Plancherel formula for Berezin deformation of \(L^2\) on Riemannian symmetric space

Abstract

Consider the space B of complex \(p\times q\) matrces with norm <1. There exists a standard one-parameter family \(S_a\) of unitary representations of the pseudounitary group U(p,q) in the space of holomorphic functions on B (i.e. scalar highest weight representations). Consider the restriction \(T_a\) of \(S_a\) to the pseudoorthogonal group O(p,q). The representation of O(p,q) in \(L^2\) on the symmetric space \(O(p,q)/O(p)\times O(q)\) is a limit of the representations \(T_a\) in some precise sence. Spectrum of a representation \(T_a\) is comlicated and it depends on \(\alpha\). We obtain the complete Plancherel formula for the representations \(T_a\) for all admissible values of the parameter \(\alpha\). We also extend this result to all classical noncompact and compact Riemannian symmetric spaces.