Berezin quantization and representation theory

© V. F. Molchanov

Tambov State University named after G. R. Derzhavin

Internatsionalnaya 33, Tambov, Russian Federation, 392000

E-mail<EMAIL_ADDRESS>
to memory of Gerrit van Dijk

Abstract. We present an approach to Berezin quantization (a variant of
quantization in the spirit of Berezin) on para-Hermitian symmetric spaces
using the notion of an ”overgroup”. This approach gives covariant and
contravariant symbols and the Berezin transform in a natural and transparent
way

Keywords: Lie groups and Lie algebras; para-Hermitian symmetric spaces;
Berezin quantization; covariant and contravariant symbols; Berezin transform

The basic notion of the Berezin quantization on a manifold $M$ is a
correspondence which to an operator $A$ from a class assigns the pair of
functions $F$ and $F^{\natural}$ defined on $M$. These functions are called
covariant and contravariant symbols of $A$. We are interested in homogeneous
space $M=G/H$ (namely, a para-Hermitian symmetric space) and classes of
operators related to the representation theory.

The most algebraic version of quantization – we call it the polynomial
quantization – is obtained when operators belong to the algebra of operators
corresponding in a representation $T$ of $G$ to elements $X$ of the universal
enveloping algebra ${\rm Env}\,(\mathfrak{g})$ of the Lie algebra
$\mathfrak{g}$ of $G$. In this case symbols turn out to be polynomials on
$\mathfrak{g}$.

Now we offer a new theme in the Berezin quantization on $G/H$: as an initial
class of operators we take operators corresponding to elements of the group
$G$ itself in a representation $T$ of this group. We call it the group
quantization.

§ 1. Para-Hermitian symmetric spaces

Let $G/H$ be a semisimple symmetric space. Here $G$ is a connected semisimple
Lie group with an involutive automorphism $\sigma\neq 1$, and $H$ is an open
subgroup of $G^{\sigma}$, the subgroup of fixed points of $\sigma$. We
consider that groups act on their homogeneous spaces from the right, so that
$G/H$ consists of right cosets $Hg$.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of $G$ and of $H$
respectively. Let $B_{\mathfrak{g}}$ be the Killing form of $G$. There is a
decomposition of $\mathfrak{g}$ into direct sums of $+1,\,-1$-eigenspaces of
the involution $\sigma$:

${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{q}}.$

The subspace $\mathfrak{q}$ is invariant with respect to $H$ in the adjoint
representation ${\rm Ad}$. It can be identified with the tangent space to
$G/H$ at the point $x^{0}=He$.

The dimension of Cartan subspaces of $\mathfrak{q}$ (maximal Abelian
subalgebras in $\mathfrak{q}$ consisting of semisimple elements) is called the
rank of $G/H$.

Now let $G/H$ be a symplectic manifold. Then $\mathfrak{h}$ has a non-trivial
centre $Z({\mathfrak{h}})$. For simplicity we assume that $G/H$ is an orbit
${\rm{Ad}}\,G\cdot Z_{0}$ of an element $Z_{0}\in{\mathfrak{g}}$. In
particular, then $Z_{0}\in Z(\mathfrak{h})$.

Further, we can also assume that $G$ is simple. Such spaces $G/H$ are divided
into 4 classes (see [3],[4]):

$(a)$ Hermitian symmetric spaces;

$(b)$ semi-Kählerian symmetric spaces;

$(c)$ para-Hermitian symmetric spaces;

$(d)$ complexifications of spaces of class (a).

Dimensions of $Z({\mathfrak{h}})$ are 1,1,1,2, respectively. Spaces of class
$(a)$ are Riemannian, of other three classes are pseudo-Riemannian (not
Riemannian).

We focus on spaces of class $(c)$. Here the center $Z(\mathfrak{h})$ is one-
dimensional, so that $Z(\mathfrak{h})={\mathbb{R}}Z_{0}$, and $Z_{0}$ can be
normalized so that the operator $I=(\rm{ad}Z_{0})_{\mathfrak{q}}$ on
$\mathfrak{q}$ has eigenvalues $\pm 1$. A symplectic structure on $G/H$ is
defined by the bilinear form $\omega(X,Y)=B_{\mathfrak{g}}(X,IY)$ on
$\mathfrak{q}$.

The $\pm 1$-eigenspaces ${\mathfrak{q}}^{\pm}\subset{\mathfrak{q}}$ of $I$ are
Lagrangian, $H$-invariant, and irreducible. They are Abelian subalgebras of
$\mathfrak{g}$. So $\mathfrak{g}$ becomes a graded Lie algebra:

${\mathfrak{g}}={\mathfrak{q}}^{-}+{\mathfrak{h}}+{\mathfrak{q}}^{+},$

with commutation relations $[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h}$,
$[\mathfrak{h},\mathfrak{q}^{-}]\subset\mathfrak{q}^{-}$,
$[\mathfrak{h},\mathfrak{q}^{+}]\subset\mathfrak{q}^{+}$.

The pair $({\mathfrak{q}}^{+},{\mathfrak{q}}^{-})$ is a Jordan pair with
multiplication

$\\{XYZ\\}=\frac{1}{2}\,[[X,Y],\,Z],$

see [5]. Let $r$ and $\varkappa$ be the rank and the genus of this Jordan
pair. The rank $r$ coincides with the rank of $G/H$.

Set $Q^{\pm}=\exp{\mathfrak{q}}^{\pm}$. The subgroups
$P^{\pm}=HQ^{\pm}=Q^{\pm}H$ are maximal parabolic subgroups of $G$. One has
the following decompositions:

$\displaystyle G$ $\displaystyle=$ $\displaystyle\overline{Q^{+}HQ^{-}}$ (1.1)
$\displaystyle=$ $\displaystyle\overline{Q^{-}HQ^{+}},$ (1.2)

where bar means closure and the sets under the bar are open and dense in $G$.
Let us call (1.1) and (1.2) the Gauss decomposition and (allowing some slang)
the anti-Gauss decomposition respectively. Decompositions (1.1), (1.2) mean
that almost any element $g\in G$ can be decomposed as ( the Gauss
decomposition):

$g=\exp\eta\cdot h\cdot\exp\xi,$ $None$

or (the anti-Gauss decomposition):

$g=\exp\xi\cdot h\cdot\exp\eta,$ $None$

where $h\in H$, $\xi\in{\mathfrak{q}^{-}}$, $\eta\in{\mathfrak{q}^{+}}$, all
three factors in (1.3) and (1.4) are defined uniquely. We also use the Gauss
decomposition (1.3) in a little different form:

$g=\exp\eta\cdot\exp\xi\cdot h,$ $None$

where $\eta$ and $h$ are the same as in (1.3), and $\xi$ is obtained from
$\xi$ in (1.3) by ${\rm Ad}\,h$.

Decompositions (1.3) and (1.4) generate actions of $G$ on ${\mathfrak{q}}^{-}$
and ${\mathfrak{q}}^{+}$ respectively, namely, $\xi\mapsto\mathaccent
869{\xi}=\xi\bullet g$ and $\eta\mapsto\mathaccent 866{\eta}=\eta\circ g$:

$\displaystyle{\rm{exp}}\,\xi\cdot g$ $\displaystyle=$
$\displaystyle{\rm{exp}}\,Y\cdot\mathaccent 869{h}\cdot{\rm{exp}}\,\mathaccent
869{\xi},$ (1.3) $\displaystyle{\rm{exp}}\,\eta\cdot g$ $\displaystyle=$
$\displaystyle{\rm{exp}}\,X\cdot\mathaccent 866{h}\cdot{\rm{exp}}\,\mathaccent
866{\eta},$ (1.4)

where $X\in{\mathfrak{q}^{-}},\,Y\in{\mathfrak{q}^{+}}$. These actions are
defined on open and dense sets depending on $g$. Therefore, $G$ acts on
$\mathfrak{q}^{-}\times\mathfrak{q}^{+}:\,(\xi,\eta)\mapsto(\mathaccent
869{\xi},\mathaccent 866{\eta})$. The stabilizer of the point
$(0,0)\in{\mathfrak{q}^{-}\times\mathfrak{q}^{+}}$ is $P^{+}\cap P^{-}=H$, so
that we get an embedding

$\mathfrak{q}^{-}\times\mathfrak{q}^{+}\hookrightarrow G/H.$ $None$

It is defined on an open and dense set, its image is also an open and dense
set. Therefore, we can consider
$(\xi,\eta)\in\mathfrak{q}^{-}\times\mathfrak{q}^{+}$ as coordinates on $G/H$,
let us call them horospherical coordinates.

