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Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Jian Ding and Chao-Ping Dong
From the journal Forum Mathematicum


Let G be a connected complex simple Lie group, and let G^d be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G^d consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G^d come from L^d via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G^d requires a finite calculation in total. As an application, we report a complete description of F^4d.

MSC 2010: 22E46

Communicated by Freydoon Shahidi

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11571097

Funding statement: The second-named author is supported by NSFC grant 11571097 and the China Scholarship Council.

A Appendix

In this appendix, we index all the involutions s in the Weyl group of F4 by presenting the weight sρ; see Table 17.

Table 17

Involutions in the Weyl group of F4.



The second-named author thanks the math department of MIT for offering excellent working conditions during October 2016 and September 2017. He is deeply grateful to the atlas mathematicians for many things, and to the referee for offering nice suggestions.


[1] J. Adams, M. van Leeuwen, P. Trapa and D. Vogan, Unitary representations of real reductive groups, preprint (2012), Search in Google Scholar

[2] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. 10.1007/978-94-009-8961-0_7Search in Google Scholar

[3] D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. 10.1007/BF01393972Search in Google Scholar

[4] D. Barbasch, C.-P. Dong and K. D. Wong, A multiplicity one result for spin-lowest K-types, preprint. Search in Google Scholar

[5] 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. 10.1090/conm/546/10782Search in Google Scholar

[6] D. Barbasch and P. Pandžić, Dirac cohomology of unipotent representations of Sp(2n,) and U(p,q), J. Lie Theory 25 (2015), no. 1, 185–213. Search in Google Scholar

[7] D. Barbasch and P. Pandžić, Twisted Dirac index and applications to characters, Affine, Vertex and W-algebras, Springer INdAM Ser. 37, Springer, Cham (2019), 23–36. 10.1007/978-3-030-32906-8_2Search in Google Scholar

[8] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, American Mathematical Society, Providence, 2000. 10.1090/surv/067Search in Google Scholar

[9] J. Ding and C.-P. Dong, Spin norm, K-types, and tempered representations, J. Lie Theory 26 (2016), no. 3, 651–658. Search in Google Scholar

[10] P. Dirac, The quantum theory of the electron, Proc. Roy. Soc. Lond. Ser. A 117 (1928), 610–624. 10.1016/B978-0-08-006995-1.50017-XSearch in Google Scholar

[11] C.-P. Dong, Erratum to: On the Dirac cohomology of complex Lie group representations [mr3022758], Transform. Groups 18 (2013), no. 2, 595–597. 10.1007/s00031-013-9226-9Search in Google Scholar

[12] C.-P. Dong, Spin norm, pencils, and the u-small convex hull, Proc. Amer. Math. Soc. 144 (2016), no. 3, 999–1013. 10.1090/proc/12798Search in Google Scholar

[13] C.-P. Dong, Unitary representations with Dirac cohomology: Finiteness in the real case, Int. Math. Res. Not. IMRN (2019), 10.1093/imrn/rny293. 10.1093/imrn/rny293Search in Google Scholar

[14] C.-P. Dong, Unitary representations with non-zero Dirac cohomology for complex E6, Forum Math. 31 (2019), no. 1, 69–82. 10.1515/forum-2018-0132Search in Google Scholar

[15] C.-P. Dong and J.-S. Huang, Jacquet modules and Dirac cohomology, Adv. Math. 226 (2011), no. 4, 2911–2934. 10.1016/j.aim.2010.09.024Search in Google Scholar

[16] C.-P. Dong and J.-S. Huang, Dirac cohomology of cohomologically induced modules for reductive Lie groups, Amer. J. Math. 137 (2015), no. 1, 37–60. 10.1353/ajm.2015.0007Search in Google Scholar

[17] C.-P. Dong and K. D. Wong, Scattered representations of SL(n,), preprint (2019), Search in Google Scholar

[18] M. Duflo, Réprésentation unitaires irréductibles des groupes simples complexes de rang deux, Bull. Soc. Math. France 107 (1979), no. 1, 55–96. 10.24033/bsmf.1885Search in Google Scholar

[19] Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. 10.1007/BF02391779Search in Google Scholar

[20] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. 10.1007/BF02392813Search in Google Scholar

[21] J.-S. Huang, Y.-F. Kang and P. Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009), no. 1, 163–173. 10.1007/s00031-008-9036-7Search in Google Scholar

[22] 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. 10.1090/S0894-0347-01-00383-6Search in Google Scholar

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

[24] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972. 10.1007/978-1-4612-6398-2Search in Google Scholar

[25] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. Éc. Norm. Supér. (4) 9 (1976), no. 1, 1–29. 10.24033/asens.1302Search in Google Scholar

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

[27] A. W. Knapp and D. A. Vogan, Jr., Cohomological Induction and Unitary Representations, Princeton Math. Ser. 45, Princeton University, Princeton, 1995. 10.1515/9781400883936Search in Google Scholar

[28] W. M. McGovern, Rings of regular functions on nilpotent orbits. II. Model algebras and orbits, Comm. Algebra 22 (1994), no. 3, 765–772. 10.1080/00927879408824874Search in Google Scholar

[29] K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967), 383–429. 10.2307/1970351Search in Google Scholar

[30] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. 10.2307/1970892Search in Google Scholar

[31] 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. 10.1007/BF02881021Search in Google Scholar

[32] S. A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the Ag(λ) modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521–546. 10.1215/S0012-7094-99-09616-3Search in Google Scholar

[33] 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. 10.2307/121036Search in Google Scholar

[34] D. Vogan, Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997. Search in Google Scholar

[35] 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. 10.1007/BFb0090421Search in Google Scholar

[36] D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. 10.2307/2007074Search in Google Scholar

[37] D. A. Vogan, Jr., The unitary dual of GL(n) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. 10.1007/BF01394418Search in Google Scholar

[38] D. P. Zhelobenko, Harmonic analysis on complex semisimple Lie groups, Mir, Moscow, 1974. Search in Google Scholar

[39] Atlas of Lie Groups and Representations, version 1.0, January 2017, for more about the software. Search in Google Scholar

Received: 2019-10-28
Revised: 2020-02-09
Published Online: 2020-03-19
Published in Print: 2020-07-01

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