Decomposition of regular representations for U(H)_∞

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.