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 ρto the subgroup O(p,q). This restriction has a complicated spectrum consisting of representations having different types. We construct a decomposition of ρto a finite direct sum of representations τ_j such that each summand τ_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.

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