The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group

TL;DR

This paper parametrizes irreducible tempered characters of possibly disconnected reductive real Lie groups and describes them using Kirillov's formulas, extending classical methods to nonconnected groups.

Contribution

It extends the orbit method and character formulas to possibly disconnected reductive real algebraic groups, providing a new parametrization framework.

Findings

Parametrization of irreducible tempered characters for nonconnected groups

Kirillov's formulas adapted for descent near elliptic elements

Connection to Knapp-Zuckerman parameters for connected linear groups

Abstract

Let G be a possibly disconnected reductive real Lie group. In this paper, I parametrize the set of irreductible tempered characters of G. I then describe these characters using certain ``Kirillov's formulas,'' based on the descent method near each elliptic element in G. If G is linear and connected, the parameters I use are ``final basic'' parameters in the sense of Knapp and Zuckerman.