Let us write explicit formula for embedding (1.8). We use a redecomposition
”anti-Gauss” to ”Gauss”. We take $\xi\in\mathfrak{q}^{-}$,
$\eta\in\mathfrak{q}^{+}$ and decompose the anti-Gauss product
${\rm{exp}}\,\xi\cdot{\rm{exp}}\,(-\eta)$ according to formula (1.5) (the
”Gauss”):

${\rm{exp}}\,\xi\cdot{\rm{exp}}\,(-\eta)={\rm{exp}}\,Y\cdot{\rm{exp}}\,X\cdot
h,$ $None$

where $X\in{\mathfrak{q}^{-}}$, $Y\in{\mathfrak{q}^{+}}$. The obtained element
$h\in H$ depends on $\xi$ and $\eta$ only, denote it by $h(\xi,\eta)$. Using
(1.9), let us form the following element $g\in G$:

$g={\rm{exp}}\,Y{\rm{exp}}\,\xi={\rm{exp}}\,X\cdot h\cdot{\rm{exp}}\,\eta.$
$None$

Then the pair $\xi,\eta$ goes just to the point $x=x^{0}g$ where $g$ is
defined by (1.10).

Under the action of the group $G$ the element $h(\xi,\eta)$ is transformed as
follows:

$h(\mathaccent 869{\xi},\mathaccent 866{\eta})={\mathaccent 869{h}}^{-1}\cdot
h(\xi,\eta)\cdot{\mathaccent 866{h}},$ $None$

where $\mathaccent 869{h}$ and $\mathaccent 866{h}$ are taken from (1.6) and
(1.7) respectively.

For $h\in H$, denote

$b(h)={\rm det\,(Ad}\,h)|_{{\mathfrak{q}}^{+}}\,.$

The function $k(\xi,\eta)=b(h(\xi,\eta))$ is $N(\xi,\eta)^{-\varkappa}$, where
$N(\xi,\eta)$ is an irreducible polynomial $N(\xi,\eta)$ of degree $r$ in
$\xi$ and in $\eta$ separately. Considered as a function on $G/H$, the
function $k(\xi,\eta)$ is an analogue of the Bergman kernel for Hermitian
symmetric spaces. It follows from (1.11) that under action of $g\in G$ the
function $N(\xi,\eta)$ is transformed as follows:

$N(\,\mathaccent 869{\xi}\,,\,\mathaccent 866{\eta}\,)=N(\xi,\eta)\cdot
b(\,{\mathaccent 869{h}}\,)^{1/\varkappa}\cdot b(\,{\mathaccent
866{h}}\,)^{-1/\varkappa}$ $None$

In horospherical coordinates the $G$-invariant measure on $G/H$ is:

$dx=dx(\xi,\eta)=|N(\xi,\eta)|^{-\varkappa}\,d\xi\,d\eta,$

where $d\xi$ and $d\eta$ are Euclidean measures on ${\mathfrak{q}^{-}}$ and
${\mathfrak{q}^{+}}$ respectively.

§ 2. Maximal degenerate series representations

In this Section we introduce two series of representations induced by
characters of maximal parabolic subgroups $P^{\pm}$ of $G$ (maximal degenerate
series representations).

First we take the character $\omega_{\lambda}(h)$, $\lambda\in\mathbb{C}$, of
$H$:

$\omega_{\lambda}(h)=\big{|}b(h)\big{|}^{-\lambda/\varkappa}$

and then we extend this character to the subgroups $P^{\pm}$, setting it equal
to 1 on $Q^{\pm}$.

Let us consider induced representations $\pi^{\pm}_{\lambda}$ of $G$:

$\pi^{\pm}_{\lambda}={\rm{Ind}}\,\left(G,\,P^{\mp},\,\omega_{\mp\lambda}\right).$

They act on the space ${\cal{D}}^{\pm}_{\lambda}(G)$ of functions $f\in
C^{\infty}(G)$ having the uniformity property:

$f(pg)=\omega_{\mp\lambda}(p)f(g),\ \ p\in P^{\mp},$

by translations from the right:

$\left(\pi^{\pm}_{\lambda}(g)f\right)(g_{1})=f(g_{1}g).$

Realize them in the non-compact picture: we restrict functions from
${\cal{D}}^{\pm}_{\lambda}(G)$ to the subgroups $Q^{\pm}$ and identify them
(as manifolds) with ${\mathfrak{q}}^{\pm}$, we obtain

$\left(\pi^{-}_{\lambda}(g)f\right)(\xi)=\omega_{\lambda}(\mathaccent
869{h})f(\mathaccent 869{\xi}),\ \ \
\left(\pi^{+}_{\lambda}(g)f\right)(\eta)=\omega_{\lambda}({\mathaccent
866{h}}^{-1})f(\mathaccent 866{\eta}),$

where $\mathaccent 869{\xi}$, $\mathaccent 869{h}$, $\mathaccent 866{\eta}$,
$\mathaccent 866{h}$ are taken from decompositions (1.6), (1.7).

Let us write intertwining operators. Introduce operators $A_{\lambda}$ and
$B_{\lambda}$ by:

$\displaystyle(A_{\lambda}\varphi)(\eta)$ $\displaystyle=$
$\displaystyle\int_{\mathfrak{q}^{-}}|N(\xi,\eta)|^{-\lambda-\varkappa}\
\varphi(\xi)\,d\xi,$ $\displaystyle(B_{\lambda}\varphi)(\xi)$ $\displaystyle=$
$\displaystyle\int_{\mathfrak{q}^{+}}|N(\xi,\eta)|^{-\lambda-\varkappa}\
\varphi(\eta)\,d\eta.$

The operator $A_{\lambda}$ intertwines $\pi^{-}_{\lambda}$ with
$\pi^{+}_{-\lambda-\varkappa}$ and the operator $B_{\lambda}$ intertwines
$\pi^{+}_{\lambda}$ with $\pi^{-}_{-\lambda-\varkappa}$ . Their composition is
a scalar operator:

$B_{\lambda}A_{-\lambda-\varkappa}=c(\lambda)^{-1}\cdot{\rm id},$ $None$

where $c(\lambda)$ is a meromorphic function of $\lambda$, it is invariant
with respect to the change $\lambda\mapsto-\lambda-\varkappa$.

We can extend $\pi^{\pm}_{\lambda}$, $A_{\lambda}$ and $B_{\lambda}$ to
distributions on ${\mathfrak{q}^{-}}$ and ${\mathfrak{q}^{+}}$.

The representation $\pi^{-}_{\lambda}$ of the Lie algebra $\mathfrak{g}$ is
given by some differential operators of the first order. This representation
can be considered on different spaces of functions of $\xi$: for example, the
space $C^{\infty}({\mathfrak{q}}^{-})$, the space
${\rm{Pol}}\,({\mathfrak{q}}^{-})$ of polynomials in $\xi$, the space
${\cal{D}}^{\prime}({\mathfrak{q}}^{-})$ of distributions on
${\mathfrak{q}}^{-}$, in particular, the space $Z$ of distributions on
${\mathfrak{q}}^{-}$ concentrated at the origin, etc. The same concerns to
$\pi^{+}_{\lambda}$.

