Some remarks on quasi-invariant actions of loop groups and the group of diffeomorphisms of the circle

Abstract

We construct the series of quasi-invariant actions of the group Diff of diffeomorphisms of the circle and loop groups on the functional spaces provided by non-Wiener Gauss measures. We construct some measures which can be considered as analogues of Haar measure for loop groups and the group Diff. These constructions allow us to construct series of representations of these groups including all known types of representations (higest weight representations, energy representations, almost invariant structures, etc.)