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Abstract. We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group.

Abstract. This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the EvensLu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU (2) case. Applications include integral formulas and factorizations for Toeplitz determinants.

The purpose of this paper is to describe some conjectures and results on the existence and uniqueness of invariant measures on formal completions of Kac-Moody groups and associated homogeneous spaces. The basic invariant measure is a natural generalization of Haar measure for the unitary form of a finite type Kac-Moody group, and its projection to flag spaces is a generalization of the normalized invariant volume element. The other "invariant measures" are actually measures having values in line bundles over these spaces; these bundle-valued measures heuristically arise from coupling the basic invariant measure to hermitian structures on associated line bundles, but in the examples that we consider, they are in fact singular with respect to the basic invariant measure. The main examples considered are infinite classical groups (i.e. affine Kac-Moody groups of infinite rank), central extensions of loop groups, and the Virasoro group (which is not technically a Kac-Moody group).

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