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

Authors:
Jing-Song Huang and Pavle Pandzic

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 be a connected semisimple Lie group with finite center. Let be the maximal compact subgroup of corresponding to a fixed Cartan involution . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary -module contains a -type with highest weight , then has infinitesimal character . Here is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary -modules with non-zero Dirac cohomology, provided 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

Email:
mahuang@ust.hk

**Pavle Pandzic**

Affiliation:
Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia

Email:
pandzic@math.hr

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

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