha\otimes\ov H_\alpha$
by the operators
$ T_\alpha(g_1)\otimes \ov T_\alpha(g_2)$
given
by
\begin{multline*}
T_\alpha
\begin{pmatrix} a_1&b_1\\\ov b_1&\ov a_1
\end{pmatrix}\otimes
 \ov T_\alpha
\begin{pmatrix} a_2&b_2\\\ov b_2&\ov a_2
\end{pmatrix}
 f(z,u)=\\=
f\Bigl(\frac{a_1z+b_1}{\ov b_1 z+\ov a_1}\, ,\,
\frac{\ov  a_2\,\ov u+ \ov b_2}{ b_2 \ov u+ a_2}\Bigr)
(\ov b_1 z+\ov a_1)^{-\alpha}
     ( b_2 \ov u+ a_2)^{-\alpha}
.\end{multline*}
We restrict this representation
to the diagonal
subgroup $\SL_2(\R)\subset\SL_2(\R)\times\SL_2(\R)$,
this corresponds to the substitution
$a_1=a_2=a$, $b_1=b_2=b$ to the last formula.
This  representation
of  $\SL_2(\R)$ is a linear
(nonprojective)
 representation.
Indeed,
$$
(\ov b z+\ov a)^{-\alpha}
     ( b \ov u+ a)^{-\alpha}=
 (1+\ov a^{-1}\ov b z)^{-\alpha}
  (1+ a^{-1}b \ov u)^{-\alpha}
(a\ov a)^{-\alpha}
,$$
and this expression
is a single-valued function
for $|z|<1$, $|u|<1$.

\smallskip

{\bf 4. Principal series of representations.}
 Fix $s\in \R$.
Consider the representation $\rho_s$
of the group $\SL(2,\R)$ in $L^2$
on the circle $S^1$
given by
$$
\rho_s\matritsa f(e^{i\phi})=
f\Bigl(\frac{a e^{i\phi}+b}
{\ov b e^{i\phi}+\ov a}\Bigr)
|\ov b e^{i\phi}+\ov a|^{-1-2is}
.$$
These unitary representations are
the so-called representations
of the principal series.
 Recall, that the representation $T_s$
 is equivalent  to $T_{-s}$
(see  \cite{Bar}).

\smallskip

{\bf 5. Spectral decomposition.} Let $\alpha$
be the same as in
pp.2--3.
Consider the kernel
$$
K_\alpha(\phi, s; z,u):=
\frac{(1-\bar z e^{i\phi})^{-1/2-is}
                          (1- u e^{-i\phi})^{-1/2-is} }
    {(1-\bar z u)^{\alpha-1/2-is}}
,$$
where $|z|<1, |u|<1$, $\phi\in[0,2\pi]$, $s\in\R$.
Consider the integral operator $J_\alpha$,
 that takes
a function $f\in H_\alpha\otimes\ov H_\alpha$
to the function
$F(\phi,s)$
given by
\begin{multline}
F(\phi,s)=
\iint\limits_{\D\times D}
 K_\alpha(\phi, s; z,u)  f(z,\ov u)
  (1-z\ov z)^{\alpha-2}(1-u\ov u)^{\alpha-2} \{dz\}\{du\}
.
\end{multline}

Consider the action of the group $\SL(2,\R)$
 in space of functions in the variables
$(\phi,s)$ defined by
$$
R\matritsa F(e^{i\phi},s)=
F\Bigl(\frac{a e^{i\phi}+b}
{\ov b e^{i\phi}+\ov a},\, s\Bigr)
|\ov b e^{i\phi}+\ov a|^{-1-2is}
.$$
Note, that for a fixed $s\in \R$
the function $F(e^{i\phi},s)$
(as a function in $\phi$)
transforms by the formula
for principal series representations.

Simple calculation show,
that the operator $J_\alpha$
intertwines the representations
$T_\alpha(g)\otimes \ov T_\alpha(g)$ and $R$
of $\SL(2,\R)$:
\begin{equation}
 J_\alpha \cdot \bigl(T_\alpha(g)\otimes
 \ov T_\alpha(g)\bigr)=
   R(g)\cdot J_\alpha
\end{equation}

{\sc Theorem 1.} {\it Operator $J_\alpha$
is a unitary
operator from $H_\alpha\otimes \ov H_\alpha$
to the space
$L^2$ on $\phi\in[0,2\pi]$, $s\ge 0$
with respect to the measure}
\begin{multline}
\frac
{|\Gamma(\alpha-1/2+is)|^2}
{\Gamma(\alpha)^2}\,\, s\,\frac{\sh (\pi s)}
 {\ch (\pi s)}
\,ds\,d\phi=\Bigl|
\frac{\Gamma(\alpha-1/2+is)\Gamma(1/2+is)}
{\Gamma(\alpha)\Gamma(is)}
\Bigr|^2
\,\,
\,ds\,d\phi
.\end{multline}

Thus $T_\alpha\otimes \ov T_\alpha$
is a multiplicity free integral
over principal series representations
(this fact was obtained by Pukanszky \cite{Puk}).
Various ways for obtaining the
Plancherel measure (9)
are contained in \cite{VGG}, \cite{tensor},
 \cite{vD}, \cite{Nerindex}.



