Polynomial quantization on para-Hermitian symmetric spaces Текст научной статьи по специальности «Математика»

POLYNOMIAL QUANTIZATION ON PARA-HERMITIAN SYMMETRIC SPACES 1

V. F. Molchanov, N. B. Volotova G. R. Derzhavin Tambov State University, Russia

In this paper we give the construction of a variant of quantization (we call it the polynomial quantization) in the spirit of Berezin (see [1], [2]) on para-Hermitian symmetric spaces of the first category (we use the terminology from [6]). These spaces belong to a very spacious class of semisimple symmetric spaces. Moreover, they are symplectic manifolds.

There are 4 classes of symplectic semisimple symmetric spaces G/H:

(a) Hermitian symmetric spaces;

(b) semi-Kahlerian symmetric spaces;

(c) para-Hermitian symmetric spaces of the first category;

(d) para-Hermitian symmetric spaces of the second category;

These names are taken from [4], [3], [6], [6] respectively. Spaces of class (a) are Riemannian, spaces of other three classes are pseudo-Riemannian (non-Riemannian). For spaces of classes

(b), (c), (d), their Riemannian forms belong to class (a).

If G is a simple Lie group, then these 4 classes give a classification of these spaces (from the local point of view).

We consider spaces of class (c). These spaces have two linearly independent polarizations which means that the tangent space at any point splits into the direct sum of two Lagrangian subspaces invariant with respect to the stabilizer of this point.

The classification of them (from the local point of view) can be obtained by means of the list of Berger [3]: one has to take there spaces G/H with the reducible action of H, see also [6] and Section 2 below.

We can consider that G/H is an G-orbit in the Lie algebra g of G under the adjoint representation. For the supercomplete system we take the kernel $^(£,77) of an operator intertwining representations 7of maximal degenerate series. Here /j € C, e = 0, 1, vectors £ and 77 run Lagrangian subspaces q- and q+ at the initial point x° = He (e the identity element of G), Pairs (£, 77) give local coordinates on G/H. Representations 7r“e and 7r+e act on some spaces of functions tp(£) and 'ip(rj) respectively. Functions tp(£) form an analog of the Fock space.

For the initial algebra of operators, we take the algebra of operators D = 7r“e(X), where X belongs to the universal enveloping algebra Env(g) of the Lie algebra g.

Covariant and contravariant symbols are defined similarly to Berezin theory for Hermitian symmetric spaces. These symbols turn out to be polynomials on G/H, i.e. restrictions to G/H of polynomials on g. It is why we call our version the polynomial quantization.

In Section 1 we recall the notion of Berezin quantization [1], [2].

Sections 2 and 3 contain some material concerning para-Hermitian symmetric spaces, we lean on [8].

In Section 4 we construct the polynomial quantization on para-Hermitian symmetric spaces G/H of arbitrary rank. We give general constructions and definitions, formulate some statements, sometimes give outlines of their proofs, put open problems. Detailed proofs for rank one

Supported by the Russian Foundation for Basic Research (grant No. 05-01-00074a), the Scientific Programs ’’Universities of Russia” (grant No. ur.04.01.465) and ”DeveL Sci. Potent, High, School”, (Templan, No. 1.2.02).

spaces are given in [11] (see also [10] for dimG/H = 2).

Let us introduce some notation and agreements.

By N we denote {0,1,2,...}.

We use the following notation for a character of the group R* = M \ {0}:

tx’£ = |i|A(sgni)e,

where i gR*, A £ C, £ = 0, 1.

For a manifold M, let V(M) denote the Schwartz space of compactly supported infinitely differentiable C-valued functions on M, with a usual topology, and T>'(M) the space of distributions on M - of linear continuous functionals on V(M).

For a Lie group G, we denote by Ge the connected component of the identity element.

If ip is an automorphism of a Lie group G, then Glp denotes the set of fixed elements of (p. For a Lie algebra g, we denote by Env(g) the universal enveloping algebra of g.

For a differentiable representation of the Lie group G, we retain the same symbol for the corresponding representations of the Lie algebra g of G and of the universal enveloping algebra Env(g) .

Let a bilinear form on V(M):

(F,f)= [ F(x)f(x)dx, (0.1)

Jm

where dx is a measure on M, is invariant with respect to a pair (T, S) of representations of a group G acting on V(M):

(:T(g)FJ) = (F,S(g~1)f). (0.2)

Then we can extend the representation T to V'(M) by means of (0.2), where (F, /) denotes the value of a distribution F £ V'(M) at a test function / € V(M). It is really an extension, since V(M) is embedded into V'(M) by attaching to / G T>(M) the functional h (f,h) by means of (0.1). We retain the symbol (in this case T) for this extension.