Notice

$\omega_{\lambda}\big{(}h(\xi,\eta)\big{)}=|N(\xi,\eta)|^{\lambda}\,,$ (2.2)

hence formula (1.12) gives

$|N(\mathaccent 869{\xi},\mathaccent
866{\eta})|^{\lambda}=|N(\xi,\eta)|^{\lambda}\cdot\omega_{\lambda}({\mathaccent
869{h}})^{-1}\cdot\omega_{\lambda}({\mathaccent 866{h}})\,,$

which can be interpreted as an invariance property of the function
$|N(\xi,\eta)|^{\lambda}$ :

$\Big{[}\pi^{-}_{\lambda}(g)\otimes\pi^{+}_{\lambda}(g)\Big{]}|N(\xi,\eta)|^{\lambda}=|N(\xi,\eta)|^{\lambda}.$

§ 3. Symbols and transforms

In this Section we apply to a para-Hermitian symmetric space $G/H$ the scheme
of quantization in the spirit of Berezin offered in [6]. We consider a variant
of the quantization, we call the polynomial quantization. For an initial
algebra of operators we take here the algebra of operators
$\pi^{-}_{\lambda}({\rm Env}\,(\mathfrak{g}))$, where $\lambda\in\mathbb{C}$
and ${\rm Env}\,(\mathfrak{g})$ is the universal enveloping algebra of
$\mathfrak{g}$. In contrast to [6], we use the non-compact picture for
representations $\pi^{\pm}_{\lambda}$, see § 2. The role of the Fock space is
played by a space of functions $\varphi(\xi)$, $\xi\in{\mathfrak{q}}^{-}$, so
that our operators act in functions $\varphi(\xi)$. We introduce covariant and
contravariant symbols of operators, the Berezin transform etc.

As a (an analogue of) supercomplete system we take the kernel of the
intertwining operators $A_{-\lambda-\varkappa}$ from § 2, namely, the function

$\Phi(\xi,\eta)=\Phi_{\lambda}(\xi,\eta)=|N(\xi,\eta)|^{\lambda}\,.$ $None$

For an operator $D=\pi^{-}_{\lambda}(X)$, $X\in{\rm Env}\,(\mathfrak{g}))$,
the function

$F(\xi,\eta)=\frac{1}{\Phi(\xi,\eta)}\,(\pi^{-}_{\lambda}(X)\otimes
1)\Phi(\xi,\eta)$

is called the covariant symbol of $D$. Since $\xi$, $\eta$ are horospherical
coordinates on $G/H$, covariant symbols become functions on $G/H$ and,
moreover, polynomials on $G/H\subset{\mathfrak{g}}$. It is why we call this
variant of quantization the polynomial quantization. Denote the space of
covariant symbols by ${\cal{A}}_{\lambda}$. For generic $\lambda$, this space
is the space $S(G/H)$ of all polynomials on $G/H$.

In particular, the covariant symbol of the identity operator is the function
on $G/H$ equal to $1$ identically. If $X$ belongs to the Lie algebra
$\mathfrak{g}$ itself, then the covariant symbol of the operator
$\pi^{-}_{\lambda}(X)$ is a linear function $B_{\mathfrak{g}}(X,x)$ of $x\in
G/H\subset{\mathfrak{g}}$ with coordinates $\xi$, $\eta$, up to a factor
depending on $\lambda$.

The operator $D$ is recovered by its covariant symbol $F$:

$(D\varphi)(\xi)=c\,\int_{G/H}\,F(\xi,v)\,\frac{\Phi(\xi,v)}{\Phi(u,v)}\,\varphi(u)\,dx(u,v),$
$None$

where $c=c(\lambda)$ is taken from (2.1). Indeed, the function $\Phi$ has a
reproducing property

$\varphi(\xi)=c(\lambda)\int_{G/H}\ \frac{\Phi(\xi,\eta)}{\Phi(u,v)}\
\varphi(u)\,dx(u,v).$

which is nothing but formula (2.1) written in another form.

Let $U$ be the representation of the group $G$ by translations in functions on
$G/H$ (quasiregular representation), for example, on the space
$C^{\infty}(G/H)$, and $U$ the corresponding representation of the Lie algebra
$\mathfrak{g}$. The correspondence $D\mapsto F$, assigning to an operator its
covariant symbol, is $\mathfrak{g}$-equivariant, it means that if $F$ is the
covariant symbol of an operator
$D=\pi^{-}_{\lambda}(X),\,X\in{\rm{Env}}\,(\mathfrak{g})$, then $U(L)F$, where
$L\in{\mathfrak{g}}$, is the covariant symbol of the operator
$\pi^{-}_{\lambda}({\rm{ad}}\,L\cdot X)$.

The multiplication of operators gives rise to the multiplication of covariant
symbols, denote it by $*$. Namely, let $F_{1}$, $F_{2}$ be covariant symbols
of operators $D_{1}$, $D_{2}$ respectively. Then the covariant symbol
$F_{1}*F_{2}$ of the product $D_{1}D_{2}$ is

$(F_{1}*F_{2})(\xi,\eta)=\frac{1}{\Phi(\xi,\eta)}\,(D_{1}\otimes
1)\left(\Phi(\xi,\eta)F_{2}(\xi,\eta)\right)\,.$

Putting in (3.1) $D=D_{1}$, $F=F_{1}$ and
$\varphi(u)=\Phi(u,\eta)F_{2}(u,\eta)$, we get

$(F_{1}*F_{2})(\xi,\eta)=\int_{G/H}\,F_{1}(\xi,v)F_{2}(u,\eta)\,{\cal{B}}(\xi,\eta;u,v)\,dx(u,v),$
$None$

where

${\cal{B}}(\xi,\eta;u,v)=c\,\frac{\Phi(\xi,v)\Phi(u,\eta)}{\Phi(\xi,\eta)\Phi(u,v)}\,.$

Let us call this function ${\cal{B}}$ the Berezin kernel. It can be regarded
as a function ${\cal B}(x,y)$ on $G/H\times G/H$. It is invariant with respect
to $G$:

${\cal{B}}({\rm{Ad}}\,g\cdot x,\,{\rm{Ad}}\,g\cdot y)={\cal{B}}(x,y).$

Define a transform $x\mapsto x\breve{\hskip 5.0pt}$ of the space $G/H$, that
in horospherical coordinates $\xi,\eta$ is the permutation
$\xi\leftrightarrow\eta$, i. e. $(\xi,\eta)\mapsto(\eta,\xi)$. This transform
induces the transform $F\mapsto F\,\breve{\hskip 5.0pt}$ of functions in
$S(G/H)$: $F\,\breve{\hskip 5.0pt}(\xi,\eta)=F(\eta,\xi)$. The Berezin kernel
is invariant with respect to the simultaneous permutation
$\xi\leftrightarrow\eta$ and $u\leftrightarrow v$. By (3.3) it implies that
the transform $F\mapsto F\,\breve{\hskip 5.0pt}$ is an anti-involution with
respect to the multiplication of symbols:

$({F_{1}*F_{2}})\breve{\hskip 5.0pt}=F_{2}\breve{\hskip
5.0pt}*F_{1}\breve{\hskip 5.0pt}.$

For operators $D$, the transform $F\mapsto F\,\breve{\hskip 5.0pt}$ of symbols
means the conjugation $D\mapsto D\,\breve{\hskip 5.0pt}$ with respect to the
bilinear form generated by the operator $A_{\lambda}$:

$(A_{\lambda}\varphi,\psi)=\int|N(\xi,\eta)|^{-\lambda-\varkappa}\varphi(\xi)\psi(\eta)\,d\xi\,d\eta.$

Moreover, if $D=\pi^{-}_{\lambda}(X)$, then

$D\,\breve{\hskip 5.0pt}=\pi^{+}_{\lambda}(X^{\vee})\,,$

where $X\mapsto X^{\vee}$ is the transform of the algebra
${\rm{Env}}\,(\mathfrak{g})$ induced by the transform $g\mapsto g^{-1}$ in the
group $G$.

Thus, the space ${\cal{A}}_{\lambda}$ with multiplication $*$ is an
associative algebra with 1, the transform $F\mapsto F\,\breve{\hskip 5.0pt}$
is an anti-involution of this algebra.