\smallskip

{\bf 6. Holomorphic
 continuation of $J_\alpha f(\phi,s)$.}
Denote by $W$ the space
of all $f\in H_\alpha\otimes \ov H_\alpha$
admitting analytic  continuation
to a bidisk
 $|z|<1+\delta$, $|u|<1+\delta$.

For $f\in W$, consider its Taylor series
$$f(z,\ov u)=\sum_{k,l} c_{kl} z^k \ov u^l.$$
%converges in the bidisk
% $|z|<1+\delta$, $|u|<1+\delta$.
Obviously, its coefficients $c_{kl}$
exponentially decrease
for $k+l\to\infty$.


\smallskip

{\sc Lemma.}
{\it Fix  $\phi$.
For $f\in W$, the function $J_\alpha f(\phi,s)$
can be extended holomorphically
to the whole complex plane $s\in\C$}.

\smallskip

 {\sc Proof.}
 Fix
$s\in \C$, $\phi\in[0,2\pi]$.
The function $K_\alpha(\phi,s;  z,u)$
as a function in the variables $z$, $u$
has a
polynomial growth near
the boundary of the
bidisk $\D\times \D$:
$$
|K_\alpha(\phi,s;  z,u)|
\leqslant \exp\{4\pi\,(1+ |{\rm Im}\, s|\}
(1-|z|)^{-(1+\alpha+2\,|{\rm Re}\,s|)}
(1-|u|)^{-(1+\alpha+2\,|{\rm Re}\,s|)}
$$
Hence,
for fixed $s$, $\phi$, the coefficients
$a_{kl}=a_{kl}(\phi,s)$
of the series
$$K_\alpha(\phi,s;z,u )=\sum a_{kl}(s,\phi)\ov  z^k u^l$$
have the polynomial growth
 as $k,l\to+\infty$.

Indeed, let $q(z,u)$ be a function
in the bidisk satisfying
$$
|q(z,u)|\leqslant C\cdot \delta^{-h}
\qquad \text{for\quad $|z|\le1-\delta$, $|u|\le1-\delta$}
.$$
The Taylor coefficients $b_{kl}$
of $q(z,u)$ are given by the formula
$$b_{kl}=\frac 1{(2\pi i)^2}
\iint\limits_{|z|=1-\delta, |u|=1-\delta}
\frac{q(z)\,dz\,du}{z^{k+1} u^{l+1}}
.$$
Hence,
 $$|b_{kl}|\le {\rm const}\cdot
\delta^{-h} (1-\delta)^{-k-l-2}.
$$
for all $\delta$. We choose
$\delta=h/(h+k+l+2)$ and obtain
polynomial growth for $b_{kl}$.

For the kernel $K_\alpha(z,u)$, we obtain
in this way
the uniform estimates of the form
$|a_{kl}|\leqslant
A\cdot (1+k+l)^\tau$
in each rectangle
\begin{equation}
|{\rm Im}\, s|\leqslant N, \qquad |{\rm Re}\, s|\leqslant M
.\end{equation}


Since the Taylor coefficients $c_{kl}$ for $f\in W$
exponentially decrease,  the series
$$%\begin{equation}
J_\alpha f(\phi,s)=\sum c_{k,l} a_{k,l}(s,\phi) \cdot
$$%\end{equation}
is absolutely convergent
 (see. (4)),
 the summands are holomorphic
 with respect to
$s$, and the series
is uniformly convergent
on rectangles (10).


\smallskip

{\bf 7. Correspondence of differential
operators.}
The operators of the Lie
algebra $sl(2)\oplus sl(2)$
in the space
$H_\alpha\otimes \ov H_\alpha$
have the form
\begin{align*}
L_0^{(z)}&=
z\frac\partial {\partial z}  +\frac\alpha 2 ;&\qquad
L_1^{(z)}&=z^2\frac\partial {\partial z} +\alpha z;&\qquad
L_{-1}^{(z)}&=\frac\partial {\partial z}&\\
L_0^{(u)}&=\ov u\frac\partial {\partial \ov u}  +
\frac\alpha 2 ;&\qquad
L_{1}^{(u)}&=\frac\partial {\partial \ov u};&\qquad
L_{-1}^{(u)}&=\ov u^2\frac\partial {\partial \ov u}
 +\alpha \ov u&
\end{align*}
For us it will be more convenient the following
collection
of the operators
\begin{align*}
L_0&:=L_0^{(z)}-L_0^{(u)};& L_{-1}&:=L_1^{(u)}-L_{-1}^{(z)};
 & L_1:=L_1^{(z)}-L_{-1}^{(u)};
\\
M_0&:=L_0^{(z)}+L_0^{(u)};& M_{-1}&:=L_1^{(u)}+L_{-1}^{(z)};&
   M_1:=L_1^{(z)}+L_{-1}^{(u)};
\end{align*}
The operators $L_0, L_1, L_{-1}$
span the diagonal subalgebra
$sl_2$ in $sl_2\oplus sl_2$.
Their images under the operator $J_\alpha$
are defined by the formulas
\begin{align*}
&
 J_\alpha\circ\Bigl[z \frac\partial {\partial z}-
    \ov u \frac\partial {\partial \ov u} \Bigr] f(\phi,s)=
\bigl[ \frac\partial {i\partial \phi} \circ J_\alpha \bigr]
f(\phi,s)\\
&
J_\alpha\circ\Bigl[ \ov u^2 \frac\partial {\partial\ov u} +\alpha\ov u -
             \frac\partial {\partial z} \Bigr]f(\phi,s)=
  -\Bigl[ e^{i\phi} \frac\partial {i\partial \phi} +
    (\frac 12+is)e^{i\phi}\Bigr] J_\alpha\,\,  f(\phi,s)\\
&
J_\alpha\circ\Bigl[  z^2 \frac\partial {\partial z}  +\alpha z -
                         \frac\partial {\partial \ov u}\Bigr]=
  \Bigl[ e^{-i\phi} \frac\partial {i\partial \phi} -
    (\frac 12+is)e^{-i\phi}\Bigr] J_\alpha\,\,  f(\phi,s)
\end{align*}

