arXiv math/99110201999-11-03math.RTmath-phmath.CAmath.CVmath.MP

Plancherel formula for Berezin deformation of $L^2$ on Riemannian symmetric space

Yu. A. Neretin

发表信息：: Journal of Functional Analysis, 189(2002), p. 336-408
DOI：: 10.1006/jfan.2000.3691

英文摘要：

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 $α$. We obtain the complete Plancherel formula for the representations $T_a$ for all admissible values of the parameter $α$. We also extend this result to all classical noncompact and compact Riemannian symmetric spaces.

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