Now we define contravariant symbols. A function (a polynomial) $F(\xi,\eta)$
is the contravariant symbols for the following operator $A$ (acting on
functions $\varphi(\xi)$):

$(A\varphi)(\xi)=c\,\int_{G/H}\,F(u,v)\,\frac{\Phi(\xi,v)}{\Phi(u,v)}\,\varphi(u)\,dx(u,v)$
$None$

(notice that (3.4) differs from (3.2) by the first argument of $F$ only). This
operator is a Töplitz type operator.

Thus, we have two maps: $D\mapsto F$ (”co”) and $F\mapsto A$ (”contra”),
connecting polynomials on $G/H$ and operators acting on functions
$\varphi(\xi)$.

If a polynomial $F$ on $G/H$ is the covariant symbol of an operator
$D=\pi^{-}_{\lambda}(X)$, $X\in{\rm{Env}}\,(\mathfrak{g})$, and the
contravariant symbol of an operator $A$ simultaneously, then
$A=\pi^{-}_{-\lambda-\varkappa}(X^{\vee})$. Therefore, $A$ is obtained from
$D$ by the conjugation with respect to the bilinear form

$(F,f)=\int_{\mathfrak{q}^{-}}\,F(\xi)\,f(\xi)\,d\xi.$

In terms of kernels, it means that the kernel $L(\xi,u)$ of the operator $A$
is obtained from the kernel $K(\xi,u)$ of the operator $D$ by the
transposition of arguments and the change $\lambda\mapsto-\lambda-\varkappa$.
So, the composition ${\cal{O}}:D\mapsto A$ (”contra” $\circ$ ”co”) is

${\cal{O}}:\
\pi^{-}_{\lambda}(X)\longmapsto\pi^{-}_{-\lambda-\varkappa}(X^{\vee}).$

This map commutes with the adjoint representation. Such a map was absent in
Berezin’s theory for Hermitian symmetric spaces.

The composition ${\cal{B}}$ (”co” $\circ$ ”contra”) maps the contravariant
symbol of an operator $D$ to its covariant symbol. Let us call ${\cal{B}}$ the
Berezin transform. The kernel of this transform is just the Berezin kernel.

Main problems here are the following. One has to do explicitly: (a) to express
the Berezin transform ${\cal{B}}$ in terms of Laplacians
$\Delta_{1},\ldots,\Delta_{r}$ ($r$ being the rank), in fact, it is the same
that to decompose a canonical representation (see § 4) into irreducible
constituents; (b) to compute eigenvalues of ${\cal{B}}$ on irreducible
constituents; (c) to find a full asymptotics of ${\cal{B}}$ when
$\lambda\to-\infty$ (an analogue of the Planck constant is $h=-1/\lambda$). In
particular, two first terms of asymptotic decomposition the Berezin transform
${\cal{B}}$ when $\lambda\to-\infty$ should be

${\cal{B}}\sim 1-\frac{1}{\lambda}\,\Delta,$

where $\Delta$ is the Laplace–Beltrami operator. It gives the correspondence
principle:

$\displaystyle F_{1}*F_{2}$ $\displaystyle\longrightarrow$ $\displaystyle
F_{1}F_{2},$ $\displaystyle-\lambda\,(F_{1}*F_{2}-F_{2}*F_{1})$
$\displaystyle\longrightarrow$ $\displaystyle\\{F_{1},\,F_{2}\\},$

in right hand sides the pointwise multiplication and the Poisson bracket
stand.

These problems are solved for spaces of rank one and for spaces with the group
$G={\rm SO}_{0}(p,q)$, for latter spaces the rank is equal to 2, see [7], [8].

§ 4. Canonical representations and quantization

The main tool for studying quantization is the so-called canonical
representations (this term was introduced in [9]). For Hermitian symmetric
spaces $G/K$, they were introduced by Vershik, Gelfand and Graev [9] (for the
Lobachevsky plane) and Berezin [1], [2] (in classical case). These
representations act by translations in functions on $G/K$ and are unitary with
respect to some non-local inner product (now called a Berezin form).

We define canonical representations of a group $G$ in a more general setting.
We give up the condition of unitarity (as too narrow) and let these
representations act on sufficiently extensive spaces, in particular, on spaces
of distributions. Moreover, we permit also non-transitive actions of a group
$G$. Our approach uses the notion of an ”overgroup” and consists in the
following.

Let $G$ and $\mathaccent 869{G}$ be semisimple Lie groups such that $G$ is a
spherical subgroup of the $\mathaccent 869{G}$ (i. e. $G$ is the fixed point
subgroup of an involution of $\mathaccent 869{G}$). We call $\mathaccent
869{G}$ an ”overgroup” for $G$. Let $\mathaccent 869{P}$ be a maximal
parabolic subgroup of $\mathaccent 869{G}$, such that $\mathaccent 869{P}\cap
G=H$. Let ${\mathaccent 869{R}}_{\lambda}$, $\lambda\in\mathbb{C}$, be a
series of representations of $\mathaccent 869{G}$ induced by characters of
$\mathaccent 869{P}$. They can depend on some discrete parameters, we do not
write them. As a rule, representations ${\mathaccent 869{R}}_{\lambda}$ are
irreducible. They act on a compact manifold $\Omega$ (a flag manifold for
$\mathaccent 869{G}$). The series $R_{\lambda}$ has an intertwining operator
$Q_{\lambda}$. Restrictions $R_{\lambda}$ of ${\mathaccent 869{R}}_{\lambda}$
to $G$ are called canonical representations of $G$. They act on functions on
$\Omega$. In general, $\Omega$ is not a homogeneous space for $G$, there are
several open $G$-orbits on $\Omega$. They are semisimple symmetric spaces
$G/H_{i}$. The manifold $\Omega$ is the closure of the union of open orbits.
The intertwining operator $Q_{\lambda}$ called the Berezin transform.

One can consider a some different version of canonical representations,
namely, the restriction of canonical representations in the first sense (on
functions on $\Omega$) to some open orbit $G/H_{i}$. Both variants deserve to
be investigated. The first variant is in some sense more natural. But for
quantization we need just the second variant.

Recall, see § 1, classification (a), (b), (c), (d) for symplectic symmetric
spaces $G/H$. As an overgroup $\mathaccent 869{G}$ for $G$ for classes (a),
(b) we take the complexification $G^{\,\mathbb{C}}$ of $G$, and for classes
(c), (d) we take the direct product $G\times G$.

§ 5. Polynomial quantization and the overgroup

Now let $G/H$ be a para-Hermitian symmetric space, see § 2. As an overgroup
for $G$ we take the direct product ${\mathaccent 869{G}}=G\times G$. It
contains $G$ as the diagonal $\\{(g,g),g\in G\\}$.

First we describe a series of representations ${\mathaccent 869{R}}_{\lambda}$
of $\mathaccent 869{G}$.

Let $\mathaccent 869{P}$ be a parabolic subgroup $\mathaccent 869{P}$
consisting of pairs $(zh,hn)$, $z\in Q^{-}$, $h\in H$, $n\in Q^{+}$. Let
${\mathaccent 869{\omega}}_{\lambda}$ be a character of $\mathaccent 869{P}$
equal to $\omega_{\lambda}(h)$ at these pairs. The representation of
$\mathaccent 869{G}$ induced by the character ${\mathaccent
869{\omega}}_{\lambda}$ of the subgroup $\mathaccent 869{P}$ is denoted
${\mathaccent 869{R}}_{\lambda}$.

Let us give some realizations of representations ${\mathaccent
869{R}}_{\lambda}$.

Denote by ${\cal C}$ the manifold of ”double” cosets

$y=s^{-1}_{1}Q^{-}Q^{+}s_{2},\ \ s_{1},s_{2}\in G.$

This manifold is an analogue of a cone for representations of the pseudo-
orthogonal group associated with a cone. The group $\mathaccent 869{G}$ acts
on ${\cal C}$ as follows:

$y\mapsto g^{-1}_{1}yg_{2},\ \ g_{1},g_{2}\in G.$ $None$

Denote by ${\cal D}_{\lambda}(\cal C)$ the space of functions $f$ on ${\cal
C}$ of class $C^{\infty}$ satisfying the following homogeneity condition

$f(s^{-1}_{1}\,h\,Q^{-}Q^{+}s_{2})=\omega_{\lambda}(h)\,f(s^{-1}_{1}\,Q^{-}Q^{+}s_{2}).$
$None$

The representation ${\mathaccent 869{R}}_{\lambda}$ acts on ${\cal
D}_{\lambda}(\cal C)$ by

$\big{(}\,{\mathaccent
869{R}}_{\lambda}(g_{1},\,g_{2})\,f\big{)}(y)=f(g^{-1}_{1}yg_{2}),\ \
g_{1},g_{2}\in G.$

Let us take in ${\cal C}$ two sections: ”hyperbolic” section $\cal X$ and
”parabolic” section $\Gamma$.

