Representations of the Infinite-Dimensional Affine Group

TL;DR

This paper introduces an infinite-dimensional affine group, constructs its irreducible unitary representation, and adapts existing methods to accommodate the group's unique properties, expanding the understanding of infinite-dimensional group representations.

Contribution

It develops a new framework for representing an infinite-dimensional affine group, extending techniques used for diffeomorphism groups with necessary modifications.

Findings

Constructed irreducible unitary representation of the affine group

Demonstrated the group's action on certain classes of functions

Extended existing representation methods to new infinite-dimensional context

Abstract

We introduce an infinite-dimensional affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.