Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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 $\mathfrak{n}$-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 $L^{2}(\Gamma \backslash G)$, 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 $K$-types in the irreducible unitary $\mathfrak{g}$-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 $A_{\mathfrak q}(\lambda )$ 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

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 22E46, 22E47

Retrieve articles in all journals with MSC (2000): 22E46, 22E47


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

American Mathematical Society