The manifold $\cal X\subset{\cal C}$ consists of cosets

$x=s^{-1}Q^{-}Q^{+}s,\ \ s\in G.$

The group $G$ acts on $\cal X$ by $x\mapsto g^{-1}xg$. The stabilizer of the
initial point $x^{0}=Q^{-}Q^{+}$ is $H$, so that $\cal X$ can be identified
with $G/H$.

The manifold $\Gamma\subset{\cal C}$ consists of cosets

$\gamma={\rm{exp}}\,(-\eta)\,Q^{-}Q^{+}{\rm{exp}}\,\xi,\ \
\xi\in{\mathfrak{q}^{-}},\ \eta\in{\mathfrak{q}^{+}}.$ $None$

This manifold can be identified with
${\mathfrak{q}^{-}}\times{\mathfrak{q}^{+}}$.

Let us embed $\Gamma\hookrightarrow\cal X$. It is the embedding
$\mathfrak{q}^{-}\times\mathfrak{q}^{+}\hookrightarrow G/H$, see (1.8), (1.9),
(1.10) and further, in terms of ${\cal C}$.

Let a point $x=s^{-1}Q^{-}Q^{+}s$, $s\in G$, has horospherical coordinates
$\xi,\eta$. By (1.9) we find the element $h(\xi,\eta)$:

${\rm{exp}}\,\xi\cdot{\rm{exp}}\,(-\eta)={\rm{exp}}\,(-Y)\cdot{\rm{exp}}\,X\cdot
h_{0},\ \ h_{0}=h(\xi,\eta),$

where $X\in{\mathfrak{q}^{-}}$, $Y\in{\mathfrak{q}^{+}}$, and by (1.10) we
obtain

$s={\rm{exp}}\,Y\cdot{\rm{exp}}\,\xi={\rm{exp}}\,X\cdot
h_{0}\cdot{\rm{exp}}\,\eta,$ $None$

so that

$\displaystyle x$ $\displaystyle=$ $\displaystyle s^{-1}Q^{-}Q^{+}s$ (5.5)
$\displaystyle=$ $\displaystyle{\rm{exp}}\,(-\eta)\cdot
h^{-1}_{0}Q^{-}Q^{+}{\rm{exp}}\,\xi.$

Thus, the embedding above assigns to a point $\gamma\in\Gamma$, given by
(5.3), the point $x\in\cal X$, given by (5.5) where $h_{0}=h(\xi,\eta)$.

The representation ${\mathaccent 869{R}}_{\lambda}$ can be realized in
functions on these manifolds $\cal X$ and $\Gamma$.

First consider $\cal X$. A point $x=s^{-1}Q^{-}Q^{+}s$ in $\cal X$ under
action (5.1) goes to the point
$g^{-1}_{1}xg_{2}=g^{-1}_{1}s^{-1}Q^{-}Q^{+}sg_{2}$ in $\cal C$. Take the
element $sg_{2}(sg_{1})^{-1}$, i. e. the element $sg_{2}g_{1}^{-1}s^{-1}$, and
decompose it ”by Gauss”:

$sg_{2}g_{1}^{-1}s^{-1}={\rm exp}(-Y^{*})\cdot{\rm exp}\,X^{*}\cdot h^{*},\
X^{*}\in{\mathfrak{q}^{-}},\ Y^{*}\in{\mathfrak{q}^{+}}.$ $None$

Here the element $h^{*}\in H$ depends on the point $x$ only and does not
depend on its representative $s$. Let us form an element $s^{*}\in G$:

$s^{*}={\rm exp}\,Y^{*}\cdot sg_{2}={\rm exp}\,X^{*}\cdot h^{*}sg_{1}.$ $None$

It gives the point $x^{*}\in\cal X$:

$x^{*}=(s^{*})^{-1}Q^{-}Q^{+}s^{*}.$

By (5.7) we have

$x^{*}=g_{1}^{-1}s^{-1}(h^{*})^{-1}Q^{-}Q^{+}sg_{2}.$

Therefore,

$f(x^{*})=\omega_{\lambda}((h^{*})^{-1})\,f(g_{1}^{-1}xg_{2}),$

so that ${\mathaccent 869{R}}_{\lambda}$ acts in functions on ${\cal X}=G/H$
as follows:

$\big{(}{\mathaccent
869{R}}_{\lambda}\,(g_{1},g_{2})\,f\big{)}(x)=\omega_{\lambda}(h^{*})\,f(x^{*}).$
$None$

###### Theorem 5.1

In horospherical coordinates $\xi,\eta$ on $G/H$ the representation
${\mathaccent 869{R}}_{\lambda}$ is

$\big{(}{\mathaccent
869{R}}_{\lambda}\,(g_{1},g_{2})\,f\big{)}(\xi,\eta)=\frac{\Phi_{\lambda}(\xi\bullet
g_{2},\eta\circ g_{1})}{\Phi_{\lambda}(\xi,\eta)}\
\omega_{\lambda}({\mathaccent 869{h}}_{2})\ \omega_{\lambda}({\mathaccent
866{h}}_{1}^{-1})\ f(\xi\bullet g_{2},\,\eta\circ g_{1}),$ $None$

where $\mathaccent 869{h}_{2}$ and $\mathaccent 866{h}_{1}$ are taken from
decompositions $(1.6)$ and $(1.7)$ with $g=g_{2}$ and $g=g_{1}$ respectively.

Proof. Let a point $x=s^{-1}Q^{-}Q^{+}s$, $s\in G$, has horospherical
coordinates $\xi,\eta$. By (5.4) and (1.6), (1.7) we have

$\displaystyle sg_{2}={\rm{exp}}\,Y\cdot{\rm{exp}}\,\xi\,\cdot
g_{2}={\rm{exp}}\,Y\cdot{\rm{exp}}\,Y_{2}\,\cdot{\mathaccent
869{h}}_{2}\cdot{\rm{exp}}\,{\mathaccent 869{\xi}}_{2},$ $\displaystyle
sg_{1}={\rm{exp}}\,X\cdot h_{0}\cdot{\rm{exp}}\,\eta\,\cdot
g_{1}={\rm{exp}}\,X\cdot h_{0}\cdot{\rm exp}\,X_{1}\cdot{\mathaccent
866{h}}_{1}\cdot{\rm{exp}}\,{\mathaccent 866{\eta}}_{1}.$

where ${\mathaccent 869{\xi}}_{2}=\xi\bullet g_{2}$, ${\mathaccent
866{\eta}}_{1}=\eta\circ g_{1}$. Hence

$\displaystyle s^{*}={\rm{exp}}\,Y^{*}\cdot
sg_{2}={\rm{exp}}\,Y_{3}\,\cdot{\mathaccent
869{h}}_{2}\cdot{\rm{exp}}\,{\mathaccent 869{\xi}_{2}},$ (5.10) $\displaystyle
s^{*}={\rm{exp}}\,X^{*}\cdot sg_{1}={\rm{exp}}\,X_{3}\cdot h^{*}\cdot
h_{0}\cdot{\mathaccent 866{h}}_{1}\cdot{\rm{exp}}\,{\mathaccent
866{\eta}}_{1}.$ (5.11)

