On classifying unitary modules by their Dirac cohomology

Let G be a connected real reductive group with maximal compact subgroup K of the same rank as G. Dirac cohomology of an A_q（λ） module can be identified with a geometric object—the k-dominant part of a face of the convex hull of the Weyl group orbit of the parameter λ ＋ ρ. We show how Dirac cohomology can be used as a parameter to classify the A_q（λ） modules.