Representation theory for infinite classical motion groups is formulated in terms of invariant measure classes and cocycle cohomology. It is shown that invariant measure classes are always represented by invariant probability measures, and these classes are determined for Cartan motion groups. The existence of "induced" cocycle cohomology is established in this ergodic setting. Also it is shown that the continuity properties of representations are rather rigidly determined.

In previous work we proved that for a SU (2, C) valued loop having the critical degree of smoothness (one half of a derivative in the L 2 Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a triangular factorization, and (3) the loop has a root subgroup factorization. For a loop g satisfying these conditions, the Toeplitz determinant det(A(g)A(g −1 )) and shifted Toeplitz determinant det(A 1 (g)A 1 (g −1 )) factor as products in root subgroup coordinates. In this paper we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in SL(2, C).The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set, and (2) the non-compactness of SL(2, C) entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.

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 I? 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 (L 2 normalized) finite dimensional spherical functions.

We investigate Birkhoff (or triangular) factorization and (what we propose to call) root subgroup factorization for elements of a noncompact simple Lie group G 0 of inner type. For compact groups root subgroup factorization is related to Bott-Samelson desingularization, and many striking applications have been discovered by Lu ([3]). In this paper, in the inner noncompact case, we obtain parallel characterizations of the Birkhoff components of G 0 and an analogous construction of root subgroup coordinates for the Birkhoff components. As in the compact case, we show that the restriction of Haar measure to the top Birkhoff component is a product measure in root subgroup coordinates.