The Character of the Exceptional Series of Representations of SU(1,1)

TL;DR

This paper determines the character of the exceptional series of SU(1,1) representations using Bargmann's realization, involving complex integral kernel constructions in both position and momentum spaces.

Contribution

It introduces a method to compute the characters of exceptional series representations of SU(1,1) within Bargmann's framework, handling non-local metrics and integral kernels.

Findings

Successfully constructed the integral kernel of the group ring in the momentum space.

Extended the analysis framework from principal to exceptional series.

Demonstrated the advantage of using Bargmann's canonical framework for such representations.

Abstract

The character of the exceptional series of representations of SU(1,1) is determined by using Bargmann's realization of the representation in the Hilbert space $H_\sigma$ of functions defined on the unit circle. The construction of the integral kernel of the group ring turns out to be especially involved because of the non-local metric appearing in the scalar product with respect to which the representations are unitary. Since the non-local metric disappears in the `momentum space' $i.e.$ in the space of the Fourier coefficients the integral kernel is constructed in the momentum space, which is transformed back to yield the integral kernel of the group ring in $H_\sigma$. The rest of the procedure is parallel to that for the principal series treated in a previous paper. The main advantage of this method is that the entire analysis can be carried out within the canonical framework of Bargmann.