Similarly, if an operator A is symmetric with respect to the form (0.1):

(AF,f) = {F,Af), (0.3)

then we can extend it to V(M) by means of (0.3).

1 Berezin quantization

Let us recall the concept of quantization proposed by Berezin, see [1], [2]. We do not give it in its full generality, but restrict ourselves to a simplified version.

Let M be a symplectic manifold. Then C°°(M) is a Lie algebra with respect to the Poisson bracket {A,B}, A,B e C°°(M).

Quantization in the sense of Berezin consists of the following two steps.

rTVl O -fi ref ■ r\n Vi U O PAnofvni’t Ct "Fq TTI iltr O'f QCCAPI of UrO D 1 rTDKrQO A ( nr\r\^ Q 1 1 n ( A/f}

J. 11\_| U1UU UllVy HUlL* W V^UIIUUJ. U< i.Ullllllj’ UJ. UlUUV VylUlUi V V_, V ^ / V J V/VliUWllilVU ill J

and depending on a parameter h > 0 (called the Planck constant), a multiplication in A{h) is

denoted by * (it also depends on h). These algebras must satisfy the following conditions (a) -(d):

(a) lim Ai * A2 = AjA2; (1.1)

h-> 0

f h) lim — ( At ^ An — Ao * At) = {At . An\: i 1 9)

v_/ L-*’-*-”

the multiplication in the right hand side of (1.1) is the usual pointwise multiplication, in the right hand side of (1.2) the Poisson bracket stands, conditions (a) and (b) together are called the correspondence principle;

(c) the function Ao = 1 (equal to 1 identically) is the unit element of each algebra A(h);

(d) the complex conjugation A A is an anti-involution of each algebra A(h).

The second step: one has to construct representations A h-> A of algebras A{h) by operators on a Hilbert space.

Berezin mainly considered the case when M is a Hermitian symmetric space G/K. Therefore, it has a complex structure. Let it be realized as a bounded domain in Cm. Then functions A are functions A(z,z), z E M, analytic in z and z separately. The complex conjugation comes to the permutation of arguments: A(z,z) = A(z,z).

Let B(z,z) be the corresponding Bergman kernel. The starting-point of the Berezin construction is the so-called supercoplete system (a system of coherent states):

$*(*) = $(z,w) = %(z,w) =

where n < /in (/in is some number), x is the genus of the corresponding Jordan algebra. Let T,, be the Fock space on M. It is a Hilbert space of analutic functions on M square-integrable with respect to the measure

c(/z) B(z, z)II^>idu(z),

where c{n) is a normalized factor (depending on ¡j, analytically), du(z) is an invariant measure on M. As a function of z, the function $>w{z) belongs to and has a reproducing property:

(f,$w) = /M,

the inner product is taken in

Let A be a bounded operator in Let us assign to it a function of two variables:

Mz^) =

Its restriction to the diagonal, i.e. the function A(z,z), is a function on M, it is called the covariant symbol of the operator A. The function A(z,w) is reconstructed from A(z,z) by analyticity. The operator A is comletely defined by its covariant symbol:

(Af)(z) = c J^A(z,w)^^f(w)di/(w),

where c = c(/i). The multiplication of operators gives raise to a multiplication of symbols:

{A\ -k A2){z, z) = c / Ai(z,w)A2{w,z)B(z,z',w,w)dv(w), (1-3)

Jm

where

n/ - —x ${z,w)${w,z)

B(z,zm,w,w) = c

$(z, 2r)$(l0, w)

The kernel B is called the Berezin kernel, an operator with this kernel is called the Berezin transform, it acts on functions on M. Berezin found ([2], see also [12]) an expression on the transform B in terms of Laplace operators on M (generators in the algebra of invariant differential operators on M) and determined the asymptotics of B when ¡j, —> —oo:

B~ 1--A, (1.4)

where A is the Laplace-Beltrami operator on M. It solves the problem of the construction of quantization on M: for the Planck constant one has to take h = —1/n, algebras A(h) consist of covariant symbols of bounded operators on the Fock space with the multiplication * given by (1.3), the correspondence principle follows from (1.4).

