Construction of Quasi Invariant Probability Measures on Some Current Groups of Continuous Sections of a Bundle of Compact Semisimple Lie Groups

a) In the seventies, several papers of A.M. Vershik, I.M. Gelfand and M.I. Graev ([19], [20], [21]) initiated the research of non located continuous irreducible unitary representations of current groups of the type C k (M, G) where M is a smooth Riemannian manifold, and G a finite dimensional Lie group. But, while the knowledge of the unitary dual of a finite dimensional Lie group is motivated by problems coming from the harmonic analysis and the fact that such a group has a Haar measure, because of the absence of an invariant measure, the motivations for the investigation of the unitary dual of the current groups C k (M,G) were connected with the so-called theory of non commutative multiplicative distributions on a the manifold M; we refer to [11] for an introduction to this theory.