The Character of the Principal Series of Representations of the Real Unimodular Group

TL;DR

This paper evaluates the character of the principal series of representations of SL(n,R) using Gel'fand and Naimark's definition, simplifying the determination of the integral kernel and highlighting the non-contribution of elliptic elements.

Contribution

It provides a new approach to compute characters of principal series representations by realizing them on coset spaces, simplifying the integral kernel determination.

Findings

Elliptic elements do not contribute to the character.

Representation realization on coset spaces simplifies calculations.

Integral kernel determination is facilitated by this approach.

Abstract

The character of the principal series of representations of SL(n,R) is evaluated by using Gel'fand and Naimark's definition of character. This representation is realized in the space of functions defined on the right coset space of SL(n,R) with respect to the subgroup of real triangular matrices. This form of the representations considerably simplifies the problem of determination of the integral kernel of the group ring which is fundamental in the Gel'fand-Naimark theory of character. An important feature of the principal series of representations is that the `elliptic' elements of SL(n,R) do not contribute to its character.