Stochastic processes on geometric loop groups, diffeomorphism groups of connected manifolds, and associated unitary representations

Abstract

This paper is devoted to the investigation of semidirect products of loop groups and homeomorphism or diffeomorphism groups of finite-and infinite-dimensional real, complex, and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. It is shown that these groups exist and have the structure of infinite-dimensional Lie groups, i.e., they are continuous or differentiable manifolds and the composition (f, g) ↦ f −1 g is continuous or differentiable depending on the smoothness class of groups. Moreover, it is proved that in the cases of complex and quaternion manifolds, these groups have the structures of complex and quaternion manifolds, respectively. Nevertheless, it is proved that these groups do not necessarily satisfy the Campbell-Hausdorff formula even locally outside of the exceptional case of a group of holomorphic diffeomorphisms of a compact complex manifold. Unitary representations of these groups G′, including irreducible ones, are constructed by using quasi-invariant measures on groups G relative to dense subgroups G′. It is proved that this procedure provides a family of cardinality card(ℝ) of pairwise nonequivalent, irreducible, unitary representations. The differentiabilty of such representations is studied.