Unitary representations of real reductive groups

Abstract

We present an algorithm for computing the 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 a unitary representation from Harish-Chandra’s Plancherel formula. The behavior of these deformations was approximately determined in the KazhdanLusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. Our algorithm follows the signature of the form through this deformation, counting changes at reducibility points. An important tool is Weyl’s “unitary trick:” replacing the classical invariant Hermitian form (where Lie(G) acts by skew-adjoint operators) by a new one where a compact form of Lie(G) acts by skew-adjoint operators. ∗All of the authors were supported in part by NSF grant DMS-0968275. The first author was supported in part by NSF grant DMS-0967566. †The third author was supported in part by NSF grant DMS-0968060. ‡The fourth author was supported in part by NSF grant DMS-0967272.