These three formulas easily follow
from
 (8).


 \smallskip

{\sc Theorem 2.}
 {\it The unitary operator $J_\alpha$
  transform the operator $M_0$
 to the operator
\begin{multline}
Q_0f(\phi,s)
=-\frac{(-\tfrac12+is)
 (-\alpha+\tfrac12+is)}{2is}f(\phi,s+i)+
\\+
\frac{(\tfrac12+is)(\alpha-\tfrac12+is)}{2is}f(\phi,s-i)
-
\frac{-\alpha+\tfrac12+is}{2is(-\tfrac12+is)}
     \frac{\partial^2}{\partial\phi^2}f(\phi,s+i)
,\end{multline}
i.e., for any $f\in W$ {\rm(}see our
Section {\rm 6 )},
$$Q_0J_\alpha f=J_\alpha M_0f.$$
The operator $M_1$ transforms to
\begin{multline}
Q_1f(\phi,s)=\\
=e^{i\phi}\biggl[\frac{(\tfrac12+is) (-\alpha+\tfrac12+is)}{2is}f(\phi,s+i)
+\frac{(\tfrac12+is)(\alpha-\tfrac12+is)}{2is}f(\phi,s-i)-\\
\frac{-\alpha+\tfrac12+is}{2is(-\tfrac12+is)}\frac{d^2}{d\phi^2}f(\phi,s+i)+
\frac{-\alpha+\tfrac12+is}{-\tfrac12+is}
      \frac{\partial}{i \partial\phi}f(\phi,s+i)\biggr]
,\end{multline}
and the operator $M_{-1}$ transforms to}
\begin{multline}
Q_{-1}f(\phi,s)=\\
=e^{-i\phi}\biggl[-\frac{(\tfrac12+is) (-\alpha+\tfrac12+is)}{2is}f(\phi,s+i)
+\frac{(\tfrac12+is)(\alpha-\tfrac12+is)}{2is}f(\phi,s-i)-\\
\frac{-\alpha+\tfrac12+is}{2is(-\tfrac12+is)}\frac{d^2}{d\phi^2}f(\phi,s+i)
-\frac{-\alpha+\tfrac12+is}{-\tfrac12+is}
      \frac{\partial}{i \partial\phi}f(\phi,s+i)\biggr]
.\end{multline}



{\sc Proof.}
These formulas can be checked by direct calculations.
For instance, let us consider $M_0$.


Obviously, the operator $M_0$
 is selfadjoint in $H_\alpha\otimes\ov H_\alpha$.
Hence
\begin{multline*}
J_\alpha M_0 f(\phi,s)= \\
\iint\limits_{\D\times \D}
  K(\phi,s;\ov z, u)\Bigl[
 \Bigl(z\frac\partial{\partial z}+
 \ov u\frac\partial{\partial \ov u}+\alpha\Bigr)
\Bigr]
  f(z,\ov u)\cdot
   (1-z\ov z)^{\alpha-2} (1-u\ov u)^{\alpha-2}
 \,\{dz\}\,\{du\}=                          \\=
\iint\limits_{\D\times \D}             \Bigl[
 \Bigl(\ov z\frac\partial{\partial\ov z}+
 u\frac\partial{\partial u}+\alpha\Bigr)
\Bigr]
 K(\phi,s;\ov z, u)\cdot
  f(z,\ov u)  (1-z\ov z)^{\alpha-2} (1-u\ov u)^{\alpha-2}
\,\{dz\}\,\{du\}
\end{multline*}
Thus the first statement of the theorem
is equivalent to the identity
\begin{equation}
 \Bigl(u\frac\partial{\partial u}+
\ov z\frac\partial{\partial \ov z}+\alpha\Bigr)
  K(\phi,s;\ov z, u)-
Q_0\bigl[
 K(\phi,s;\ov z, u)\bigr]=0
\end{equation}
After division by $K$, this  identity
transforms to the form
\begin{multline}
K^{-1}
\Bigl\{u\frac\partial{\partial u}+
z{\frac {\partial}{\partial\ov z}\Bigr\} K}
 +\alpha
+K^{-1}\frac{(-\tfrac12+is)
 (-\alpha+\tfrac12+is)}{2is}K(s+i)-
\\-
K^{-1}
\frac{(\tfrac12+is)(\alpha-\tfrac12+is)}{2is}K(s-i)
+
K^{-1}
\frac{-\alpha+\tfrac12+is}{2is(-\tfrac12+is)}
     \frac{\partial^2}{\partial\phi^2}K(s+i)
\end{multline}
The function in the left side
is a long rational expression
 in $z$, $\ov u$, $e^{i\phi}$, $s$;
  the identity can be easily verified by MAPLE.

