Dirac cohomology of Wallach representations

Jing-Song Huang, Pavle Pandžić and Victor Protsak

Vol. 250 (2011), No. 1, 163–190

DOI: 10.2140/pjm.2011.250.163

Abstract

Let G be either the metaplectic double cover of Sp(2n, ℝ), or SO^∗(2n), or SU(p,q). Let g be the complexified Lie algebra of G and let K be a maximal compact subgroup of G. Let X be one of the Wallach modules for the pair (g,K). In other words, X corresponds to a discrete point in the classification of unitary lowest weight modules with scalar lowest K-type. The purpose of this paper is to calculate the Dirac cohomology of X. Our approach is based on the explicit knowledge of the K-types of X. We establish a bijection between certain K-types E_i of X and certain K-types F_i of the spin module, where K is the spin double cover of K. The Dirac cohomology is then realized as the set of Parthasarathy–Ranga-Rao–Varadarajan components of E_i ⊗ F_i.

Keywords

reductive Lie group, unitary representation, Dirac cohomology, Wallach representation

Mathematical Subject Classification 2000

Primary: 22E46

Milestones

Received: 10 August 2009

Revised: 15 April 2010

Accepted: 16 April 2010

Published: 1 March 2011