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Unitary representations with Dirac cohomology: a finiteness result for complex Lie groups. (English) Zbl 1442.22013
Summary: Let \(G\) be a connected complex simple Lie group, and let \(\widehat{G}^{\text{d}}\) be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that \(\widehat{G}^{\text{d}}\) consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of \(\widehat{G}^{\text{d}}\) come from \(\widehat{L}^{\text{d}}\) via cohomological induction and they are all in the good range. Here \(L\) runs over the Levi factors of proper \(\theta\)-stable parabolic subgroups of \(G\). It follows that figuring out \(\widehat{G}^{\text{d}}\) requires a finite calculation in total. As an application, we report a complete description of \(\widehat{F}_4^{\text{d}}\).
MSC:
22E46 Semisimple Lie groups and their representations
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[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] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1-62. · Zbl 0373.22001
[3] D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103-176. · Zbl 0692.22006
[4] D. Barbasch, C.-P. Dong and K. D. Wong, A multiplicity one result for spin-lowest K-types, preprint.
[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. · Zbl 1238.22008
[6] D. Barbasch and P. Pandžić, Dirac cohomology of unipotent representations of Sp(2n,\mathbbR) and U(p,q), J. Lie Theory 25 (2015), no. 1, 185-213. · Zbl 1387.22016
[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.
[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. · Zbl 0980.22015
[9] J. Ding and C.-P. Dong, Spin norm, K-types, and tempered representations, J. Lie Theory 26 (2016), no. 3, 651-658. · Zbl 1350.22012
[10] P. Dirac, The quantum theory of the electron, Proc. Roy. Soc. Lond. Ser. A 117 (1928), 610-624. · JFM 54.0973.01
[11] C.-P. Dong, Erratum to: On the Dirac cohomology of complex Lie group representations [mr3022758], Transform. Groups 18 (2013), no. 2, 595-597. · Zbl 1267.22004
[12] C.-P. Dong, Spin norm, pencils, and the u-small convex hull, Proc. Amer. Math. Soc. 144 (2016), no. 3, 999-1013. · Zbl 1397.17008
[13] C.-P. Dong, Unitary representations with Dirac cohomology: Finiteness in the real case, Int. Math. Res. Not. IMRN (2019), 10.1093/imrn/rny293.
[14] C.-P. Dong, Unitary representations with non-zero Dirac cohomology for complex E_6, Forum Math. 31 (2019), no. 1, 69-82. · Zbl 1429.22014
[15] C.-P. Dong and J.-S. Huang, Jacquet modules and Dirac cohomology, Adv. Math. 226 (2011), no. 4, 2911-2934. · Zbl 1221.22014
[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. · Zbl 1321.22014
[17] C.-P. Dong and K. D. Wong, Scattered representations of \(\text{SL}(n,\mathbb{C})\), preprint (2019), https://arxiv.org/abs/1910.02737.
[18] M. Duflo, Réprésentation unitaires irréductibles des groupes simples complexes de rang deux, Bull. Soc. Math. France 107 (1979), no. 1, 55-96. · Zbl 0407.22014
[19] Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241-318. · Zbl 0152.13402
[20] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111. · Zbl 0199.20102
[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. · Zbl 1179.22013
[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. · Zbl 0980.22013
[23] J.-S. Huang and P. Pandžić, Dirac Operators in Representation Theory, Math. Theory Appl., Birkhäuser, Boston, 2006. · Zbl 1103.22008
[24] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.
[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. · Zbl 0346.17008
[26] A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002. · Zbl 1075.22501
[27] A. W. Knapp and D. A. Vogan, Jr., Cohomological Induction and Unitary Representations, Princeton Math. Ser. 45, Princeton University, Princeton, 1995. · Zbl 0863.22011
[28] W. M. McGovern, Rings of regular functions on nilpotent orbits. II. Model algebras and orbits, Comm. Algebra 22 (1994), no. 3, 765-772. · Zbl 0813.22006
[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. · Zbl 0177.18004
[30] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1-30. · Zbl 0249.22003
[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. · Zbl 0434.22011
[32] S. A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the A_g(\lambda) modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521-546. · Zbl 0941.22014
[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. · Zbl 0918.22009
[34] D. Vogan, Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997.
[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.
[36] D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141-187. · Zbl 0561.22010
[37] D. A. Vogan, Jr., The unitary dual of \rm GL(n) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449-505. · Zbl 0598.22008
[38] D. P. Zhelobenko, Harmonic analysis on complex semisimple Lie groups, Mir, Moscow, 1974.
[39] Atlas of Lie Groups and Representations, version 1.0, January 2017, http://www.liegroups.org for more about the software.
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