0

Besides it, Berezin defines contravariant symbols of operators: a function A (z, z) on M is called a contravariant symbol of an operator A defined by formula:

{Af)(z)=c [ A (w,w) ^Z,W]rf(w)du(w).

~ o

It turns out that, for an operator A, the passage At-t A is given by the Berezin transform.

In this paper we consider a variant of quantization after Berezin - the so-called polynomial quantization - for para-Hermitian symmetric spaces G/H. We have to change slightly conditions mentioned above. For example, the factor i has to be omitted, instead of complex conjugation one has to take some permutation of arguments, finally, we abandon the Hilbert structure in representation spaces.

2 Para-Hermitian symmetric spaces

In this Section we expound some material from [8]. Let G be a connected semisimple Lie group. Let a be a nontrivial involutive automorphism (involution) of G. Let H be a subgroup of G lying between (Ga)e and Ga:

(Ga)e cHcGa

(i.e. H is an open subgroup of Ga). The homogeneous space G/H is called a semisimple symmetric space

As a rule, we shall consider that groups act on their homogeneous spaces from the right, so that G/H is the space of right cosets Hg.

There exists a Cartan involution r of the group G commuting with a. Let us denote 6 = or = to. Set K = GT.

Automorphisms of the Lie algebra g, induced by involutions a and r, will be denoted by the same letters a and t.

The Lie algebra g decomposes into the direct sum g = fj + q of eigensubspaces of a with eigenalues +1 and —1 respectively. A similar decomposition g = t + p takes place for the involution t. Since a and r commute, thereis a joint decomposition

g = enf) + 6nq-|-pnf) + pnq.

The subspace q can be identified with the tangent space to G/H at the x° = He (e is the identity element of G). It is invariant with respect to the group H and its Lie algebra f) in the adjoint representation in g.

A Cartan subspace in q is a maximal Abelian subalgebra in q, consisting of semisimple elements. All these subspaces have the same dimension. This dimension is called the rank of the symmetric space G/H. Assume that the pair (g, fj) is effective, i.e. 1) contains no nontrivial ideal of the algebra g.

Now assume, in addition, that G/H is a symplectic manifold. Then F) has a non-trivial centre Z(\j). For simplicity we assume that G/H is a G-orbit of an element Zq 6 g in the adjoint representation of the group G, i.e. G/H = AdG ■ Zq. In particular, Zq 6 Z(f)).

We can assume that G is simple. Symplectic semisimple symmetric spaces G/H with G simple are divided into four classes:

(a) Hermitian symmetric spaces;

(b) semi-Kahlerian symmetric spaces;

(c) para-Hermitian symmetric spaces of the first category;

(d) para-Hermitian symmetric spaces of the second category;

These names are taken from [4], [3], [6], [6] respectively. Spaces of class (a) are Riemannian, spaces of other three classes are pseudo-Riemannian (non-Riemannian). For spaces of classes

(b), (c), (d), their Riemannian forms belong to class (a). For spaces of class (a), the Lie algebra f) coinsides with the Lie algebra t. For spaces of classes (a), (b), (c), the centre Z(t)) is onedimensional, so that Z((]) — MZq . For spaces of of classes (b), (c), the element Zq lies in t fl f) and p fl f) respectively. Spaces of class (d) are complexifications of spaces of class (a).

We study spaces of class (c). In this case, the element Zq can be normalized in such a way that the operator I = (adZo)q in q is an involution. A symplectic structure on G/H is defined by the bilinear form

u(Z,Y) = Ba(X,IY)

on q, where BB is the Killing form of the Lie algebra g.

Let q* be eigensubspaces of I in q with eigenvalues ±1, respectively. Both these spaces are Abelian subalgebras in g, they are invariant with respect to H and irreducible. Moreover, they are Lagrangian.

Thus, the Lie algebra 0 is a graded algebra:

0 = q + (} + q+ = 0-i + 0o + 0+i-

Let dimension of the space q (it is the same as dimension of the space G/H) is equal to 2m. Then a bilinear form on q, the restriction of the Killing form Bg, has signature (m, m).

The involution 0 is a linear isomorphism of q+ onto q- and inversely, the same is true for the involution t. Let us identify the space q~ with Rm in some way, for example, we take a basis in q~ so that Bg(Ei,9Ej) = cSij, where c is a factor, and identify this basis with the standard basis in MTO). Then the space q+ is identified with Rm by means the isomorphism 6 : q+ -> q .

