Symbols in Berezin quantization for representation operators

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 ♮ defined on M . These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H 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 Env g of the Lie algebra g of G . In this case symbols turn out to be polynomials on the Lie algebra g . In this paper 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. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G= SL (2;R) , H - the subgroup of diagonal matrices, G/H - a hyperboloid of one sheet in R3 ; b) G - the pseudoorthogonal group SO0 (p ; q ) , the subgroup H covers with finite multiplicity the group SO0 (p -1, q -1)× SO0 (1;1) ; the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G .

Lie groups and Lie algebras, pseudo-orthogonal groups, representations of Lie groups, para-Hermitian symmetric spaces, Berezin quantization, covariant and contravariant symbols