Let G denote the infinite
dimensional group consisting of all unitary operators which are compact
perturbations of the identity (on a fixed separable Hilbert 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
L2 limit of the corresponding (L2 normalized) finite dimensional spherical
functions.
