Matrix analogs of B-functions and Plancherel formula for Berezin kernel representations

Abstract

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of representations). The Berezin representations are deformations of L^2 on Riemann noncompact symmetric spaces G/K defined by Berezin kernels, i.e powers of |det(1-zu^*)| (in matrix ball models). These representations also can be obtained by restrictions of holomorphic representations of some group Q containing G as symmetric subgroup. We also discuss models of noncompact Riemann symmetric spaces: matrix balls, matrix cones, matrix wedges, and sections of wedges.