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

By Y. Neretin. 2002

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Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L2 on noncompact symmetric space O(p, q)/O(p)×O(q) and L2 on compact symmetric space O(p+q)/O(p)×O(q).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/90f09f43887a0855fd5bc2ca5d17dce14ac02c85

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Harmonic analysis on the infinite symmetric group (2003, background): “ined are close to those 6Notice a certain similarity between our family {Tz} and Neretin’s deformation of the natural representation on L2(G/K), where G/K is a Riemannian symmetric space, see Neretin [Ner3]. 5 arising in random matrix theory. The link with random matrix theory seems to be especially”
重複度１の表現と複素多様体上の可視的な作用Multiplicity-free representations and visible actions on complex manifolds (2005, background): “There has been recent progress in this direction, including Ben Säıd ([3]), Ørsted-Zhang [52], van Dijk-Hille [9] and Neretin [49].”
Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs (2006, methodology): “on so far, for G = SL(2,R) by V. F. Molchanov [64]; for G = SU(2,2) by B. Ørsted and G. Zhang [70]; for G = SU(n,1) by G. van Dijk and S. Hille [10]; for G = SU(p,q) by Y. Neretin and G. Ol’shanski˘ı [66, 67]. See also G. van Dijk–M. Pevzner [11], M. Pevzner [72] and G. Zhang [92]. Their results sho”