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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 31, Issue 1


Unitary representations with non-zero Dirac cohomology for complex E 6

Chao-Ping Dong
Published Online: 2018-09-10 | DOI: https://doi.org/10.1515/forum-2018-0132


This paper classifies the equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology for complex E6. This is achieved by using our finiteness result, and by improving the computing method. According to a conjecture of Barbasch and Pandžić, our classification should also be helpful for understanding the entire unitary dual of complex E6.

Keywords: Dirac cohomology; spin norm; unitary representation

MSC 2010: 22E46


  • [1]

    J. Adams, M. van Leeuwen, P. Trapa and D. Vogan, Unitary representations of real reductive groups, preprint (2012), https://arxiv.org/abs/1212.2192.

  • [2]

    D. Barbasch and P. Pandžić, Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22. Google Scholar

  • [3]

    J. Ding and C.-P. Dong, Unitary representations with Dirac cohomology: A finiteness result, preprint (2017), https://arxiv.org/abs/1702.01876.

  • [4]

    C.-P. Dong, On the Dirac cohomology of complex Lie group representations, Transform. Groups 18 (2013), no. 1, 61–79; Erratum: Transform. Groups 18 (2013), no. 2, 595–597. Web of ScienceCrossrefGoogle Scholar

  • [5]

    C.-P. Dong, Spin norm, pencils, and the u-small convex hull, Proc. Amer. Math. Soc. 144 (2016), no. 3, 999–1013. Web of ScienceGoogle Scholar

  • [6]

    J.-S. Huang and P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185–202. CrossrefGoogle Scholar

  • [7]

    J.-S. Huang and P. Pandžić, Dirac Operators in Representation Theory, Math. Theory Appl., Birkhäuser, Boston, 2006. Google Scholar

  • [8]

    A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002. Google Scholar

  • [9]

    W. M. McGovern, Rings of regular functions on nilpotent orbits. II. Model algebras and orbits, Comm. Algebra 22 (1994), no. 3, 765–772. CrossrefGoogle Scholar

  • [10]

    R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. CrossrefGoogle Scholar

  • [11]

    R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. Sect. A Math. Sci. 89 (1980), no. 1, 1–24. Google Scholar

  • [12]

    S. A. Salamanca-Riba and D. A. Vogan, Jr., On the classification of unitary representations of reductive Lie groups, Ann. of Math. (2) 148 (1998), no. 3, 1067–1133. CrossrefGoogle Scholar

  • [13]

    D. Vogan, Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar (1997).

  • [14]

    D. A. Vogan, Jr., Singular unitary representations, Noncommutative Harmonic Analysis and Lie Groups (Marseille 1980), Lecture Notes in Math. 880, Springer, Berlin (1981), 506–535. Google Scholar

  • [15]

    D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. CrossrefGoogle Scholar

  • [16]

    D. P. Zhelobenko, Harmonic Analysis on Complex Seimsimple Lie Groups, Izdat. “Nauka”, Moscow, 1974. Google Scholar

  • [17]

    Atlas, version 1.0, January 2017 (see www.liegroups.org for more information about the software).

About the article

Received: 2018-05-30

Revised: 2018-07-03

Published Online: 2018-09-10

Published in Print: 2019-01-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571097

Dong is supported by NSFC grant 11571097, the Fundamental Research Funds for the Central Universities, and the China Scholarship Council.

Citation Information: Forum Mathematicum, Volume 31, Issue 1, Pages 69–82, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2018-0132.

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