The pair (q+,q_) is a Jordan pair [7] with the multiplication {X,Y,Z} = (1/2)[[X, Y], Z], Let r and x be rank and genus of this Jordan pair. The Jordan pair (q+,q_), the space G/H and the space K/K fl H have the same rank r (so that, in particular, G/H has the discrete series).

Set Q± = exp q±. The subgroups P± = IiQ± = Q±H are maximal parabolic subgroups of G, with H as their Levi subgroup. We have the following decompositions:

G = Q+HQ-, (2.1)

G = Q-HQ+, (2.2)

G = Q+HK, (2.3)

G = Q-HK, (2.4)

where the bar means closure and the sets under the bar are open and dense in G. Let us call (2.1) the Gauss decomposition and (2.3) the Iwasawa-type decomposition. Allowing some slang, let us call (2.2) the anti-Gauss decomposition and (2.4) the anti-Iwasawa-type decomposition. For an element in G all three factors in (2.1), (2.2) and the first factors in (2.3), (2.4) are defined

uniquely, whereas the second and the third factors in (2.3), (2.4) are defined up to an element

of K n H.

For g € G we define transformations £ i-> £ ■ g = £ of the space q_ and 77 1—>- 77 of the space q+q+ taking £ i-> £ and 77 1—>- 77 from the Gauss and the anti-Gauss decompositions:

exp£-g = expy • h ■ exp£, (2.5)

exp r) ■ g = exp X • h • exp rf, (2.6)

where X G q_, Y G q+, the dot in (2.5), (2.6) means the multiplication in the group G. These actions are defined on open and dense sets, depending on g.

Therefore, G acts on q~ x q+: (£, r/) (£,rf). Since we have identified q~ and q+ with ]Rm,

we have two actions of the group G on Wn. Applying the isomorphism 0 to (2.6), we obtain that the action of the group G on q+ ~ Mm is 77 i-> rf = 77 • g, where g = 9(g). the stabilizer of the point (0,0) 6 q' x q+ is P+ fl = H, so that we obtain an embedding

q~ x q+ ^ G/H, (2.7)

whose image is an open and dense set. Therefore, we can consider £,77 as coordinates on G/H, let us call them horospherical coordinates.

The connection between the Gauss and the anti-Gauss decompositions gives us an operator K(£, 77) and a function 6(£, 77), both very important, see below. Namely, decompose the anti-Gauss product exp£ ■ exp (—??) according to the Gauss decomposition:

exp £ • exp (—77) = exp Y • h ■ exp X,

V n+ Dpi

K(i, 7?) on q+:

where X £ q , Y G q^. Denote the obtained element h G H by h(£,Tj). Define an operator

-1

This operator is an analogue of the Bergman transform for Hermitian symmetric spaces. In terms of Jordan pairs it becomes:

K(Z, rj)T = T — 2{r/£T} + {77{£TC}r/}.

Under the action of G the operator K(^,rj) is transformed as follows:

(Adh-1) K(Z,ti) (Ad^)

where h and h are taken from (2.5) and (2.6) respectively.

The function det K(£,rj) is a polynomial in £,77. Moreover (see [7]), it is N(£,r))*, where N(£, rf) is an irreducible polynomial in £ and 77 of degree r in £ and 77 separately. In view of (2.7), the function

bfor,) = [dettf(^)]-1 = (2.8)

can be considered as a function on G/H. an analog of the Bergman kernel. It is invariant with respect to H.

Let us write G'-invariant metric ds2, symplectic form oj and measure dx on G/H. Take a basis Ei, ...,Em in q_ as above. Then0£i,..., 9Em form a basis of q+. Let and 7^ are coordinates in q_ and q+ in these bases respectively. Denote by k^(^,rj) the entries of the operator if(£,r/)_1 Then

ds2 = 2 (^, 77) (2.9)

w = 'Y^k13 (i,ri) d^i Adrij, (2.10)

dx = \b{i,r])\dii...d^mdr}i...dr]m.

Similarity of formulas (2.9) and (2.10) reflects that G/H has the structure of a manifold over the of algebra of ’’dual numbers” z — x + iy, x,y £ M., i2 = 1.

The coset spaces 5+ = G/P~, S~ = G/P+, S — K/K fl H are compact manifolds, diffeo-

morphic to each other. Namely, the diffeomorphism of S~ and S is defined by the following

correspondence:

s~k <—> svk, k G K,

where = Р+е, s° = (К П Н)е are initial points, е is the identity element of groups. The

natural actions of G on S~ and S+ yields two actions of G on S. The action on S~ gives the

action s h* s = s • g on S, namely, if s = s°k, k Є K, then S' = s°k, where k is obtained from

the Iwasawa decomposition of the product kg:

kg = exp Yx • h\ ■ k. (2-11)

The restriction of this action to K gives the natural action of the group K by right translations on S. The action on gives the action s >-* S' = s ■ g, where, recall, g = 9(g).