Therefore, using (5.10) and (5.11), we obtain

$\displaystyle x^{*}$ $\displaystyle=$
$\displaystyle(s^{*})^{-1}Q^{-}Q^{+}s^{*}$ $\displaystyle=$
$\displaystyle{\rm{exp}}\,{\mathaccent
866{\eta}}_{1}\cdot\left(h^{*}h_{0}{\mathaccent 866{h}}_{1}\right)^{-1}\cdot
Q^{-}Q^{+}\cdot{\mathaccent 869{h}}_{2}\cdot{\rm{exp}}\,{\mathaccent
869{\xi}_{2}}$ $\displaystyle=$ $\displaystyle{\rm{exp}}\,{\mathaccent
866{\eta}}_{1}\cdot\left(h^{*}h_{0}{\mathaccent
866{h}}_{1}\right)^{-1}\cdot{\mathaccent 869{h}}_{2}\cdot
Q^{-}Q^{+}\cdot{\rm{exp}}\,{\mathaccent 869{\xi}_{2}}.$

By homogeneity condition (5.2) we have

$f(x^{*})=f\left({\rm{exp}}\,{\mathaccent 866{\eta}}_{1}\cdot
Q^{-}Q^{+}\cdot{\rm{exp}}\,{\mathaccent
869{\xi}_{2}}\right)\cdot\omega_{\lambda}\left((h^{*}h_{0}{\mathaccent
866{h}}_{1})^{-1}{\mathaccent 869{h}}_{2}\right)$ $None$

On the other hand, by (5.5) we can write the point $x^{*}$ in the following
form:

$x^{*}={\rm{exp}}\,{\mathaccent
866{\eta}}_{1}\cdot(h^{*}_{0})^{-1}Q^{-}Q^{+}\cdot{\rm{exp}}\,{\mathaccent
869{\xi}_{2}},$

where $h^{*}_{0}=h\left({\mathaccent 869{\xi}}_{2},{\mathaccent
866{\eta}}_{1}\right)$. Whence again by homogeneity condition (5.2) we obtain

$f(x^{*})=f\left({\rm{exp}}\,{\mathaccent 866{\eta}}_{1}\cdot
Q^{-}Q^{+}\cdot{\rm{exp}}\,{\mathaccent
869{\xi}_{2}}\right)\cdot\omega_{\lambda}\left((h^{*}_{0})^{-1}\right).$
$None$

Comparing (5.12) and (5.13) we get

$\omega_{\lambda}\left({\mathaccent
866{h}}_{1}^{-1}h^{-1}_{0}(h^{*})^{-1}{\mathaccent
869{h}}_{2}\right)=\omega_{\lambda}\left((h^{*}_{0})^{-1}\right),$

whence

$\omega_{\lambda}(h^{*})=\frac{\omega_{\lambda}(h^{*}_{0})}{\omega_{\lambda}(h_{0})}\,\omega_{\lambda}({\mathaccent
866{h}}_{1}^{-1})\,\omega_{\lambda}({\mathaccent 869{h}}_{2}).$

Substitute it to (5.8) and remember (2.2) and (3.1), as result we obtain
(5.9). $\square$

Similarly, the representation ${\mathaccent 869{R}}_{\lambda}$ can be realized
in functions on the manifold $\Gamma$, it is given by:

$\left({\mathaccent
869{R}}_{\lambda}(g_{1},g_{2})\,f\right)\,(\xi,\eta)=\omega_{\lambda}({\mathaccent
869{h}}_{2})\ \omega_{\lambda}({\mathaccent 866{h}}_{1}^{-1})\ f(\xi\bullet
g_{2},\eta\circ g_{1}).$

It shows that ${\mathaccent 869{R}}_{\lambda}$ is equivalent to a tensor
product:

${\mathaccent
869{R}}_{\lambda}(g_{1},g_{2})=\pi^{-}_{\lambda}(g_{2})\otimes\pi^{+}_{\lambda}(g_{1}).$

The group $\mathaccent 869{G}$ contains three subgroups isomorphic to $G$. The
first one is the diagonal consisting of pairs $(g,g)$, $g\in G$. The
restriction of the representation ${\mathaccent 869{R}}_{\lambda}$ to this
subgroup is the representation $U$ by translations on $G/H$:

$\left({\mathaccent 869{R}}_{\lambda}(g,g)\,f\right)(x)=f(g^{-1}xg).$

Indeed, (5.6) and (5.7) with $g_{1}=g_{2}=g$ give $h^{*}=e$ and $s^{*}=sg$.

Two other subgroups $G_{1}$ and $G_{2}$ consist of pairs $(g,e)$ and $(e,g)$,
where $g\in G$, respectively.

By virtue of Theorem 5.1, the restriction of the representation ${\mathaccent
869{R}}_{\lambda}$ to the subgroup $G_{2}$ is given by

$\displaystyle\left({\mathaccent 869{R}}_{\lambda}(e,g)\,f\right)(\xi,\eta)$
$\displaystyle=$ $\displaystyle\frac{\Phi_{\lambda}({\mathaccent
869{\xi}},\eta)}{\Phi_{\lambda}(\xi,\eta)}\ \omega_{\lambda}({\mathaccent
869{h}})\,f({\mathaccent 869{\xi}},\eta)$ $\displaystyle=$
$\displaystyle\frac{1}{\Phi_{\lambda}(\xi,\eta)}\left(\pi^{-}_{\lambda}(g)\otimes
1\right)\Big{[}f(\xi,\eta)\Phi_{\lambda}(\xi,\eta)\Big{]}\,.$

Similarly, the restriction of the representation ${\mathaccent
869{R}}_{\lambda}$ to the subgroup $G_{1}$ is given by

$\big{(}{\mathaccent
869{R}}_{\lambda}(g,e)\,f\big{)}(\xi,\eta)=\frac{1}{\Phi_{\lambda}(\xi,\eta)}\,\big{(}1\otimes\pi^{+}_{\lambda}(g)\big{)}\Big{[}f(\xi,\eta)\Phi_{\lambda}(\xi,\eta)\Big{]}\,.$

Let us go from the group $G$ to the universal enveloping algebra ${\rm
Env}(\mathfrak{g})$ and preserve notations for representations. Let us take as
$f$ the function $f_{0}$ equal to the 1 identically. Then for $X\in{\rm
Env}(\mathfrak{g})$ we have

$\displaystyle({\mathaccent 869{R}}_{\lambda}(0,X)f_{0})(\xi,\eta)$
$\displaystyle=$
$\displaystyle\frac{1}{\Phi_{\lambda}(\xi,\eta)}(\pi^{-}_{\lambda}(X)\otimes
1)\Phi_{\lambda}(\xi,\eta),$ (5.14) $\displaystyle({\mathaccent
869{R}}_{-\lambda-\varkappa}(X,0)f_{0})(\xi,\eta)$ $\displaystyle=$
$\displaystyle\frac{1}{\Phi_{\lambda}(\xi,\eta)}(1\otimes\pi^{+}_{-\lambda-\varkappa}(X))\Phi_{-\lambda-\varkappa}(\xi,\eta).$
(5.15)

Right hand sides of formulae (5.14) and (5.15) are just covariant and
contravariant symbols of operator $D=\pi^{-}_{\lambda}(X)$ in polynomial
quantization, see § 3.

Let us change the position of arguments in ${\mathaccent 869{R}}_{\lambda}$,
then we have a new representation ${\mathaccent 866{R}}_{\lambda}$ of
${\mathaccent 869{G}}$, namely, ${\mathaccent
866{R}}_{\lambda}(g_{1},g_{2})={\mathaccent 869{R}}_{\lambda}(g_{2},g_{1})$.
Using the realization of ${\mathaccent 869{R}}_{\lambda}$ on the section
$\Gamma$, we see that the tensor product $A_{\lambda}\otimes B_{\lambda}$
intertwines the representation ${\mathaccent 869{R}}_{\lambda}$ with the
representation ${\mathaccent 866{R}}_{-\lambda-\varkappa}$. Passing from
$\Gamma$ to $\cal X$ and replacing $\lambda$ by $-\lambda-\varkappa$, we
obtain that the operator $c(\lambda)A_{-\lambda-\varkappa}\otimes
B_{-\lambda-\varkappa}$ intertwines the representation ${\mathaccent
866{R}}_{-\lambda-\varkappa}$ with the representation ${\mathaccent
869{R}}_{\lambda}$ and transfers contravariant symbols to covariant ones. It
has the kernel ${\cal B}_{\lambda}(\xi,\eta;u,v)$, i. e. it is precisely the
Berezin transform.

