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.