Let us consider the following action of G on S x S:

(s,t) (s ■ g, t-g). (2.12)

The stabilizer of the point (s°,s°) under the action (2.12) is H, so that we obtain a natural equivariant embedding G/H <-»• S x S. The image of this embedding is a single open dense orbit. Denote it by Q. Thus, S x S is a compactification of G/H. For the G-orbit structure of S x S, see [5]. Notice that G/H can be represented as the tangent bundle of the manifold S. The spaces q~ and q+ can be embedded in S:

£ Н-» S° • exp£, Tj Н-» s° ■ exp 9(rj),

with open dense images. Therefore, both £ = (£i, ...,£m) and 77 = (771, ...,77m), see above, can be regerded as coordinates on S. In these coordinates a iv-invariant measure on S is written as

ds = y/b(Z,9{Q)d£ (2.13)

= Vb(6{^)^)dr], (2.14)

where = d£i...d£m, dr] = dr)\...dr)m.

We now define an important function ||s,i|| on S x S. For s,t Є S, let us take ks, kt Є K so that s°ks = s, s°kt = t. Apply to ksk^1 the Gauss decomposition:

kskt_1 = exp Y • h ■ expX. (2-15)

It turns out that (Ad/i)q+ depends on s and t only, but not on the choice of ks and kt- We set

||s,i|| = I(det Ad/i)q+p1/,x , (2.16)

where h is taken from (2.15). Formula (2.16) defines the function \\s,t\\ on an open dense subset of S x S. This function is continuous, symmetric (i.e. ||s, i|| = j|i, s||) and invariant with respect to the diagonal action of K:

\\sk,tk\\ = ||M||.

It can be extended on the whole S x S, keeping all these properties. The orbit Q is characterized by condition ||s, i|| Ф 0.

The following table contains the list of simple symmetric Lie algebras g/f), g/(j that correspond to para-Hermitian symmetric spaces G/H with G simple, see [6]. For aesthetic reasons we denote Lie algebras by capital Latin letters instead of small Gothic ones.

s f)

SL (n,R) SL (p, R) + SL(g, R) + R

SU*(2n) SU*(2p) + SU*(2g) + R

SU (n, n) SL (n, C) + R

SO* (4 n) SU*(2n) + R

SO (n, n) SL (n, R) + R

SO (p, q) SO(p- l,g- 1) + R

Sp(n, R) SL (n, R) + R

Sp (n, n) SU*(2n) + R

Еб(б) SO (5,5) + R

E6(-26) SO (1,9) + M

^7(7) ^6(6) + ®

E7(_25) E6(-26) + ^

З Maximal degenerate series representations

In this Section we consider series of representations of the group G induced by characters of maximal parabolic subgroups P^, see Section 2. We mainly lean on [8].

For ¡1 Є C, let be a character of the subgroup H\

ui^h) = |det (Ad/i)q+| ^.

Extend this character to subgroups P± setting it equal to 1 on Q+ and Q~. For simplicity we restrict ourselves to such characters of P±.

Consider induced representations of the group G:

ftp = IndpT

The representation тг^ acts by right translations:

(^(sO/) (9i) = fiaig)

on the space T>^(G) consisting of functions / Є C°°(G), satisfying the condition:

f{pg) = wT^(p)/(g), p Є P*.

For a function / on G, denote f(g) = f(g), where g = в(д). The map / / gives a linear

isomorphism of the spaces T>^(G) and T>~(G) and intertwines representations тг^ and ir'^ о в:

(7ГЇ(^)/Г= rf(9)f-

First consider the realization of the representations 7^7 in the compact picture. They act on the space T>(S) as follows:

(s) =b>nih1)ip{s),

rf{9)

where s — s ■ k, elements k and h\ are determined by (2.11).

Let us consider the following bilinear form on V(S):

[i/j,<p)s = I" tl>(s)ip(s)ds, (3.1)

s

the measure ds is given by (2.13) or (2.14). This form is invariant with respect to the pairs (tt+, nt^) and (n~, i.e.

W, v)s = (V’, s , (3.2)

where either upper, or lower signs ± are taken.

