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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 $G$ 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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  • [AS] Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 0463358
    Michael Atiyah and Wilfried Schmid, Erratum: “A geometric construction of the discrete series for semisimple Lie groups” [Invent. Math. 42 (1977), 1–62; MR 57 #3310], Invent. Math. 54 (1979), no. 2, 189–192. MR 550183, 10.1007/BF01408936
  • [C] Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, pp. 57–71 (French). MR 0042427
  • [CO] William Casselman and M. Scott Osborne, The 𝔫-cohomology of representations with an infinitesimal character, Compositio Math. 31 (1975), no. 2, 219–227. MR 0396704
  • [H] Ryoshi Hotta, On a realization of the discrete series for semisimple Lie groups, J. Math. Soc. Japan 23 (1971), 384–407. MR 0306405
  • [HP] Ryoshi Hotta and R. Parthasarathy, A geometric meaning of the multiplicity of integrable discrete classes in 𝐿²(Γ\𝐺), Osaka J. Math. 10 (1973), 211–234. MR 0338265
  • [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 𝑔-modules with nonzero relative cohomology, Invent. Math. 59 (1980), no. 1, 1–11. MR 575078, 10.1007/BF01390311
  • [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, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 0318398
  • [SR] Susana A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the 𝐴_{𝑔}(𝜆) modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521–546. MR 1671213, 10.1215/S0012-7094-99-09616-3
  • [S] Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47–144. MR 0396854
  • [V1] David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
  • [V2] David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. MR 750719, 10.2307/2007074
  • [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] David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
  • [W] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683

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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

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

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