 Let us explain how to
verify the identity (15) "by hands".
Each summand of  (15)
can be represented as
a linear combination
$$
a(s)+b(s)\cdot \frac 1{1-z\ov u}
+c(s)\cdot\Bigl[\frac 1{1-ze^{i\phi}}+
    \frac1{1-\ov ue^{-i\phi}}\Bigr]+
    d(s)\cdot\frac{(1-ze^{i\phi})(1-\ov u e^{-i\phi})}
    {1-z\ov u}$$
After this, it remains to sum the coeffitients.

\smallskip

{\sc Remark.}
Let us apply the
operator $Q_0$ to
the functions $f(\phi,s)= g(s)$.
Then the equations
$$Q_0 g(s) = (k+\alpha)  g(s)$$
coincide with a
partial case of the difference equations for the continuous
dual Hahn polynomials
 (see \cite{AAR}, (6.10.9)).


\smallskip

{\bf 8. Some remarks.}
In the work of the author \cite{Nerindex},
there were obtained some elements
of an operational calculus for
the index hypergeometric transform
 (it is called also
 by Olevsky transform
or Jacobi transform,
see \cite{Wey}, \cite{Koo})
\begin{equation}
g(x)\mapsto \widehat g(s)=
\frac 1 {\Gamma(b+c)} \int_0^\infty
g(x)\,\,{}_2F_1(b+is,b-is;b+c;-x) x^{b+c-1}(1+x)^{b-c}    dx
,
\end{equation}
In \cite{Nerindex},
it was shown that
the index hypergeometric transform
maps
the differential operators
$$ Ag(x)=x g(x); \qquad Bg(x)=x\frac\partial{\partial x} g(x)$$
(and hence all the
 operators admitting polynomial
expression in
$x$, $x\frac{\partial}{\partial x}$)
to difference operators in imaginary direction;
see also the work of Cherednik \cite{Che}
containing some similar statements
for simmetric functions
in multidimensional case.
Existence of the formulas (11)--(13)
more or less follows
from these results, but this way
for obtaining the expressions (11)--(13)
also is not very simple.




\smallskip


There arises the following question.


\smallskip

{\sc Question.}
Is it possible to write explicitly
operators of the overalgebra
for the case of
$L^2$
on a pseudo-Riemannian symmetric space
and for the kernel representations?

 Is it possible
to do this at least for rank 1
symmetric spaces?


\smallskip


\begin{thebibliography}{cc}

\bibitem{AAR}
Andrews, G.R.,  Askey, R., Roy, R.,
{\it Special functions.}, Cambridge Univ. Press, 1999

\bibitem{Ban}
van den Ban E.P., Schlichtkrull H.
{\it  A residue calculus for root systems.}
Compositio Math. 123 (2000), no. 1, 27--72.

\bibitem{Bar}
Bargmann V.
{\it Irreducible unitary representations of
the Lorentz group.} Ann. Math, 48 (1947), 568--640

\bibitem{Ber}
%Berezin F.A. {\it  O svyazi mezhdu ko- i kontravariantnymi
%simvolami operatorov na klassicheskikh kompleksnykh
%simmetricheskikh prostranstvakh}.
%Dokl. AN SSSR, 241 (1978), 15--17.
Berezin, F.A. {\it On relations between covariant and contravariant
symbols of operators
for complex classical domains.}
 Dokl. Akad Nauk SSSR, 241, No 1 (1978), 15--17;
English translation: Sov. Math. Dokl. 19 (1978), 786--789

\bibitem{dB}
de Branges, L.
{\it Tensor product spaces.}
J. Math. Anal. Appl. 38 (1972), 109--148.


\bibitem{Che}
Cherednik I.
{\it Harish-Chandra transform and difference operators.}
Preprint, available via http://arXiv.org/abs/math/9706010

\bibitem{Del}
Delorme P. {\it  Formule de Plancherel
 pour les espaces sym\'etriques r�ductifs},
 Ann. of Math. 147
   (1998), 417--452.

\bibitem{vD}
van Dijk, H., Hille, S.C.,
 {\it Canonical representations related to hyperbolic
spaces},
J.Funct.Anal., 147, 109--139   (1997).






\bibitem{GN} Gelfand I.M., Naimark M.I.,
{\it Unitary representations of
classical groups.}
% {\it Unitarnye predstavleniya klassicheskikh grupp.}
 Trudy MIAN., t.36 (1950);
 German translation: Gelfand I.N., Neumark M.A.,
 {\it Unitare Darstellungen der klassischen gruppen.},
 Akademie-Verlag, Berlin, 1957.



