Unitary representations with Dirac cohomology: finiteness in the real case

Chao-Ping Dong

Published 2017-08-01Version 1

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. Let $\theta$ be the Cartan involution of $G$ corresponding to $\sigma$. This paper shows that, up to equivalence, for all but finitely many exceptions, any irreducible unitary Harish-Chandra module $\pi$ of $G(\mathbb{R})$ having non-zero Dirac cohomology must be cohomologically induced from an irreducible unitary Harish-Chandra module $\pi_{L(\mathbb{R})}$ of $L(\mathbb{R})$ with nonvanishing Dirac cohomology. Moreover, $\pi_{L(\mathbb{R})}$ is in the good range. Here $L(\mathbb{R})$ is a proper $\theta$-stable Levi subgroup of $G(\mathbb{R})$.