Poisson measures for topological groups and their representations

TL;DR

This paper constructs and analyzes Poisson and Gaussian quasi-invariant measures on various groups of diffeomorphisms and loop groups over real and non-Archimedean manifolds, exploring their unitary representations.

Contribution

It extends the construction of quasi-invariant measures to non-Archimedean cases and investigates associated unitary representations for diverse groups and manifolds.

Findings

Construction of Gaussian measures on groups of diffeomorphisms and loop groups.

Investigation of Poisson measures on configuration spaces of manifolds.

Analysis of unitary representations related to these measures.

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 $\Gamma_M$ 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.