Unitary Representations with Dirac Cohomology: Finiteness in the Real Case

Abstract

Abstract Let $G$ be a complex connected simple algebraic group with a fixed real form $\sigma $. Let $G({\mathbb{R}})=G^\sigma $ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of $G({\mathbb{R}})$ (up to equivalence) having nonvanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain $G({\mathbb{R}})$, with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.