\bibitem{GK1}          %\marginpar{ GK1  }
%Gindikin S.G., Karpelevich F.I. {\it
%Mera Plansherelya dlya  rimanovykh simmetricheskikh prostranstv
%nepolozhitel'noj krivizny.}
%Dokl. AN SSSR, 145 (1962), 252--255
Gindikin, S.G., Karpelevich, F.I., {\it Plancherel measure for
Riemannian symmetric spaces of non-positive curvature.}
 Dokl. Akad Nauk SSSR,
 145, 252--255(1962); English translation Sov.Mat.Dokl,3, 962--965
 (1962).

\bibitem{GK2}            %\marginpar{ GK2  }
%Gindikin S.G., Karpelevich F.I. {\it
%Ob odnom integrale, svyazannom s rimanovymi
%simmetricheskimi prostranstvami nepolozhitel'noj krivizny.}
%Izv. AN SSSR, Ser. mat., 30 (1966), 1147--1156
Gindikin, S.G., Karpelevich, F.I.,
{\it On an integral connected with symmetric
 Riemannian space of non-positive curvature.}
 Izv. Akad. Nauk SSSR, Ser Mat., 30, 1147--1156(1966)
; English translation in
 Transl.Amer.Math.Soc., 85, 249--258(1969)



\bibitem{Har1}
Harinck P. {\it Plancherel formula pour $\GL(n,\C)/\U(p,q)$.}
J. Reine Angew. Math. 428 (1992), 45--95.

\bibitem{Har2}
Harinck P.
 {\it Fonctions orbitales
  sur $G_\C/G_\R$. Formula d'inversion
des integrales orbitales et formula de Plancherel.}
J. Funct. Anal., 153 (1998), 52--107

\bibitem{HC}
Harish-Chandra,
 {\it Harmonic analysis on real semisimple groups.  III:
The Maas--Selberg relations and the Plancherel formula.}
Ann. Math.,  104 (1976), 117-201.

\bibitem{Hel}           %\marginpar{ Hel2  }
Helgason, S., {\it Groups and geometric analysis.}
 Acad. Press (1984);% Russkij perevod: Mir, 1987

\bibitem{KS}
Koekoek, R., Swarttouw, R.F.,
{\it Askey scheme of hypergeometric
orthogonal polynomials and their $q$-analogues.}
Delft University of Technology, 1994;
available via
{\tt http://aw.twi.tudelft.nl/~koekoek/research.html}.



\bibitem{Koo}
Koornwinder, T.H.,
{\it  Jacobi functions and analysis on noncompact symmetric
spaces.} in {\it Special functions: group theoretical aspects
and applications,}
eds. Askey, R., Koornwinder T.H., Schempp W., 1--85,
D. Reidel Publ. Co., Dordrecht--Boston, 1984.

\bibitem{Mak}
 Makarevich, B. O.,
 {\it Open symmetric orbits
 of reductive groups in symmetric
$R$-spaces.}
 Mat. Sb. (N.S.) 91(133) (1973), 390--401, 472;
 English transl.:
 Math. USSR-Sb., 20 (1973), 406--418.

\bibitem{tensor}
Molchanov V.F.
% {\it Tenzornye proizvedeniya unitarnykh predstavlenij
%trekhmernoj gruppy Lorentsa.} Izv. Akad. Nauk SSSR,
%ser. matem., 43 (1979), 86---891.
{\it
Tensor products of unitary representations of the
three-dimensional Lorentz group.}
 Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4,
860--891, 967.
English translation: Math USSR Izv., 15 (1980), 113--143.


\bibitem{Molhyper}
Molchanov V.F.% {\it Formula Plansherelya dlya giperboloidov.}
{\it  Plancherel's formula for hyperboloids.}
 Trudy Mat. Inst. Steklov. 147 (1980), 65--85
%Proc. Steklov Institute,?(1981),  393--396
English translation:
{\it Boundary value problems of mathematical physics. X.}
 Proc. Steklov Inst. Math. 1981, no. 2.
Amer. Math. Soc., Providence, R.I.,
 1981. pp. 63--83.


\bibitem{Molrank}
Molchanov V.F.
{\it The Plancherel formula for
pseudo-Riemannian symmetric spaces of rank $1$.}
Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 545--549.
%{\it Formula Plansherelya
% dlya psevdorimanovykh prostranstv
%ranga 1.} Dokl. Akad. nauk SSSR, 290  (1987), 545--549
English translation:
Sov. Math. Dokl. 34 (1987), 323--326

\bibitem{MolVINITI}
Molchanov V.F.
% {\it Garmonicheskij
% analiz na odnorodnykh prostranstvakh.}
%Sovrem. problemy matematiki,
%Fundamental'nye napravleniya,  59.
%Nekommutativnyj garmonicheskij analiz-I.,M.:VINITI 1990, 5--144;
%Encyclopaedia of Math. Sci., 59, Springer, 1995, 1--135
{\it Harmonic analysis on homogeneous spaces}.
 Encyclopaedia Math. Sci., 59,
Representation theory
 and noncommutative harmonic analysis, II, 1--135,
Springer, Berlin, 1995.


