Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Huang, Jing-Song; Pandžić, Pavle

doi:10.1090/S0894-0347-01-00383-6

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan
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by Jing-Song Huang and Pavle Pandžić;

J. Amer. Math. Soc. 15 (2002), 185-202

DOI: https://doi.org/10.1090/S0894-0347-01-00383-6

Published electronically: September 6, 2001

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Abstract:

Let $G$ be a connected semisimple Lie group with finite center. Let $K$ be the maximal compact subgroup of $G$ corresponding to a fixed Cartan involution $\theta$. We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary $(\mathfrak {g},K)$-module $X$ contains a $K$-type with highest weight $\gamma$, then $X$ has infinitesimal character $\gamma +\rho _{c}$. Here $\rho _{c}$ is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary $(\mathfrak {g},K)$-modules $X$ with non-zero Dirac cohomology, provided $X$ has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.

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