Description

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 the Plancherel formula. The behavior of these deformations was in part determined in the Kazhdan-Lusztig analysis of irreducible characters ; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures.

Our algorithm traces 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).