Unitary representations of a loop ax+b group, Wiener measure and Gamma-function

TL;DR

This paper constructs irreducible unitary representations of a loop affine ax+b group using Wiener measure and connects their matrix coefficients to a loop analogue of the Gamma-function.

Contribution

It introduces a novel family of representations for the loop affine group with central extension, linking them to a loop Gamma-function.

Findings

Constructed irreducible unitary representations on Wiener measure space

Connected matrix coefficients to a loop Gamma-function

Provides new insights into loop group representations

Abstract

We construct a family of irreducible unitary representations of the loop affine group of a line (ax+b group) with central extension on the Hilbert space of square integrable functions with respect to the Wiener measure. We relate the matrix coefficients of the elements of the loop ax+b group to the loop analogue of the Gamma-function.