Startseite Mathematik Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Jian Ding und Chao-Ping Dong EMAIL logo
Veröffentlicht/Copyright: 19. März 2020
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 32 Heft 4

Abstract

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.

Keywords: Dirac cohomology; good range; spin norm; unitary representation
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 s⁢ρ; see Table 17.

Table 17

Involutions in the Weyl group of F4.

Indexs⁢ρIndexs⁢ρIndexs⁢ρ
1[1,1,1,1]2[-1,2,1,1]3[2,-1,2,1]
4[1,3,-1,2]5[1,1,2,-1]6[-1,4,-1,2]
7[-1,2,2,-1]8[2,-1,3,-1]9[-1,-1,3,1]
10[5,-3,1,3]11[4,1,-2,4]12[1,5,-1,-1]
13[-1,-1,4,-1]14[-1,6,-1,-1]15[3,-5,3,3]
16[5,-1,-1,4]17[4,5,-4,2]18[-5,1,1,4]
19[-3,5,-3,5]20[7,-5,4,-2]21[6,1,1,-4]
22[-4,-1,2,4]23[-5,3,-1,5]24[-3,9,-5,3]
25[5,-7,6,-2]26[10,-5,1,1]27[7,-1,2,-4]
28[1,1,-3,7]29[6,3,-1,-3]30[-7,1,5,-3]
31[-5,7,1,-5]32[1,-4,1,6]33[9,1,-3,1]
34[-1,-3,1,6]35[-6,-1,6,-3]36[-2,1,-2,7]
37[-11,5,1,1]38[-7,5,3,-5]39[-5,9,-1,-4]
40[1,-2,-1,7]41[7,-10,4,3]42[10,-1,-2,1]
43[10,-3,1,-2]44[5,5,-7,5]45[9,1,-2,-1]
46[1,1,4,-8]47[-1,-1,-1,7]48[-9,11,-4,1]
49[1,-6,8,-6]50[10,-1,-1,-1]51[-7,-4,5,2]
52[-1,-5,8,-6]53[-11,9,-2,1]54[-11,7,1,-2]
55[-2,1,5,-8]56[-6,11,-7,4]57[-9,11,-3,-1]
58[5,-10,8,-4]59[1,-2,6,-8]60[9,-4,-3,6]
61[3,7,-3,-5]62[-1,-1,6,-8]63[-11,9,-1,-1]
64[1,-11,7,1]65[1,10,-8,1]66[-1,-10,7,1]
67[-5,-6,8,-3]68[-9,6,-4,6]69[-1,11,-8,1]
70[-4,11,-4,-4]71[4,-11,4,4]72[1,-11,8,-1]
73[9,-6,4,-6]74[5,6,-8,3]75[1,10,-7,-1]
76[-1,-10,8,-1]77[-1,11,-7,-1]78[11,-9,1,1]
79[1,1,-6,8]80[-3,-7,3,5]81[-9,4,3,-6]
82[-1,2,-6,8]83[-5,10,-8,4]84[9,-11,3,1]
85[6,-11,7,-4]86[2,-1,-5,8]87[11,-7,-1,2]
88[11,-9,2,-1]89[1,5,-8,6]90[7,4,-5,-2]
91[-10,1,1,1]92[-1,6,-8,6]93[9,-11,4,-1]
94[1,1,1,-7]95[-1,-1,-4,8]96[-9,-1,2,1]
97[-5,-5,7,-5]98[-10,3,-1,2]99[-10,1,2,-1]
100[-7,10,-4,-3]101[-1,2,1,-7]102[5,-9,1,4]
103[7,-5,-3,5]104[11,-5,-1,-1]105[2,-1,2,-7]
106[6,1,-6,3]107[1,3,-1,-6]108[-9,-1,3,-1]
109[-1,4,-1,-6]110[5,-7,-1,5]111[7,-1,-5,3]
112[-6,-3,1,3]113[-1,-1,3,-7]114[-7,1,-2,4]
115[-10,5,-1,-1]116[-5,7,-6,2]117[3,-9,5,-3]
118[5,-3,1,-5]119[4,1,-2,-4]120[-6,-1,-1,4]
121[-7,5,-4,2]122[3,-5,3,-5]123[5,-1,-1,-4]
124[-4,-5,4,-2]125[-5,1,1,-4]126[-3,5,-3,-3]
127[1,-6,1,1]128[1,1,-4,1]129[-1,-5,1,1]
130[-4,-1,2,-4]131[-5,3,-1,-3]132[1,1,-3,-1]
133[-2,1,-3,1]134[1,-2,-2,1]135[1,-4,1,-2]
136[-1,-1,-2,1]137[-1,-3,1,-2]138[-2,1,-2,-1]
139[1,-2,-1,-1]140[-1,-1,-1,-1]

