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Twisted Dirac Index and Applications to Characters

  • Dan Barbasch
  • Pavle PandžićEmail author
Chapter
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Part of the Springer INdAM Series book series (SINDAMS, volume 37)

Abstract

We present recent joint work with Peter Trapa on the notion of twisted Dirac index and its applications to (twisted) characters and to extensions of modules in a short and informal way. We also announce some further generalizations with applications to Lefschetz numbers and automorphic forms.

Keywords

Open image in new window-module Dirac cohomology Dirac index Characters Twisted characters Euler-Poincaré pairing Lefschetz numbers 
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References

  1. 1.
    Alekseev, A., Meinrenken, E.: Lie theory and the Chern-Weil homomorphism. Ann. Sci. Ecole. Norm. Sup. 38, 303–338 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arthur, J.: A Paley-Wiener theorem for real reductive groups. Acta Math. 150(1–2), 1–89 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arthur, J.: The invariant trace formula II. Global theory. J. Am. Math. Soc. 1(3), 501–554 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42, 1–62 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Barbasch, D., Ciubotaru, D., Trapa, P.: Dirac cohomology for graded affine Hecke algebras. Acta Math. 209(2), 197–227 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barbasch, D., Pandžić, P.: Dirac cohomology and unipotent representations of complex groups. In: Connes, A. Gorokhovsky, A. Lesch, M. Pflaum, M. Rangipour, B. (eds.) Noncommutative Geometry and Global Analysis. Contemporary Mathematics, vol. 546, pp. 1–22. American Mathematical Society, Providence (2011)Google Scholar
  7. 7.
    Barbasch, D., Pandžić, P.: Dirac cohomology of unipotent representations of \(Sp(2n,{\mathbb{R}})\) and \(U(p, q)\). J. Lie Theory 25(1), 185–213 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barbasch, D., Pandžić, P., Trapa, P.: Dirac index and twisted characters. Trans. Am. Math. Soc. 371(3), 1701–1733 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Barbasch, D., Pandžić, P.: Dirac index, Lefschetz numbers and the trace formula, in preparationGoogle Scholar
  10. 10.
    Barbasch, D., Speh, B.: Cuspidal representations of reductive groups. arXiv:0810.0787
  11. 11.
    Bouaziz, A.: Sur les charactères des groupes de Lie réductifs non connexes. J. Funct. Anal. 70(1), 1–79 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ciubotaru, D.: Dirac cohomology for symplectic reflection algebras. Sel. Math. 22(1), 111–144 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Enright, T., Howe, R., Wallach, N.: A classification of unitary highest weight modules. Representation theory of reductive groups (Park City, Utah). Progress in Mathematics, vol. 40, pp. 97–143. Birkhäuser, Boston (1982)CrossRefGoogle Scholar
  14. 14.
    Goette, S.: Equivariant \(\eta \)-invariants on homogeneous spaces. Math. Z. 232, 1–42 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Harish-Chandra, The Characters of Semisimple Groups, Trans. Am. Math. Soc. 83(1), 98–163 (1956)Google Scholar
  16. 16.
    Huang, J.-S., Wong, K.D.: A Casselman-Osborne theorem for rational Cherednik algebras. Transform. Groups 23(1), 75–99 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Huang, J.-S., Kang, Y.-F., Pandžić, P.: Dirac cohomology of some Harish-Chandra modules. Transform. Groups 14(1), 163–173 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Huang, J.-S., Pandžić, P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan. J. Am. Math. Soc. 15, 185–202 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Huang, J.-S., Pandžić, P.: Dirac Operators in Representation Theory. Mathematics: Theory and Applications. Birkhäuser, Boston (2006)Google Scholar
  20. 20.
    Huang, J.-S., Pandžić, P.: Dirac cohomology for Lie superalgebras. Transform. Groups 10, 201–209 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Huang, J.-S., Pandžić, P., Protsak, V.: Dirac cohomology of Wallach representations. Pac. J. Math. 250(1), 163–190 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Huang, J.-S., Pandžić, P., Renard, D.: Dirac operators and Lie algebra cohomology. Represent. Theory 10, 299–313 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Huang, J.-S., Pandžić, P., Zhu, F.: Dirac cohomology, K-characters and branching laws. Am. J. Math. 135(5), 1253–1269 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kac, V.P., Möseneder Frajria, P., Papi, P.: Multiplets of representations, twisted Dirac operators and Vogan’s conjecture in affine setting. Adv. Math. 217, 2485–2562 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kac, V.P., Möseneder Frajria, P., Papi, P.: Dirac operators and the very strange formula for Lie superalgebras. In: Advances in Lie Superalgebras, Springer INdAM Series, vol. 7, pp. 121–148zbMATHCrossRefGoogle Scholar
  26. 26.
    Kostant, B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100, 447–501 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kostant, B.: Dirac cohomology for the cubic Dirac operator. Studies in Memory of Issai Schur. Progress in Mathematics, vol. 210, pp. 69–93 (2003)Google Scholar
  28. 28.
    Kumar, S.: Induction functor in non-commutative equivariant cohomology and Dirac cohomology. J. Algebra 291, 187–207 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Labesse, J.P.: Pseudo-coefficients très cuspidaux et K-theorie. Math. Ann. 291, 607–616 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Labesse, J.-P.: Cohomologie, stabilization et changement de base. Astérisque (257), (1999)Google Scholar
  31. 31.
    Mehdi, S., Pandžić, P., Vogan, D.: Translation principle for Dirac index. Am. J. Math. 139(6), 1465–1491 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Mehdi, S., Pandžić, P., Vogan, D., Zierau, R.: Dirac index and associated cycles of Harish-Chandra modules. arXiv:1712.04169
  33. 33.
    Mehdi, S., Pandžić, P., Vogan, D., Zierau, R.: Computing the associated cycles of certain Harish-Chandra modules. Glas. Mat. 53(73), 2, 275–330 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Pandžić, P., Renard, D.: Dirac induction for Harish-Chandra modules. J. Lie Theory 20(4), 617–641 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Pandžić, P., Somberg, P.: Dirac operator and its cohomology for the quantum group \(U_q(\mathfrak{sl}{(2))}\). J. Math. Phys. 58(4), 041702 (2017), 13 ppGoogle Scholar
  36. 36.
    Parthasarathy, R.: Dirac operator and the discrete series. Ann. Math. 96, 1–30 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Prlić, A.: Algebraic Dirac induction for nonholomorphic discrete series of \(SU(2,1)\). J. Lie Theory 26(3), 889–910 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Prlić, A.: Construction of discrete series representations of \(SO_e(4,1)\) via algebraic Dirac induction. arXiv:1811.01610
  39. 39.
    Rohlfs, J., Speh, B.: Automorphic representations and Lefschetz numbers. Ann. Sci. Ec. Norm. Super. \(4^e\) sér. 22, 473–499(1989)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Vogan, D.: Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, (Fall 1997)Google Scholar
  41. 41.
    Vogan, D., Zuckerman, G.: Unitary representations with nonzero cohomology. Compos. Math. 53, 51–90 (1984)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Waldspurger, J.-L.: Les facteurs de transfert pour les groupes classiques: une formulaire. Manuscr. Math. 133(1–2), 41–82 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Waldspurger, J.-L.: La formule des traces locale tordue. arXiv:1205.1100
  44. 44.
    Xiao, W.: Dirac operators and cohomology for Lie superalgebra of type I. J. Lie Theory 27(1), 111–121 (2017)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

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