Define an operator A^ on T>(S):

{An<p)(s) = J ||s, s0^|rM_^(i)o?i,

s

where kt is an element of K, such that .s°A;t = t. The integral converges absolutely for Re/i < 1 — >t and can be extended to the //-plane as a meromorphic function. This operator intertwines tt^ with

A^(g) = *-»-x(g)An,

where either upper, or lower signs ± are taken. The operator is symmetric with respect to the form (3.1):

[A^, <p)s = (tf>, Aptfs. (3.3)

Further, we have

A—fj—xA^ = ujq(h)E, (3-4)

where uo(fj) is a meromorphic function.

Let us extend the representations and the operator A^ to the space T>'(S) of distributios on S by means of formulas (3.2) and (3.3).

Now we consider the non-compact picture. We restrict functions in V^(G) to subgroups Q^,

identify these subgroups (as manifolds) with , which intheir tien are identified with Wn. We

obtain

fc(g)f) (0 = w,, (£)/(£),

where £ and h are given by formula (2.5) (so that £ = £ ■ g), and also

(^(sO/) (v) = ^(g)f(v),

where rf=r]-g, g = 9{g).

The form (3.1) becomes the form

(F,f) = jF(t)mdZ (3.5)

(the integral is taken over q~ = Rm), the operator AM is

(A»f)(v) = J (3.6)

The representation ir~ of the Lie algebra g is given by some differential operators. Originally it is defined on functions of £, which are restrictions of functions in V~ (G) to exp q~. But we can consider these representations acting on other spaces functions of £: for example, on functions of class C00 on q~ = Mm, on the space Pol(q_) of polynomials in £, on the space T>'(q~) of

distributions on q~, in particular, on the space U'0(q~) of distributions of £, concentrated at

zero, etc. The same is true for 7r+.

For representations of the Lie algebra g, defined on G,00(q:t), together with the intertwining operator Afj,, given by (3.6), we can write another intertwining operator, multiplying the integrand in (3.6) by sgn N(£,r)). Therefore, we have two intertwining operators A^, e = 0,1:

(A... f\(n\ = f

\- j / \ -1 / j -■ \'sr// JV'i»/ 5

see (0.11). The operator A^ is A^q. As A^, the operator A^ intertwines with

Similarly to (3.4), for this operator we have

A — fl — X, = ^o(/^J (3-7)

where wo(M)£) is a meromorphic function of /j. The operator AM)£ can be also considered on distributions q± (= Rm). Apparently, the operator A/h£ can appear as an intertwining operator for a series of representations of a group G, covering (one or two times) the group Ad G. We lay

aside the clearing up this question. For rank one it is true, see [11].

4 Polynomial quantization on para-Hermitian symmetric spaces

In thus Section we apply to the space G/H (a para-Hermitian symmetric space of arbitrary rank) the scheme of quantization offered in [8] and give basic constructions.

US6 another _ in comparison with [8] — initial algebra of operators, namely, for such an algebra, we take the algebra of operators 7r~(X), X G Env(g), with the parameter fi, acting on functions <£>(£), £ G q~ = Mm. In contrast to [8], we use the noncompact picture for representations 7rJ.

Since our algebras of functions on G/H are algebras of polynomials on GjH, we call our version of quantization the polynomial quantization. Polynomials on G/H are understood to be the restrictions of polynomials on 0 to G/H (recall that G/H is a manifold in g, namely, it is

AJ/i r7 „„„ o\

AUU ‘ J, SCC OCL-11U11

For a supercomplete system we take the kernel of the operator namely,

mv) = $,A^v) = N(t,rlr’£,

where £,77 are vectors in Mm, the function iV(£,7?) is given by formula (2.8), fj, G C, e = 0,1.

Let us extend the operators X G Env(g), to functions /(£,??) in C°°(q~ x q+) as

follows:

*p(x)t = ^{X) ® 1, ■K^{X)r] = 1 <g>7r+(X). (4.1)

Let denote the space of functions /(£,??), that are obtained by applying of all operators

7r~(X)^, X G Env(g), to the function $^(£,77). Functions in W^e are linear combinations of functions N(£,rj)IJ,~r’£~r, r G N, whose coefficients are polynomials in /i and £,77. The space W^e is invariant with respect to operators (4.1).

