On invariant measures for the group of diffeomorphisms of the circle

TL;DR

This paper explores the construction of invariant measures for the group of circle diffeomorphisms, aiming to advance geometric understanding of their representations.

Contribution

It extends previous work on biinvariant measures to the diffeomorphism group of the circle, providing new geometric insights.

Findings

Constructed biinvariant measures for circle diffeomorphisms.

Connected measure construction to positive energy representations.

Extended geometric methods from loop groups to diffeomorphism groups.

Abstract

In a previous paper the author constructed biinvariant measures (possibly having values in a line bundle) for a loop group LK (with compact simply connected K) acting on the formal completion of its complexification LG. One motivation for this was to find a geometric construction for the unitary structure for the positive energy representations of LK. In this paper we pursue an analogous construction for the group of diffeomorphisms of the circle.