Acknowledgements

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.

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. Suche 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_7Suche in Google Scholar

[3] D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. 10.1007/BF01393972Suche in Google Scholar

[4] D. Barbasch, C.-P. Dong and K. D. Wong, A multiplicity one result for spin-lowest K-types, preprint. Suche 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/546/10782Suche 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. Suche 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_2Suche 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/067Suche 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. Suche 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-XSuche 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/s00031-013-9226-9Suche 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/12798Suche 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/rny293Suche 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-2018-0132Suche 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.2010.09.024Suche 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.2015.0007Suche 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. Suche 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.1885Suche 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/BF02391779Suche 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/BF02392813Suche 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/s00031-008-9036-7Suche 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/S0894-0347-01-00383-6Suche in Google Scholar

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

[24] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972. 10.1007/978-1-4612-6398-2Suche 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.1302Suche in Google Scholar

[26] A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser, Boston, 2002. Suche 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. 10.1515/9781400883936Suche 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. 10.1080/00927879408824874Suche 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. 10.2307/1970351Suche in Google Scholar

[30] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. 10.2307/1970892Suche 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/BF02881021Suche 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/S0012-7094-99-09616-3Suche 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. 10.2307/121036Suche in Google Scholar

[34] D. Vogan, Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997. Suche 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/BFb0090421Suche in Google Scholar

[36] D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. 10.2307/2007074Suche 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/BF01394418Suche in Google Scholar

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

[39] Atlas of Lie Groups and Representations, version 1.0, January 2017, http://www.liegroups.org for more about the software. Suche in Google Scholar

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

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Decomposition rank of approximately subhomogeneous C*-algebras
  3. Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation
  4. Degree bounds for modular covariants
  5. The Kobayashi–Royden metric on punctured spheres
  6. Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces
  7. Artin L-functions of almost monomial Galois groups
  8. Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups
  9. Weighted sum formulas of multiple t-values with even arguments
  10. Long time behavior of solutions to 3D generalized MHD equations
  11. Bilinear forms on non-homogeneous Sobolev spaces
  12. Weyl and Zariski chambers on projective surfaces
  13. Rankin–Selberg L-functions via good sections
  14. Archimedean domains of skew generalized power series
https://doi.org/10.1515/forum-2019-0295
Schlagwörter für diesen Artikel
Dirac cohomology; good range; spin norm; unitary representation

Artikel in diesem Heft

  1. Frontmatter
  2. Decomposition rank of approximately subhomogeneous C*-algebras
  3. Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation
  4. Degree bounds for modular covariants
  5. The Kobayashi–Royden metric on punctured spheres
  6. Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces
  7. Artin L-functions of almost monomial Galois groups
  8. Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups
  9. Weighted sum formulas of multiple t-values with even arguments
  10. Long time behavior of solutions to 3D generalized MHD equations
  11. Bilinear forms on non-homogeneous Sobolev spaces
  12. Weyl and Zariski chambers on projective surfaces
  13. Rankin–Selberg L-functions via good sections
  14. Archimedean domains of skew generalized power series
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2026 De Gruyter Brill
Heruntergeladen am 3.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0295/pdf
Button zum nach oben scrollen