UNITARY REPRESENTATIONS OF REAL REDUCTIVE GROUPS

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 a variant of Weyl's "unitary trick": replacing the classical invariant Hermitian form (where Lie(G) acts by skewadjoint operators) by a new one (where a compact form of Lie(G) acts by skew-adjoint operators).