TY - JOUR

T1 - Decomposition of regular representations for U(H)∞

AU - Pickrell, Doug

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1987/6

Y1 - 1987/6

N2 - 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.

AB - 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.

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U2 - 10.2140/pjm.1987.128.319

DO - 10.2140/pjm.1987.128.319

M3 - Article

AN - SCOPUS:84972562790

VL - 128

SP - 319

EP - 332

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -