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Graded Hecke Algebras for Disconnected Reductive Groups

  • Anne-Marie Aubert
  • Ahmed Moussaoui
  • Maarten Solleveld
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

We introduce graded Hecke algebras \(\mathbb H\) based on a (possibly disconnected) complex reductive group G and a cuspidal local system \({\mathcal L}\) on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G.

We develop the representation theory of the algebras \(\mathbb H\), obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data \((G,M,{\mathcal L})\) and they are closely related to Langlands parameters.

Our main motivation for considering these graded Hecke algebras is that the space of irreducible \(\mathbb H\)-representations is canonically in bijection with a certain set of “logarithms” of enhanced L-parameters. Therefore, we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.

Keywords

Hecke algebras Reductive groups Cuspidal local systems Langlands programme 

References

  1. [Art]
    J. Arthur, “A note on L-packets”, Pure Appl. Math. Quaterly 2.1 (2006), 199–217.MathSciNetCrossRefGoogle Scholar
  2. [ABPS1]
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “Geometric structure in smooth dual and local Langlands correspondence”, Japan. J. Math. 9 (2014), 99–136.CrossRefGoogle Scholar
  3. [ABPS2]
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “The principal series of p-adic groups with disconnected centre”, Proc. London Math. Soc. 114.5 (2017), 798–854.MathSciNetCrossRefGoogle Scholar
  4. [ABPS3]
    A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, “Conjectures about p-adic groups and their noncommutative geometry”, Contemp. Math. 691 (2017), 15–51.MathSciNetCrossRefGoogle Scholar
  5. [AMS]
    A.-M. Aubert, A. Moussaoui, M. Solleveld, “Generalizations of the Springer correspondence and cuspidal Langlands parameters”, Manus. Math. (2018), 1–72.Google Scholar
  6. [AMS2]
    A.-M. Aubert, A. Moussaoui, M. Solleveld, “Affine Hecke algebras for Langlands parameters”, arXiv:1701.03593, 2017.Google Scholar
  7. [BeLu]
    J. Bernstein, V. Lunts, “Equivariant Sheaves and Functors”, Lecture Notes in Mathematics 1578.Google Scholar
  8. [Bor]
    A. Borel, “Automorphic L-functions”, Proc. Symp. Pure Math 33.2 (1979), 27–61.MathSciNetCrossRefGoogle Scholar
  9. [Car]
    R.W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Pure and Applied Mathematics, John Wiley & Sons, New York NJ, 1985.Google Scholar
  10. [Ciu]
    D. Ciubotaru, “On unitary unipotent representations of p-adic groups and affine Hecke algebras with unequal parameters”, Representation Theory 12 (2008), 453–498.MathSciNetCrossRefGoogle Scholar
  11. [Hai]
    T.J. Haines, “The stable Bernstein center and test functions for Shimura varieties”, pp. 118–186 in: Automorphic forms and Galois representations, London Math. Soc. Lecture Note Ser. 415, Cambridge University Press, 2014.Google Scholar
  12. [Kal]
    T. Kaletha, “Global rigid inner forms and multiplicities of discrete automorphic representations”, arXiv:1501.01667, 2015.Google Scholar
  13. [KaLu]
    D. Kazhdan, G. Lusztig, “Proof of the Deligne–Langlands conjecture for Hecke algebras”, Invent. Math. 87 (1987), 153–215.MathSciNetCrossRefGoogle Scholar
  14. [Lus1]
    G. Lusztig, “Intersection cohomology complexes on a reductive group”, Invent. Math. 75.2 (1984), 205–272.MathSciNetCrossRefGoogle Scholar
  15. [Lus2]
    G. Lusztig, “Character sheaves V”, Adv. in Math. 61 (1986), 103–155.MathSciNetCrossRefGoogle Scholar
  16. [Lus3]
    G. Lusztig, “Cuspidal local systems and graded Hecke algebras”, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 145–202.MathSciNetCrossRefGoogle Scholar
  17. [Lus4]
    G. Lusztig, “Affine Hecke algebras and their graded version”, J. Amer. Math. Soc 2.3 (1989), 599–635.MathSciNetCrossRefGoogle Scholar
  18. [Lus5]
    G. Lusztig, “Cuspidal local systems and graded Hecke algebras. II”, pp. 217–275 in: Representations of groups, Canadian Mathematical Society Conference Proceedings 16, 1995.Google Scholar
  19. [Lus6]
    G. Lusztig, “Classification of unipotent representations of simple p-adic groups”, Int. Math. Res. Notices 11 (1995), 517–589.MathSciNetCrossRefGoogle Scholar
  20. [Lus7]
    G. Lusztig, “Cuspidal local systems and graded Hecke algebras. III”, Represent. Theory 6 (2002), 202–242.MathSciNetCrossRefGoogle Scholar
  21. [Lus8]
    G. Lusztig, “Classification of unipotent representations of simple p-adic groups. II”, Represent. Theory 6 (2002), 243–289.MathSciNetCrossRefGoogle Scholar
  22. [RaRa]
    A. Ram, J. Ramagge, “Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory”, pp. 428–466 in: A tribute to C.S. Seshadri (Chennai 2002), Trends in Mathematics, Birkhäuser, 2003.Google Scholar
  23. [Ree]
    M. Reeder, “Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations”, Representation Theory 6 (2002), 101–126.MathSciNetCrossRefGoogle Scholar
  24. [Sol1]
    M. Solleveld, “Parabolically induced representations of graded Hecke algebras”, Algebras and Representation Theory 15.2 (2012), 233–271.MathSciNetCrossRefGoogle Scholar
  25. [Sol2]
    M. Solleveld, “Homology of graded Hecke algebras”, J. Algebra 323 (2010), 1622–1648.MathSciNetCrossRefGoogle Scholar
  26. [Spr]
    T.A. Springer, Linear algebraic groups 2nd ed., Progress in Mathematics 9, Birkhäuser, 1998.Google Scholar
  27. [Wit]
    S. Witherspoon, “Twisted graded Hecke algebras”, J. Algebra 317 (2007), 30–42.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Anne-Marie Aubert
    • 1
  • Ahmed Moussaoui
    • 2
  • Maarten Solleveld
    • 3
  1. 1.Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C.ParisFrance
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  3. 3.IMAPP, Radboud Universiteit NijmegenNijmegenThe Netherlands

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