§ 6. Polynomial quantization on rank one spaces

We consider here the spaces $G/H$, where $G={\rm{SL}}\,(n,\mathbb{R})$,
$H={\rm{GL}}\,(n-1,\mathbb{R})$. They have dimension $2n-2$, rank $r=1$ and
genus $\varkappa=n$. These spaces $G/H$ exhaust all para-Hermitian symmetric
spaces of rank one up to the covering. Further we assume $n\geqslant 3$.

Let ${\rm Mat}\,(n,\mathbb{R})$ denote the space of real $n\times n$ matrices
$x$. The Lie algebra $\mathfrak{g}$ of $G$ consists of $x$ with ${\rm
tr}\,x=0$. By § 1, the space $G/H$ is a $G$-orbit in $\mathfrak{g}$.

But now it is more convenient for us to change a little the realization of
$G/H$.

The group $G$ acts on ${\rm{Mat}}\,(n,\mathbb{R})$ by $x\mapsto g^{-1}xg$. Let
us write matrices $x$ in the block form according to the partition
$n=(n-1)+1$:

$x=\left(\begin{array}[]{cc}\alpha&\beta\\\ \gamma&\delta\end{array}\right)$

where $\alpha\in{\rm{Mat}}\,(n-1,\mathbb{R})$, $\beta$ is a vector-column in
${\mathbb{R}}^{n-1}$, $\gamma$ is a vector-row in ${\mathbb{R}}^{n-1}$ and
$\delta$ is a number.

Let $x^{0}$ be the following matrix:

$x^{0}=\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right)$

The $G$-orbit of $x^{0}$ is just $G/H$. This manifold is the set of matrices
$x$ whose trace and rank are equal to 1. The stabilizer $H$ of $x^{0}$
consists of matrices ${\rm diag}\\{a,b\\}$, where
$a\in{\rm{GL}}\,(n-1,\mathbb{R})$, $b=({\det}\ a)^{-1}$, so that
$H={\rm{GL}}\,(n-1,\mathbb{R})$.

Subalgebras ${\mathfrak{q}}^{-}$ and ${\mathfrak{q}}^{+}$ consist respectively
of matrices

$X=\left(\begin{array}[]{cc}0&0\\\ \xi&0\end{array}\right),\ \ \
Y=\left(\begin{array}[]{cc}0&\eta\\\ 0&0\end{array}\right),$

where $\xi$ is a row $(\xi_{1},\ldots,\xi_{n-1})$, and $\eta$ is a column
$(\eta_{1},\ldots,\eta_{n-1})$ in ${\mathbb{R}}^{n-1}$. Embedding (1.5) is

$x=\frac{1}{N(\xi,\eta)}\,\left(\begin{array}[]{cc}-\eta\xi&-\eta\\\
\xi&1\end{array}\right),$

where $N(\xi,\eta)=1-\xi\eta=1-(\xi_{1}\eta_{1}+\ldots+\xi_{n-1}\eta_{n-1})$.

A $G$-invariant metric $ds^{2}$ on $G/H$ up to a factor is
${\rm{tr}}\,(dx^{2})$. It generates the measure $dx$, the Laplace–Beltrami
operator $\Delta$ and the Poisson bracket $\\{f,h\\}$. In coordinates
$\xi,\eta$ we have:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-2N(\xi,\eta)^{-2}\Bigl{\\{}\sum\xi_{i}\,d\eta_{i}\sum\eta_{i}d\xi_{i}+N(\xi,\eta)\sum
d\xi_{i}\,d\eta_{i}\Bigr{\\}}.$ $\displaystyle dx$ $\displaystyle=$
$\displaystyle\left|N(\xi,\eta)\right|^{-n}\,d\xi\,d\eta\ \ \
(d\xi=d\xi_{1}...d\xi_{n-1}),$ $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle
N(\xi,\eta)\,\sum\,(\delta_{ij}-\xi_{i}\eta_{j})\,\frac{\partial^{2}}{\partial\xi_{i}\,\partial\eta_{j}}\,,$
$\displaystyle\\{f,h\\}$ $\displaystyle=$ $\displaystyle
N(\xi,\eta)\,\sum\,(\delta_{ij}-\xi_{i}\eta_{j})\left(\frac{\partial
f}{\partial\eta_{i}}\,\frac{\partial h}{\partial\xi_{j}}-\frac{\partial
f}{\partial\xi_{i}}\,\frac{\partial h}{\partial\eta_{j}}\right).$

The Berezin kernel is

${\mathcal{B}}(x,y)=c(\lambda)\,\frac{\Phi(\xi,v)\Phi(u,\eta)}{\Phi(\xi,\eta)\Phi(u,v)}\,=\,c(\lambda)\,|{\rm
tr}(xy)|^{\lambda},$

where

$c(\lambda)=\left\\{2^{n+1}\pi^{n-2}\Gamma(-\lambda-n+1)\Gamma(\lambda+1)\left[\cos\left(\lambda+\frac{n}{2}\right)\pi-\cos\frac{n\pi}{2}\right]\right\\}^{-1}.$

The Berezin transform is written in terms of the Laplace–Beltrami operator
$\Delta$ as follows

${\mathcal{B}}=\frac{\Gamma(-\lambda+\sigma)\,\Gamma(-\lambda-\sigma-n+1)}{\Gamma(-\lambda)\,\Gamma(-\lambda-n+1)}\,,$
$None$

the right-hand side should be regarded as a function of
$\Delta=\sigma(\sigma+n-1)$.

Now let $\lambda\to-\infty$. Then (6.1) gives

${\mathcal{B}}\sim 1-\frac{1}{\lambda}\,\Delta.$

Hence we have

$F_{1}*F_{2}\sim F_{1}F_{2}-\frac{1}{\lambda}\,N^{2}\,\frac{\partial
F_{1}}{\partial\xi}\,\frac{\partial F_{2}}{\partial\eta},$

so that for $\lambda\to-\infty$ we have

$F_{1}*F_{2}\longrightarrow F_{1}F_{2},$ $None$
$-\lambda\,(F_{1}*F_{2}-F_{2}*F_{1})\longrightarrow\\{F_{1},\,F_{2}\\},$
$None$

in the right hand sides of (6.2) and (6.3) the pointwise multiplication and
the Poisson bracket stand, respectively. Relations (6.2) and (6.3) show that
for the family of algebras of covariant symbols the correspondence principle
is true. As the Planck constant, one has to take $h=-1/\lambda$.

Moreover, we can write not only two terms of the asymptotics but also a full
asymptotic decomposition (a deformation decomposition) of ${\mathcal{B}}$
explicitly. In order to have a transparent formula, one has to expand not in
powers of $h=-1/\lambda$ but use ”generalized powers” of $-\lambda-n$. Then
decomposition turns out to be a series terminating on any irreducible subspace
of polynomials on $G/H$.