\bibitem{pse}
Neretin Yu.A.
% {\it Psevdorimanovy simmetricheskie prostranstva:
%odnotipnye realizatsii i otkrytye vlozheniya v grassmaniany.}
%Zapiski nauchn. semin. POMI RAN, 256, 145--167
%Neretin, Yu.A.
%{\it Pseemannian symmetric spaces: one type realizations
%and open embeddings to Grassmannians.},
%Zapiski Nauchn. Semin. POMI, v. 256, 145--167;
%Neretin, Yu. A.(RS-IUM)
{\it Pseudo-Riemannian symmetric spaces:
 single-type realizations and open embeddings into
Grassmannians.}
Zap. Nauchn. Sem. S.-Peterburg.
 Otdel. Mat. Inst. Steklov. (POMI) 256 (1999),
   145--167,
English translation,  J.Math.Sci, New York;
preprint version is available via
 {\tt arXiv.org/abs/math/9905014}





\bibitem{Nerpl}
Neretin, Yu.A.,
{\it Plancherel formula for Berezin deformation
 of $L^2$ on Riemannian
symmetric space}, to appear in J. Funct. Anal.,
 preprint   version is available via
 {\tt  http://arXiv.org/abs/math/9911020}



\bibitem{Nerindex}
Neretin Yu.A.
% {\it Indeksnoe gipergeometricheskoe preobrazovanie
%i imitatsiya  analiza yader Berezina na giperbolicheskikh prostransvtakh},
%Mat. Sbornik, 2001, Preprint version is available via ????
%Neretin, Yu.A.,
% {\it Index hypergeometric transform
% and imitation of
%analysis of Berezin kernels on hyperbolic spaces,}
%Sbornik Math., to appear.,
{\it Index hypergeometric transformation
 and an imitation of the analysis of Berezin
kernels on hyperbolic spaces.}
Mat. Sb. 192 (2001), no. 3, 83--114;
English translation in Sb. Math. 192 (2001),
 no. 3-4, 403--432;
 is available via
 {\tt http://arXiv.org/abs/math/0104035}


\bibitem{Nerhyper}
Neretin Yu.A.
{\it Matrix balls, radial analysis of Berezin
kernels and hypergeometric determinants.}
 Moscow Math.J.,  vol.1, 157 -- 220;
preprint version
 is available via
 {\tt http://arXiv.org/abs/math/0012220}

\bibitem{Puk}
Pukanszky, L., {\it On the Kronecker products of irreducible
 unitary representations  of the
 $2\times2$ real unimodular  group.}
  Trans. Amer. Math. Soc., 100
(1961), 116--152

%\bibitem{Ros}
%Rosengren, H.
%{\it Multilinear Hankel forms
% of higher order and orthogonal polynomials.}
%Math. Scand. 82 (1998), no. 1, 53--88.


\bibitem{UU}
Unterberger, A., Upmeier, H., {\it The Berezin transform and
 invariant differential operators}.
Comm.Math.Phys.,164, 563--597(1994)


\bibitem{VGG}
Vershik A.M., Gelfand I.M., Graev M.I.,
%{\it Predstavleniya $SL_2(R)$, gde $R$ -- kol'tso funktsij.}
%Uspekhi mat. nauk, 28 (1973), No 5,83--128
%Vershik, A.M., Gelfand, I.M., Graev, M.I.,
{\it Representations of , where $R$ is function ring.}
Uspehi Mat. Nauk 28 (1973), No 5,83--128;
English translation: Russian Math.
Surveys 28 (1973), No 5, 87--132











\bibitem{Wey}
Weyl, H.,
{\it Uber gewonliche lineare Differentialgleichungen
mis singularen Stellen und ihre Eigenfunktionen {\rm 2} Note}.
Nachr. Konig. Gess. Wissen. Gottingen, Math.-Phys.,
%Est' napisanie Nachr. Gottingen
1910, 442-467; Reprinted in
Weyl, H., {\it Gessamelte Abhandlungen}, Bd. 1, 224--247,
Springer, 1968.


 \end{thebibliography}

\sf Math.Phys. Group,
Institute of Theretical and Experimental Physics,

B.Cheremushkinskaya, 25, Moscow 117259, Russia

\tt neretin@main.mccme.rssi.ru

\end{document}



math0202018.extracted_bib
\begin{thebibliography}{cc}

\bibitem{AAR}
Andrews, G.R.,  Askey, R., Roy, R.,
{\it Special functions.}, Cambridge Univ. Press, 1999

\bibitem{Ban}
van den Ban E.P., Schlichtkrull H.
{\it  A residue calculus for root systems.}
Compositio Math. 123 (2000), no. 1, 27--72.

\bibitem{Bar}
Bargmann V.
{\it Irreducible unitary representations of
the Lorentz group.} Ann. Math, 48 (1947), 568--640

\bibitem{Ber}
%Berezin F.A. {\it  O svyazi mezhdu ko- i kontravariantnymi
%simvolami operatorov na klassicheskikh kompleksnykh
%simmetricheskikh prostranstvakh}.
%Dokl. AN SSSR, 241 (1978), 15--17.
Berezin, F.A. {\it On relations between covariant and contravariant
symbols of operators
for complex classical domains.}
 Dokl. Akad Nauk SSSR, 241, No 1 (1978), 15--17;
English translation: Sov. Math. Dokl. 19 (1978), 786--789

\bibitem{dB}
de Branges, L.
{\it Tensor product spaces.}
J. Math. Anal. Appl. 38 (1972), 109--148.


