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

Huang, Jing-Song¹ ; Pandžić, Pavle²

¹ Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
² Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia

Journal of the American Mathematical Society, Tome 15 (2002) no. 1, pp. 185-202

Résumé

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.

Détail

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

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