Poisson measures for topological groups and their representations

Abstract

Gaussian quasi-invariant measures on groups of diffeomorphisms and loop groups G relative to dense subgroups G' were constructed. In the non-Archimedean case the wider class of measures was investigated, than in the real case. The cases of Riemann and non-Archimedean manifolds were considered. This article is related with unitary representations of G' associated with Poisson measures on $G^{\bf N}$ and uses quasi-invariant measures on G from the previous works. Several groups are considered: (1) (a) diffeomorphisms and (b) loop groups of real manifolds, (2) (a) diffeomorphisms and (b) loop groups of non-Archimedean manifolds over local fields. Besides these four cases further the fifth and the sixth cases are considered: for (3) (a) real and (b) non-Archimedean groups of diffeomorphisms Diff(M) representations associated with Poisson measures on configuration spaces $Γ_M$ contained in products of manifolds $M^{\bf N}$ are investigated. Here the cases of infinite-dimensional Banach manifold M (3) (a), non-Archimdean locally compact and non-locally compact Banach manifolds (3) (b) are investigated.