Unitary representations of real reductive groups

TL;DR

This paper introduces a finite algorithm to compute all irreducible unitary representations of real reductive groups by analyzing Hermitian form signatures during deformation, leveraging advanced representation theory techniques.

Contribution

It provides a novel finite algorithm for classifying irreducible unitary representations of real reductive groups using deformation and signature analysis.

Findings

Algorithm effectively computes unitary representations.

Signature behavior tracks deformations through singularities.

Utilizes forms with compact Lie algebra action for analysis.

Abstract

We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algorithm is to follow the behavior of the signature of the Hermitian form through this deformation, counting changes through singularities of the form at reducibility points. An important technical tool is replacing the classical invariant form (in which the real form of the Lie algebra acts by skew-adjoint operators) by forms in which the compact form of the Lie algebra acts by skew-adjoint operators.