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V. F. Molchanov G. R. Derzhavin Tambov State University Tambov, Russia molchanov@tsu.tmb.ru

Polynomial quantization on para-Hermitian symmetric spaces
The author was supported by the Russian Foundation for Basic Research: grants No. 05-01-00074a, No. 05-01-00001a and 07-01-91209 YaF a, the Netherlands Organization for Scientific Research (NWO): grant 047-017-015, the Scientific Program "Devel. Sci. Potent. High. School": pro ject RNP.2.1.1.351 and Templan No. 1.5.07. We construct a variant of quantization (symbol calculus) in the spirit of Berezin on paraHermitian symmetric spaces. A general scheme of quantization was presented in [1]. There are 4 classes of symplectic semisimple symmetric spaces G/H : (a) Hermitian symmetric spaces; (b) semi-KЁ ahlerian symmetric spaces; (c) para-Hermitian symmetric spaces; (d) complexifications of Hermitian symmetric spaces. Spaces of class (a) are Riemannian, spaces of other three classes are pseudo-Riemannian (non-Riemannian). Let us assume that G is a simple Lie group. Then these 4 classes give a classification. Berezin constructed quantization for spaces of class (a). We consider spaces of class (c). We can assume that G/H is an adjoint G-orbit. The Lie algebra g of G splits into the direct orthogonal (in sense of the Killing form) sum: g = h + q where h is the Lie algebra of H . The space q splits into the direct sum of Lagrangian subspaces q- and q+ of the tangent space to G at the initial point H . The subspaces q± are invariant and irreducible with respect to H , they are Abelian subalgebras of g. The pair (q+ , q- ) is a Jordan pair. Let r and be rank and genus of it, r being also rank of G/H . Set Q± = exp q± . The subgroups P ± = H Q± = Q± H are maximal parabolic subgroups of G. We have the following decompositions (the Gauss and "anti-Gauss" decompositions): G = Q+ H Q- , G = Q- H Q+ , where the bar means closure. The group G acts on q- and q+ : , , where and are taken from the Gauss and the anti-Gauss decompositions: exp · g = exp Y · h · exp , exp · g = exp X · h · exp , (1)

Therefore, G acts on q- в q, the stabilizer of the point (0, 0) is H , so that we obtain an embedding q- в q+ G/H with an open and dense image. Let us call , horospherical coordinates on G/H . For q- and q+ , let us decompose the anti-Gauss product exp · exp (- ) according to the Gauss decomposition and denote by h( , ) the corresponding element in H . For h H , let us denote b(h) = det(Adh)|q+ . The function k ( , ) = b(h( , )) is an analogue of the Bergman kernel for Hermitian symmetric spaces. It is N ( , )- , where N ( , ) is an irreducible polynomial in and of degree r in and separately. ± Representations , C, of G of a maximal degenerate series associated with G/H ± are defined as induced representations = Ind(G, P , ), where (h) = |b(h)|-/ and ± = 1 on Q . In noncompact picture, these representations act on functions ( ) and ( ) on q- and q+ respectively by (see (1)):
- + ( (g ))( ) = (h)( ), ( (g ) )( ) = (h) ( ). ± An operator A-- with the kernel ( , ) = |N ( , )| intertwines -- with . -1 The product A A-- is c() · id, where c() is a meromorphic function of . - For the initial algebra of operators, we take the algebra of operators D = (X ), where X belongs to the universal enveloping algebra Env(g) of g. This algebra acts on functions - + ( ) and ( ) by representations and respectively. Spaces of these functions form analogues of the Fock space. For the supercomplete system we take the kernel ( , ). Let - us call the covariant symbol of the operator D = (X ), X Env (g), the following function F on G/H which in horosherical coordinates is given by

F ( , ) = ( , )

-1

- ( (X ) 1) ( , ).

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These functions are polynomials on G/H . It is why we call our version quantization the polynomial quantization. For generic, the space of covariant symbols is the space of all polynomials on G/H . The operator D is recovered by its covariant symbol F as follows: (D)( ) = c() F ( , v ) ( , v ) (u) dx(u, v ), (u, v ) (2)

where dx( , ) is a G-invariant measure on G/H . The correspondence D F is g-equivariant. The multiplication of operators gives raise to a multiplication of covariant symbols. It is given by the Berezin kernel B : (F1 F2 )( , ) = where B ( , ; u, v ) = c() F1 ( , v )F2 (u, )B ( , ; u, v ) dx(u, v ), ( , v ) (u, ) . ( , ) (u, v )

A function (a polynomial) F ( , ) is the contravariant symbol for an operator A such that (A)( ) is given by the right hand side of (2) with F ( , v ) replaced by F (u, v ). Thus, we have two maps: O = (contra) (co) and the Berezin transform B = (co) (contra). The map O (it was absent in Berezin's theory) assigns to an operator D with the covariant symbol F the operator A for which F is the contravariant symbol. The kernel of A is obtained from the kernel of D by the permutation of arguments and replacing by - - . The map B assigns to the contravarint symbol F of an operator D the covariant symbol F of the same D. It is given just by the Berezin kernel. Let us formulate open problems for arbitrary rank r: find an expression of the Berezin transform B in terms of Laplace operators, find eigenvalues of B on irreducible constituents, find its full asymptotic expansion of B when -. These problems are solved for r = 1, see [2], and for spaces with G = SO0 (p, q ) (then r = 2). There is another approach to the polynomial quantization using representation theory. It gives co- and contravariant symbols and the Berezin transform in a natural and transparent way. These symbols are obtained under the restriction of a representation R of the overgroup - + G = G в G to the component subgroups G в e and e в G (R (g1 , g2 ) = (g2 ) (g1 )). References [1] V. F. Molchanov, Quantization on para-Hermitian symmetric spaces. Amer. Math. Soc. Transl., Ser. 2, 1996, vol. 175 (Adv. in Math. Sci.­31), 81­95. [2] V. F. Molchanov, N. B. Volotova. Polynomial quantization on rank one para-Hermitian symmetric spaces. Acta Appl. Math., 2004, vol. 81, Nos. 1­3, 215­222.

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