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

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

Published electronically:
September 6, 2001

MathSciNet review:
1862801

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Abstract | References | Similar Articles | Additional Information

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.

**[AS]**M. Atiyah and W. Schmid,*A geometric construction of the discrete series for semisimple Lie groups*, Invent. Math.**42**(1977), 1-62. MR**57:3310**; erratum MR**81d:22015****[C]**H. Cartan,*La transgression dans un groupe de Lie et dans un espace fibré principal*, Colloque de Topologie algébrique, C.B.R.M. Bruxelles (1950), 57-71. MR**13:107f****[CO]**W. Casselman and M. S. Osborne,*The**-cohomology of representations with an infinitesimal character*, Comp. Math.**31**(1975), 219-227. MR**53:566****[H]**R. Hotta,*On a realization of the discrete series for semisimple Lie groups*, J. of Math. Soc. of Japan**23**(1971), 384-407. MR**46:5531****[HP]**R. Hotta and R. Parthasarathy,*A geometric meaning of the multiplicities of integrable discrete classes in*, Osaka J. Math.**10**(1973), 211-234. MR**49:3031****[K]**B. Kostant,*A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups*, Duke Math. Jour.**100**(1999), 447-501. CMP**2000:05****[K2]**B. Kostant,*Dirac cohomology for the cubic Dirac operator*, in preparation.**[Ku]**S. Kumaresan,*On the canonical**-types in the irreducible unitary**-modules with non-zero relative cohomology*, Invent. Math.**59**(1980), 1-11. MR**83c:17011****[L]**J.-S. Li,*On the first eigenvalue of Laplacian on locally symmetric manifolds*, First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001, 271-278. CMP**2001:12****[P]**R. Parthasarathy,*The Dirac operator and the discrete series*, Ann. of Math.**96**(1972), 1-30. MR**47:6945****[SR]**S. A. Salamanca-Riba,*On the unitary dual of real reductive Lie groups and the**modules: the strongly regular case*, Duke Math. Jour.**96**(1998), 521-546. MR**2000a:22023****[S]**W. Schmid,*On the characters of the discrete series. The Hermitian symmetric case*, Invent. Math.**30**(1975), 47-144. MR**53:714****[V1]**D. A. Vogan, Jr.,*Representations of real reductive Lie groups*, Birkhäuser, Boston-Basel-Stuttgart, 1981. MR**83c:22022****[V2]**D. A. Vogan, Jr.,*Unitarizability of certain series of representations*, Ann. of Math.**120**(1984), 141-187. MR**86h:22028****[V3]**D. A. Vogan, Jr.,*Dirac operator and unitary representations*, 3 talks at MIT Lie groups seminar, Fall of 1997.**[V4]**D. A. Vogan, Jr.,*On the smallest eigenvalue of the Laplacian on a locally symmetric space*, Lecture at the Midwest Conference on Representation Theory and Automorphic Forms, Chicago, June, 2000.**[VZ]**D. A. Vogan, Jr. and G. J. Zuckerman,*Unitary representations with non-zero cohomology*, Comp. Math.**53**(1984), 51-90. MR**86k:22040****[W]**N. R. Wallach,*Real Reductive Groups, Volume I*, Academic Press, 1988. MR**89i:22029**

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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