\bibitem{Che}
Cherednik I.
{\it Harish-Chandra transform and difference operators.}
Preprint, available via http://arXiv.org/abs/math/9706010

\bibitem{Del}
Delorme P. {\it  Formule de Plancherel
 pour les espaces sym\'etriques r�ductifs},
 Ann. of Math. 147
   (1998), 417--452.

\bibitem{vD}
van Dijk, H., Hille, S.C.,
 {\it Canonical representations related to hyperbolic
spaces},
J.Funct.Anal., 147, 109--139   (1997).






\bibitem{GN} Gelfand I.M., Naimark M.I.,
{\it Unitary representations of
classical groups.}
% {\it Unitarnye predstavleniya klassicheskikh grupp.}
 Trudy MIAN., t.36 (1950);
 German translation: Gelfand I.N., Neumark M.A.,
 {\it Unitare Darstellungen der klassischen gruppen.},
 Akademie-Verlag, Berlin, 1957.



\bibitem{GK1}          %\marginpar{ GK1  }
%Gindikin S.G., Karpelevich F.I. {\it
%Mera Plansherelya dlya  rimanovykh simmetricheskikh prostranstv
%nepolozhitel'noj krivizny.}
%Dokl. AN SSSR, 145 (1962), 252--255
Gindikin, S.G., Karpelevich, F.I., {\it Plancherel measure for
Riemannian symmetric spaces of non-positive curvature.}
 Dokl. Akad Nauk SSSR,
 145, 252--255(1962); English translation Sov.Mat.Dokl,3, 962--965
 (1962).

\bibitem{GK2}            %\marginpar{ GK2  }
%Gindikin S.G., Karpelevich F.I. {\it
%Ob odnom integrale, svyazannom s rimanovymi
%simmetricheskimi prostranstvami nepolozhitel'noj krivizny.}
%Izv. AN SSSR, Ser. mat., 30 (1966), 1147--1156
Gindikin, S.G., Karpelevich, F.I.,
{\it On an integral connected with symmetric
 Riemannian space of non-positive curvature.}
 Izv. Akad. Nauk SSSR, Ser Mat., 30, 1147--1156(1966)
; English translation in
 Transl.Amer.Math.Soc., 85, 249--258(1969)



\bibitem{Har1}
Harinck P. {\it Plancherel formula pour $\GL(n,\C)/\U(p,q)$.}
J. Reine Angew. Math. 428 (1992), 45--95.

\bibitem{Har2}
Harinck P.
 {\it Fonctions orbitales
  sur $G_\C/G_\R$. Formula d'inversion
des integrales orbitales et formula de Plancherel.}
J. Funct. Anal., 153 (1998), 52--107

\bibitem{HC}
Harish-Chandra,
 {\it Harmonic analysis on real semisimple groups.  III:
The Maas--Selberg relations and the Plancherel formula.}
Ann. Math.,  104 (1976), 117-201.

\bibitem{Hel}           %\marginpar{ Hel2  }
Helgason, S., {\it Groups and geometric analysis.}
 Acad. Press (1984);% Russkij perevod: Mir, 1987

\bibitem{KS}
Koekoek, R., Swarttouw, R.F.,
{\it Askey scheme of hypergeometric
orthogonal polynomials and their $q$-analogues.}
Delft University of Technology, 1994;
available via
{\tt http://aw.twi.tudelft.nl/~koekoek/research.html}.



\bibitem{Koo}
Koornwinder, T.H.,
{\it  Jacobi functions and analysis on noncompact symmetric
spaces.} in {\it Special functions: group theoretical aspects
and applications,}
eds. Askey, R., Koornwinder T.H., Schempp W., 1--85,
D. Reidel Publ. Co., Dordrecht--Boston, 1984.

\bibitem{Mak}
 Makarevich, B. O.,
 {\it Open symmetric orbits
 of reductive groups in symmetric
$R$-spaces.}
 Mat. Sb. (N.S.) 91(133) (1973), 390--401, 472;
 English transl.:
 Math. USSR-Sb., 20 (1973), 406--418.

\bibitem{tensor}
Molchanov V.F.
% {\it Tenzornye proizvedeniya unitarnykh predstavlenij
%trekhmernoj gruppy Lorentsa.} Izv. Akad. Nauk SSSR,
%ser. matem., 43 (1979), 86---891.
{\it
Tensor products of unitary representations of the
three-dimensional Lorentz group.}
 Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4,
860--891, 967.
English translation: Math USSR Izv., 15 (1980), 113--143.


\bibitem{Molhyper}
Molchanov V.F.% {\it Formula Plansherelya dlya giperboloidov.}
{\it  Plancherel's formula for hyperboloids.}
 Trudy Mat. Inst. Steklov. 147 (1980), 65--85
%Proc. Steklov Institute,?(1981),  393--396
English translation:
{\it Boundary value problems of mathematical physics. X.}
 Proc. Steklov Inst. Math. 1981, no. 2.
Amer. Math. Soc., Providence, R.I.,
 1981. pp. 63--83.


