Decomposition of regular representations for U(H)

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Abstract

Let G denote the infinite dimensional group consisting of all unitary operators which are compact perturbations of the identity (on a fixed separable Hubert space). Kirillov showed that G has a discrete spectrum (as a compact group does). The point of this paper is to show that there are analogues of the Peter-Weyl theorem and Frobenius reciprocity for G. For the left regular representation, the only reasonable candidate for Haar measure is a Gaussian measure. The corresponding L2 decomposition is analogous to that for a compact group. If X is a flag homogeneous space for G, then there is a unique invariant probability measure on (a completion of) X. Frobenius reciprocity holds, for our surrogate Haar measure fibers over X precisely as in finite dimensions (this is the key observation of the paper). When X is a symmetric space, each irreducible summand contains a unique invariant direction, and this direction is the Lr limit of the corresponding (L2 normalized) finite dimensional spherical functions.

Original languageEnglish (US)
Pages (from-to)319-332
Number of pages14
JournalPacific Journal of Mathematics
Volume128
Issue number2
DOIs
StatePublished - Jun 1987

ASJC Scopus subject areas

  • Mathematics(all)

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