Plancherel Formula for Berezin Deformation of L2 on Riemannian Symmetric Space

Abstract

Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L2 on noncompact symmetric space O(p, q)/O(p)×O(q) and L2 on compact symmetric space O(p+q)/O(p)×O(q).