\bibitem{Molrank}
Molchanov V.F.
{\it The Plancherel formula for
pseudo-Riemannian symmetric spaces of rank $1$.}
Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 545--549.
%{\it Formula Plansherelya
% dlya psevdorimanovykh prostranstv
%ranga 1.} Dokl. Akad. nauk SSSR, 290  (1987), 545--549
English translation:
Sov. Math. Dokl. 34 (1987), 323--326

\bibitem{MolVINITI}
Molchanov V.F.
% {\it Garmonicheskij
% analiz na odnorodnykh prostranstvakh.}
%Sovrem. problemy matematiki,
%Fundamental'nye napravleniya,  59.
%Nekommutativnyj garmonicheskij analiz-I.,M.:VINITI 1990, 5--144;
%Encyclopaedia of Math. Sci., 59, Springer, 1995, 1--135
{\it Harmonic analysis on homogeneous spaces}.
 Encyclopaedia Math. Sci., 59,
Representation theory
 and noncommutative harmonic analysis, II, 1--135,
Springer, Berlin, 1995.


\bibitem{pse}
Neretin Yu.A.
% {\it Psevdorimanovy simmetricheskie prostranstva:
%odnotipnye realizatsii i otkrytye vlozheniya v grassmaniany.}
%Zapiski nauchn. semin. POMI RAN, 256, 145--167
%Neretin, Yu.A.
%{\it Pseemannian symmetric spaces: one type realizations
%and open embeddings to Grassmannians.},
%Zapiski Nauchn. Semin. POMI, v. 256, 145--167;
%Neretin, Yu. A.(RS-IUM)
{\it Pseudo-Riemannian symmetric spaces:
 single-type realizations and open embeddings into
Grassmannians.}
Zap. Nauchn. Sem. S.-Peterburg.
 Otdel. Mat. Inst. Steklov. (POMI) 256 (1999),
   145--167,
English translation,  J.Math.Sci, New York;
preprint version is available via
 {\tt arXiv.org/abs/math/9905014}





\bibitem{Nerpl}
Neretin, Yu.A.,
{\it Plancherel formula for Berezin deformation
 of $L^2$ on Riemannian
symmetric space}, to appear in J. Funct. Anal.,
 preprint   version is available via
 {\tt  http://arXiv.org/abs/math/9911020}



\bibitem{Nerindex}
Neretin Yu.A.
% {\it Indeksnoe gipergeometricheskoe preobrazovanie
%i imitatsiya  analiza yader Berezina na giperbolicheskikh prostransvtakh},
%Mat. Sbornik, 2001, Preprint version is available via ????
%Neretin, Yu.A.,
% {\it Index hypergeometric transform
% and imitation of
%analysis of Berezin kernels on hyperbolic spaces,}
%Sbornik Math., to appear.,
{\it Index hypergeometric transformation
 and an imitation of the analysis of Berezin
kernels on hyperbolic spaces.}
Mat. Sb. 192 (2001), no. 3, 83--114;
English translation in Sb. Math. 192 (2001),
 no. 3-4, 403--432;
 is available via
 {\tt http://arXiv.org/abs/math/0104035}


\bibitem{Nerhyper}
Neretin Yu.A.
{\it Matrix balls, radial analysis of Berezin
kernels and hypergeometric determinants.}
 Moscow Math.J.,  vol.1, 157 -- 220;
preprint version
 is available via
 {\tt http://arXiv.org/abs/math/0012220}

\bibitem{Puk}
Pukanszky, L., {\it On the Kronecker products of irreducible
 unitary representations  of the
 $2\times2$ real unimodular  group.}
  Trans. Amer. Math. Soc., 100
(1961), 116--152

%\bibitem{Ros}
%Rosengren, H.
%{\it Multilinear Hankel forms
% of higher order and orthogonal polynomials.}
%Math. Scand. 82 (1998), no. 1, 53--88.


\bibitem{UU}
Unterberger, A., Upmeier, H., {\it The Berezin transform and
 invariant differential operators}.
Comm.Math.Phys.,164, 563--597(1994)


\bibitem{VGG}
Vershik A.M., Gelfand I.M., Graev M.I.,
%{\it Predstavleniya $SL_2(R)$, gde $R$ -- kol'tso funktsij.}
%Uspekhi mat. nauk, 28 (1973), No 5,83--128
%Vershik, A.M., Gelfand, I.M., Graev, M.I.,
{\it Representations of , where $R$ is function ring.}
Uspehi Mat. Nauk 28 (1973), No 5,83--128;
English translation: Russian Math.
Surveys 28 (1973), No 5, 87--132











\bibitem{Wey}
Weyl, H.,
{\it Uber gewonliche lineare Differentialgleichungen
mis singularen Stellen und ihre Eigenfunktionen {\rm 2} Note}.
Nachr. Konig. Gess. Wissen. Gottingen, Math.-Phys.,
%Est' napisanie Nachr. Gottingen
1910, 442-467; Reprinted in
Weyl, H., {\it Gessamelte Abhandlungen}, Bd. 1, 224--247,
Springer, 1968.


 \end{thebibliography}