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

References

[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. 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 S⁢p⁢(2⁢n,ℝ) 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,ℂ), preprint (2019), https://arxiv.org/abs/1910.02737. 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, http://www.liegroups.org for more about the software. Search in Google Scholar