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

Authors: Jing-Song Huang and Pavle Pandžić
Journal: J. Amer. Math. Soc. 15 (2002), 185-202
MSC (2000): Primary 22E46, 22E47
Published electronically: September 6, 2001
MathSciNet review: 1862801
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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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Additional Information

Jing-Song Huang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
MR Author ID: 304754

Pavle Pandžić
Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia
ORCID: 0000-0002-7405-4381

Keywords: Dirac operator, cohomology, unitary representation, infinitesimal character
Received by editor(s): August 28, 2000
Received by editor(s) in revised form: February 27, 2001
Published electronically: September 6, 2001
Additional Notes: The first author’s research was partially supported by RGC-CERG grants of Hong Kong SAR. A part of this work was done during his visit to the University of Zagreb
A part of this work was done during the second author’s visit to The Hong Kong University of Science and Technology
Article copyright: © Copyright 2001 American Mathematical Society