Let us divide all functions in by the function $, i.e. consider functions

F(i,^> = 4(1^)<4 2>

where X G Env (g). These functions do not depend on e, indeed,

N>*-r>e-r/<S> = N-r. (4.3)

Consider £,r] as horospherical coordinates on G/H. Then functions (4.2) become functions on G/H. Denote the space of these functions by Ap. Let us call the function F in A^, defined by formula (4.2), the covariant symbol of the operator D — 1r~(X), X G Env (0).

Covariant symbols are polynomials on the space G/H. This statement is proved for rank one case [11], for higher rank the proof has to be found.

In particular, the covariant symbol of the identity operator is 1, i.e. the function on G/H equal to 1 identically. For operators ir~(X), corresponding to elements X of the Lie algebra

g, the covariant symbol is up to a factor depending on ¡x, a linear function Bg(X,x), where x G G/H C 0.

The operator D is recovered by its covariant symbol F as follows:

(Dip)(0 = c j F{£,v) ip(u)dx{u,v), (4.4)

where (for wo, see(3.7))

c = c(n, e) = w0(/i, e)-1. (4.5)

Indeed, the function $ has a reproducing property:

^ =C / ^^)’P{u)dx{u,v)' which is nothing but formula (3.7) rewritten in another form. Applying to both sides of this equality the operator D and using (4.2), we obtain (4.4).

Let be the tensor product of the representations 7r~ and 7of the Lie algebra 0 on

R^(X)='K-(X)i + ^(X)n.

For n, satisfying some integer-valued condition (in particular, then /i = k G N), the representation R^e is the tensor product of finite dimensional subrepresentations of representations n~

and 7rj".

№

Let U denote the representation of group G by translations on a space of functions on G/H (quasiregular representation), for example, on the space Cco{G/H). By the same letter U we denote the corresponding representation of the Lie algebra g. We have

U(L) = ^(L)z + n+(L)Jl.

The map / t-> //$ of the space onto AIL intertwines RtL.e and U.

The correspondence ChF which assigns to an operator its covariant symbol is g-equiva-riant, i.e. if F is the covariant symbol of the operator D = ir~(X), X G Env(g), then U(L)F, where L G 0, is the covariant symbol of the operator 7r“(adL • X)^.

For fj, generic, the space A^ is the space S(G/H) of all polynomials on G/H.

The multiplication of operators gives raise to a multiplication of covariant symbols, the latter will be denoted by ★. Namely, let D = D\D2 and let F, F\, F2 be covariant symbols of operators D,D\,D2, respectively. Then F = F\ * F2. From (4.2) we have

F1*F2 = ^(Dl)z($F2). (4.6)

Now setting in (4.4) D = D\ and <p(u) = &(u,rj) F2(u,r]), we get

(Fi* F2){£,r)) = J Fi(^,v)F2(u,r])B(^,T]-,u,v)dx(u,v), (4.7)

where

\ $(£,u)$(u,r?)

B{£,r);u,v) = c———-----------(4.8)

Let us call this kernel B the Berezin kernel. It can be regarded as a function of two variables

on G/H\ B = B{x\y), x,y G G/H. It is invariant with respect to the diagonal action of G:

B{Adg ■ x, Adg • y) = B(x,y).

Consider the transformation x x of the space G/H, which in horospherical coordinates £, 77 is the permutation of £ and rj: (??,£). This transformation generates a transformation

F i->- F of functions on S(G/H): F(£, rj) = F(rj, £). The Berezin kernel is invariant with respect to the simultaneous permutation £ -H- 77 and u <->■ v. By (4.7), it gives that the transformation F h-» F is an anti-involution with respect to the multiplication of covariant symbols:

(Fi * F2y = F2 * Fi-

For operators D the transformation of symbols F i-» F gives the conjugation D D with respect to a bilinear form generated by the operator A^:

= J N(Ç,rj)-ti-*’eip(Ç)ip{ri)dÇdri.

Let us define a linear map X h-» X'j on Env(g) as follows. For monomials X = X\X2 ... Xrn, where Xi G g, we set

Xv = (-Xm)...(-X2)(-X1) = (-l)mXm...X2X1

and extend to the whole Env(g) by linearity. This map is an involution. It commutes with the adjoint representation:

(adL ■ X)y = adL • Xv.

We have: if D = tt~(X), then

D = n+(Xv).

Thus, the space A^ of covariant symbols is an associative algebra (with the unit) with respect to the multiplication ★. The transformation F F is an anti-involution of this algebra.

