Separation of spectra in analysis of Berezin kernels

Abstract

Consider an unitary highest weight representation of a group U(p,q) in holomorphic functions on the symmetric space U(p,q)/U(p)\times U(q). Consider its restriction \rho to the subgroup O(p,q). This restriction has a complicated spectrum consisting of representations having different types. We construct a decomposition of \rho to a finite direct sum of representations \tau_j such that each summand \tau_j has spectrum consisting of one-type representations. Our tool is theorems about restrictions of holomorphic functions on Cartan domain U(p,q)/U(p)\times U(q) to submanifolds of the boundary. We also obtain Plancherel formula for this restriction.