Duke Mathematical Journal

Characters of Springer representations on elliptic conjugacy classes

Dan M. Ciubotaru and Peter E. Trapa

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a Weyl group W, we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of W in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles.

Article information

Source
Duke Math. J., Volume 162, Number 2 (2013), 201-223.

Dates
First available in Project Euclid: 24 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1359036934

Digital Object Identifier
doi:10.1215/00127094-1961735

Mathematical Reviews number (MathSciNet)
MR3018954

Zentralblatt MATH identifier
1260.22012

Subjects
Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
Secondary: 20C08: Hecke algebras and their representations

Citation

Ciubotaru, Dan M.; Trapa, Peter E. Characters of Springer representations on elliptic conjugacy classes. Duke Math. J. 162 (2013), no. 2, 201--223. doi:10.1215/00127094-1961735. https://projecteuclid.org/euclid.dmj/1359036934


Export citation

References

  • [BCT] D. Barbasch, D. Ciubotaru, and P. Trapa, The Dirac operator for graded affine Hecke algebras, to appear in Acta Math., preprint, arXiv:1006.3822v1 [math.RT].
  • [BW] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Stud. 94, Princeton Univ. Press, Princeton, 1980.
  • [BM] W. Borho and R. MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 707–710.
  • [Ca] R. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59.
  • [C1] D. Ciubotaru, On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters, Represent. Theory 12 (2008), 453–498.
  • [C2] D. Ciubotaru, Spin representations of Weyl groups and the Springer correspondence, J. Reine Angew. Math. 671 (2012), 199–222.
  • [CKK] D. Ciubotaru, M. Kato, and S. Kato, On characters and formal degrees of discrete series of classical affine Hecke algebras, Invent. Math. 187 (2012), 589–635.
  • [COT] D. Ciubotaru, E. M. Opdam, and P. E. Trapa, Algebraic and analytic Dirac induction for graded affine Hecke algebras, preprint, arXiv:1201.2130v2 [math.RT]
  • [CM] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Math. Ser., Van Nostrand Reinhold, New York, 1993.
  • [GS] P. Gunnels and E. Sommers, A characterization of Dynkin elements, Math. Res. Letters 10 (2003), 363–373.
  • [KL] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture, Invent. Math. 87 (1987), 153–215.
  • [L1] G. Lusztig, Character sheaves, V, Adv. Math. 61 (1986), 103–155.
  • [L2] G. Lusztig, Cuspidal local systems and graded algebras, I, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 145–202.
  • [L3] G. Lusztig, Affine Hecke algebras and their graded versions, J. Amer. Math. Soc. 2 (1989), 599–635.
  • [L4] G. Lusztig, “Cuspidal local systems and graded algebras, II” in Representations of Groups (Banff, AB, 1994), Amer. Math. Soc., Providence, 1995, 217–275.
  • [O] E. Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu 3 (2004), 531–648.
  • [OS] E. Opdam and M. Solleveld, Homological algebra for affine Hecke algebras, Adv. in Math. 220 (2009), 1549–1601.
  • [R] M. Reeder, Euler-Poincaré pairings and elliptic representations of Weyl groups and $p$-adic groups, Compositio Math. 129 (2001), 149–181.
  • [SS] P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Publ. Math. Inst. Hautes Études Sci. 85 (1997), 97–191.
  • [Sh] T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), 239–267.