Now we define contravariant symbols. According to the general scheme, see, for example, [9], a function F(£,r]) is the contravariant symbol for the following operator A (acting on functions

<p(0Y-

(Atp){£) = c [ F(u,v) ®^V^-(p(u)dx(u,v) (4.9)

J $(u,v)

(it differs from (4.4) only by the first argument of the function F).

If a polynomial F in S(G/H) is simultaneously the covariant symbol of an operator D = ir~(X), X G Env(g), and the contravariant symbol of an operator A, then A = nZu^>c(Xv). Consequently, A is obtained from D by conjugation with respect to the form (3.5). In terms of kernels it means that the kernel L(£,u) of the operator A is obtained from the kernel K{£,u) of the operator D by the permutation of arguments and replacing /i by —¡j, — x.

The correspondence F (->■ A that to any F in A-n-x assigns an operator A with the contravariant symbol F, is g-equivariant, namely, U(L)F, where L G g, is the contravariant symbol of the operator A].

Thus, we have two maps: D h* F (”co”) and (’’contra”) connecting operators acting

on functions of £ and polynomials on G/H.

The composition O = (contra) o (co) mapping an operator to an operator: D i->- A was already regarded above, we have seen that O is the map

*;(x) tt:^—„(xv).

It commutes with the adjoint representation ad.

o

The composition B = (co) o (contra) maps the contravarint symbol F of an operator D to its covarint symbol F. Let us call B the Berezin transform.

Let us formulate open problems for arbitary rank: find an expression of the Berezin transform B in terms of Laplace operators (generators in the algebra of invariant differential operators on G/H), find eigenvalues of the Berezin transform B on irreducible constituents, find its full

'loirmnf nfi/i nvnfincmn TirVirtn ii _^

vv nciii r uv.

Apparently, the first two terms of this expansion are the following:

B~l--A, (4.10)

where A is the Laplace-Beltrami operator, cf. (1.4). The expansion (4.10) implies the correspondence principle (for the ’’Planck constant”, one has to take h = —1/fj,):

Fi*F2 —► fxf2,

-fi(Fi * F2 - F2* Fi) —> {Fi, F2},

when ¡1 —> —oo.

REFERENCES

1. F. A. Berezin. Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR. Ser. matem., 1975, tom 39, No. 2, 363-402. Engl, transl.: Math. USSR, Izv., 1975, tom 9, 341-379.

2. F. A. Berezin. A connection between the co- and the contravariant symbols of operators on classical complex symmetric spaces. Dokl. Akad. Nauk SSSR, 1978, tom 241, No. 1, 15-17. Engl, transl.: Sov. Math. Dokl., 1978, tom 19, 786-789.

3. M. Berger. Les espaces symetrique non compacts. Ann. Sci. Ecole Norm. Sup. 1957. 74. C. 85-177.

4. S. Helgason. Differential Geometry and Symmetric Spaces. Acad.Press, New York Lon-

10£/1 \JLKJ 11, .LC/VJ'*.

5. S. Kaneyuki. On orbit structure of compactifications of parahermitian symmetric spaces. Japan. J. Math., 1987, vol. 13, No. 2, 333-370.

6. S. Kaneyuki, M. Kozai. Paracomplex structures and affine symmetric spaces. Tokyo J. Math., 1985, vol. 8, 81-98.

7. O. Loos. Jordan Pairs. Lect. Notes Math., 1975, 460.

o ir r? A/T-i-T____ r\_________j.:_________tj_____:j.:____________________ a ___

o. v. r. ivioiciicuiuv. un parariieriiiiuian symiiictiii; »pcujes, illlitJI'. IVlcllIl. ouc.

Transl, Ser. 2, vol. 175 (Adv. in the Math. Sci.-31), 1996, 81-95.

9. V. F. Molchanov. Quantization on symplectic symmetric spaces. In: ”Proc. Tambov Summer School-Seminar, Aug. 26-31, 1996”, Vestnik Tambov Univ., 1997, tom 2, issue 4, 372-385.

10. V. F. Molchanov, N. B. Volotova. Finite dimensional analysis and polynomial quantization on a hyperboloid of one sheet. In: ”Proc. Tambov Summer School-Seminar, Aug. 26-31, 1996”, Vestnik Tambov Univ., 1998, tom 3, issue 1, 65-78.

11. V. F. Molchanov, N. B. Volotova. Polynomial quantization on rank one para-Hermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, Nos. 1-3, 215-232.

12. A. Unterberger, H. Upmeier. Berezin transform and invariant differential operators. Comm. Math. Phys., 1994, vol. 164, 563-598.