Namely, we have the following decomposition of the Berezin transform:

${\mathcal{B}}=\sum_{k=0}^{\infty}\frac{1}{k!}\cdot\frac{\Delta\,\left[\Delta-1\cdot
n\right]\,\left[\Delta-2\cdot(n+1)\right]\,\ldots\,\left[\Delta-(k-1)(k-2+n)\right]}{(-\lambda-n)^{(k)}}\,,$

where

$a^{(m)}=a(a-1)\ldots(a-m+1).$

§ 7. Group quantization

The group $G={\rm SL}(2,\mathbb{R})$ consists of real matrices

$g=\left(\begin{array}[]{cc}\alpha&\beta\\\ \gamma&\delta\\\
\end{array}\right),\ \ \ \alpha\delta-\beta\gamma=1.$

We denote:

$t^{\lambda,\varepsilon}=|t|^{\lambda}({\rm sgn}t)^{\varepsilon},\ \
\lambda\in\mathbb{C},\ \ \varepsilon\in\\{0,1\\}.$

In this section we use a little wider set of parameters: instead of $\lambda$
we take the pair $(\sigma,\varepsilon)$ where $\sigma\in\mathbb{C}$,
$\varepsilon=0,1$, Let us denote by
${\cal{D}}_{\sigma,\varepsilon}(\mathbb{R})$ the space of functions $f\in
C^{\infty}(\mathbb{R})$ such that the function

$\mathaccent 866{f}(t)=t^{2\sigma,\varepsilon}f\left(1/t\right)$

belongs to $C^{\infty}(\mathbb{R})$ too. The representation
$T_{\sigma,\varepsilon}$ of the group $G$ acts on
${\cal{D}}_{\sigma,\varepsilon}(\mathbb{R})$ by:

$\left(T_{\sigma,\varepsilon}(g)f\right)(t)=f\left(\frac{\alpha
t+\gamma}{\beta t+\delta}\right)(\beta t+\delta)^{2\sigma,\varepsilon}.$ (7.1)

Let us denote by $\mathaccent 866{T}_{\sigma,\varepsilon}$ the contragredient
representation $g\mapsto T_{\sigma,\varepsilon}(\mathaccent 866{g})$, where

$\mathaccent 866{g}=\left(\begin{array}[]{cc}\delta&\gamma\\\ \beta&\alpha\\\
\end{array}\right),$

so that

$\left(\mathaccent
866{T}_{\sigma,\varepsilon}(g)f\right)(t)=f\left(\frac{\delta t+\beta}{\gamma
t+\alpha}\right)(\gamma t+\alpha)^{2\sigma,\varepsilon}.$ (7.2)

Representations $T_{\sigma,\varepsilon}$ and $\mathaccent
866{T}_{\sigma,\varepsilon}$ are equivalent by means of the operator
$f\mapsto\mathaccent 866{f}$.

An operator $A_{\sigma,\varepsilon}$, defined by formula:

$(A_{\sigma,\varepsilon}f)(t)=\int_{-\infty}^{\infty}N^{-2\sigma-2,\,\varepsilon}f(s)\,ds,\
\ \ N=N(t,s)=1-\xi\eta,$

intertwines $T_{\sigma,\varepsilon}$ and ${\mathaccent
866{T}}_{-\sigma-1,\varepsilon}$:

${\mathaccent
866{T}}_{-\sigma-1,\varepsilon}(g)A_{\sigma,\varepsilon}=A_{\sigma,\varepsilon}T_{\sigma,\varepsilon}(g)\,,$

and also ${\mathaccent 866{T}}_{\sigma,\varepsilon}$ and
$T_{-\sigma-1,\varepsilon}$.

Consider in $\mathbb{R}^{3}$ the bilinear form

$[x,y]=-x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.$

Let ${\cal{X}}$ denotes the hyperboloid of one sheet $[x,x]=1$. Realise it as
the set of matrices

$x=\frac{1}{2}\left(\begin{array}[]{cc}1-x_{3}&x_{2}-x_{1}\\\
x_{2}+x_{1}&1+x_{3}\\\ \end{array}\right)$

with $\det x=0$. The group $G$ acts on them by conjugations $x\mapsto
g^{-1}xg$ transitively.

Let us introduce on ${\cal{X}}$ horospherical coordinates $\xi,\eta$:

$x=\frac{1}{N}\,(\xi+\eta,\,\xi-\eta,\,1+\xi\eta),\ \ $

so that:

$x=\frac{1}{N}\,\left(\begin{array}[]{cc}-\eta\xi&-\eta\\\
\xi&1\end{array}\right).$ (7.3)

Let us define for variables $\xi,\eta$ two vectors – vector-line and vector-
column:

$u=u(\xi)=\left(\begin{array}[]{cc}\xi&1\end{array}\right),\ \ \
v=v(\eta)=\left(\begin{array}[]{c}-\eta\\\ 1\end{array}\right)$

Then

$x=\frac{vu}{uv},\ \ \ N=uv.$ (7.4)

As an initial class of operators we take operators
$T_{\sigma,\varepsilon}(g)$, $g\in G$. As a supercomplete system we take the
kernel of the intertwining operator $A_{-\sigma-1,\,\varepsilon}$, namely, the
function

$\Phi(\xi,\eta)=\Phi_{\sigma,\varepsilon}(\xi,\eta)=N(\xi,\eta)^{2\sigma,\,\varepsilon}$
(7.5)

of two variables $\xi,\eta$.

For an operator $T_{\sigma,\varepsilon}(g)$, its covariant symbol and
contravariant symbol are the following functions respectively

$\displaystyle F_{g}(\xi,\eta)$
$\displaystyle=\frac{1}{\Phi_{\sigma,\varepsilon}(\xi,\eta)}\,(T_{\sigma,\varepsilon}(g)\otimes
1)\,\Phi_{\sigma,\varepsilon}(\xi,\eta),$ (7.6) $\displaystyle
F^{\natural}_{g}(\xi,\eta)$
$\displaystyle=\frac{1}{\Phi_{-\sigma-1,\varepsilon}(\xi,\eta)}\,(1\otimes
T_{-\sigma-1,\varepsilon}(\mathaccent
866{g}\,))\,\Phi_{-\sigma-1,\varepsilon}(\xi,\eta),$ (7.7)

Consider $\xi,\eta$ as horospherical coordinates on ${\cal{X}}$. Them
functions (7.6), (7.7) become functions on ${\cal{X}}$.

###### Theorem 7.1

The covariant and contravariant symbols of the operator
$T_{\sigma,\varepsilon}(g)$, $g\in~{}G$, are the following functions on the
hyperboloid $\cal{X}$ respectively:

$\displaystyle F_{g}(x)$ $\displaystyle=({\rm{tr}}(xg))^{2\sigma,\varepsilon}$
(7.8) $\displaystyle=\left(\frac{ugv}{uv}\right)^{2\sigma,\varepsilon},$ (7.9)
$\displaystyle F^{\natural}_{g}(x)$
$\displaystyle={\rm{tr}}(g^{-1}x)^{-2\sigma-2,\varepsilon}$ (7.10)
$\displaystyle=\left(\frac{ug^{-1}v}{uv}\right)^{-2\sigma-2,\varepsilon}.$
(7.11)

Proof. First we prove (7.8), (7.9). By (7.6), (7.5), (7.1) we obtain:

$\displaystyle F_{g}(\xi,\eta)$
$\displaystyle=\frac{1}{N^{2\sigma,\varepsilon}}\,\left(1-\frac{\alpha\xi+\gamma}{\beta\xi+\delta}\,\eta\right)^{2\sigma,\varepsilon}\left(\beta\xi+\delta\right)^{2\sigma,\varepsilon}$
$\displaystyle=\left(\frac{\beta\xi+\delta-\alpha\xi\eta-\gamma\eta}{N}\right)^{2\sigma,\varepsilon}$
(7.12)

It is easy to check, the nominator in (7.12) is precisely $u(\xi)gv(\eta)$, it
together with (7.4) proves (7.9). Formula (7.8) follows from (7.12) and (7.3).

Now let us prove (7.10), (7.11). By (7.7), (7.5), (7.2) we obtain:

$\displaystyle F^{\natural}_{g}(\xi,\eta)$
$\displaystyle=\frac{1}{N^{-2\sigma-2,\varepsilon}}\,\left(1-\xi\,\frac{\delta\eta+\beta}{\gamma\eta+\alpha}\right)^{-2\sigma-2,\varepsilon}\left(\gamma\eta+\alpha\right)^{-2\sigma-2,\varepsilon}$
$\displaystyle=\left(\frac{\gamma\eta+\alpha-\delta\xi\eta-\beta\xi}{N}\right)^{-2\sigma-2,\varepsilon}$
(7.13)

Since

$g^{-1}=\left(\begin{array}[]{cc}\delta&-\beta\\\ -\gamma&\alpha\\\
\end{array}\right),$

then the nominator (7.12) is precisely $u(\xi)g^{-1}v(\eta)$, it together with
(7.4) proves (7.11). Formula (7.10) follows from (7.13) and (7.3). $\square$

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