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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Dirac cohomology, unitary representations and a proof of a conjecture of Vogan
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by Jing-Song Huang and Pavle Pandžić PDF
J. Amer. Math. Soc. 15 (2002), 185-202 Request permission

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
  • Email: mahuang@ust.hk
  • Pavle Pandžić
  • Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia
  • ORCID: 0000-0002-7405-4381
  • Email: pandzic@math.hr
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1862801