Invariant measures for unitary groups associated to Kac-Moody Lie algebras

Abstract

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).

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

“…[20]), and for more results on Gaussian-like measures on infinite-dimensional curved spaces see papers by Pickrell (e.g. [22,23]). For another view of different representations of Fock space, one can look at results in the field of white noise, as presented in the book by Obata [21].…”

Loops in SU(2), Riemann Surfaces, and Factorization, I

“…where the linear coordinates are determined by θ + = g −1 + (∂g + ), θ − = (∂g − )g −1 − . The (left or right) translates of this neighborhood by elements of L fin K cover H 0 (S 1 , G); a key point is that the transition functions are functions of a finite number of variables, in an appropriate sense (see [13,Chapter 2,Part III]).…”

Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian

“…Haar measures in infinite dimension were studied by Pickrell [2] and Asada [3]. We have defined the Haar distribution on a path group by using the Hida-Streit approach of path integrals as distribution [4][5][6][7].…”