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Formule des traces sur les corps finis

  • Anne-Marie Aubert
Part of the Progress in Mathematics book series (PM, volume 141)

Abstract

James Arthur a récemment mis en lumière de nombreuses analogies entre des objets attachés à un groupe réductif p-adique et des objets attachés aux divers R-groupes (cf. [Ar2, en particulier Remarks (2) p. 118]). Nous nous intéressons ici au cas d’un groupe réductif fini, i.e., du groupe G F des points fixes sous un endomorphisme de Frobenius F d’un groupe G algébrique réductif connexe sur une clôture algébrique d’un corps fini F q et défini sur 픽 q . Le rôle des R-groupes est joué ici par les groupes de ramification
$$ W_{\text{G}^F } (\text{M,}\,\sigma \text{)}\,\text{ = }\,\text{\{ }r\, \in \,\text{N}_{\text{G}^F } (\text{M)/M}^F |{}^r\sigma = \sigma \} $$
associés à des paires (M, σ) formées d’un sous-groupe de Levi F-stable M d’un sous-groupe parabolique F-stable de G et d’une représentation irréductible cuspidale σ de M F ; ces groupes sont des extensions centrales de groupes de Coxeter.

Keywords

Element Unipotents Nous Allons Unipotent Class Character Sheave Nous Obtenons 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Références

  1. [Ar1]
    J. Arthur, A local trace formula, Publ. I.H.E.S. 73 (1991), 1–96.MathSciNetGoogle Scholar
  2. [Ar2]
    J. Arthur, On elliptic tempered characters, Acta Math. 171, 73–138.Google Scholar
  3. [Aul]
    A.-M. Aubert, Séries de Harish-Chandra de Modules et Correspondance de Howe Modulaire, J. of Algebra 165 No. 3 (1994), 576–601.MathSciNetMATHCrossRefGoogle Scholar
  4. [Au2]
    A.-M. Aubert, Systèmes de Mackey, Rapport de Recherches du LMENS 93–15 (1993).Google Scholar
  5. [BZ]
    J.N. Bernstein et A. Zelevinski, Induced representations of reductive p-adic groups I, Ann. Sci. Ec. Norm. Super. 10 (1977), 441–472.MATHGoogle Scholar
  6. [BMM]
    M. Broué, G. Malle et J. Michel, Generic Blocks of Finite Reductive Groups, Astérisque 212 (1993), 7–92.Google Scholar
  7. [C]
    R. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley-Interscience, 1985.Google Scholar
  8. [DL]
    P. Deligne et G. Lusztig, Representations of reductive groups over finite fields, Ann. of Maths 103 (1976), 103–161.MathSciNetMATHCrossRefGoogle Scholar
  9. [DH]
    D.I. Deriziotis et D.F. Holt, The Möbius function of the lattice of closed subsystems of a root system, Comm. in Algebra 21(5) (1993), 1543–1570.MathSciNetMATHCrossRefGoogle Scholar
  10. [DLM]
    F. Digne, G.I. Lehrer et J. Michel, The characters of the group of rational points of a reductive group with non– connected centre, J. reine angew. Math. 425 (1992), 155–192.MathSciNetMATHCrossRefGoogle Scholar
  11. [DM2]
    F. Digne et J. Michel, Representations of finite groups of Lie type, Cambridge University Press, Cambridge, 1990.Google Scholar
  12. [FJ]
    P. Fleischmann et I. Janiszczak, The Number of Regu lar Semisimple Elementsfor Chevalley Groups of Classical Type, J. of Algebra 155 (1993), 482–528.MathSciNetMATHCrossRefGoogle Scholar
  13. [G]
    M. Geek, A note on Harish-Chandra induction, manuscripta math. 80 (1993), 393–401.Google Scholar
  14. [GH]
    M. Geek et G. Hiss, Modular Representations of Finite Groups of Lietype in Non-defining Characteristic, ce volume.Google Scholar
  15. [GHM1]
    M. Geek, G. Hiss et G. Malle, Cuspidal unipotent Brauer characters, J. of Algebra 168 (1994), 182–220.CrossRefGoogle Scholar
  16. [GHM2]
    M. Geek, G. Hiss et G. Malle, Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type , Math.Z. 221 (1996), 353–386.MathSciNetGoogle Scholar
  17. [GM]
    M. Geek et G. Malle, Cuspidal unipotent classes and cuspidal Brauer characters, J. London Math. Soc. à paraître.Google Scholar
  18. [H]
    G. Hiss, Harish-Chandra series of Brauer characters in a finite group with a split BN-pair, J. London Math. Soc. 48 (1993), 219–228.MathSciNetMATHCrossRefGoogle Scholar
  19. [HL]
    R.B. Howlett et G.I. Lehrer, Induced cuspidal representations and generalized Hecke rings, Invent, math. 58 (1980), 37–64.MathSciNetMATHCrossRefGoogle Scholar
  20. [K]
    D. Kazhdan, Cuspidal geometry of p-adic groups, J. Analyse Math. 47 (1986), 1–36.MathSciNetMATHCrossRefGoogle Scholar
  21. [Le]
    G.I. Lehrer, Rational tori, semi-simple orbits, and the topol-ogy of hyperplanes complements, Comment. Math. Helvetici 67 (1992), 226–2516.MathSciNetMATHCrossRefGoogle Scholar
  22. [Ll]
    G. Lusztig, On the finiteness of the number of unipotent classes, Inventiones math. 34 (1976).Google Scholar
  23. [L2]
    G. Lusztig, Representations of finite Chevalley groups, Am. Math. Soc. CBMS 39 (1977).Google Scholar
  24. [L3]
    G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Stud., Princeton, 1984.Google Scholar
  25. [L4]
    G. Lusztig, Intersection cohomology complexes on a reduc tive group, Invent, math. 75 (1984), 205–272.MathSciNetMATHCrossRefGoogle Scholar
  26. G. Lusztig, Character Sheaves, Adv. in Math., 56 (1985), 193–237; 57 (1985), 226–265; 57 (1985), 266–315; 59 (1986), 1–63; 61 (1986), 103–155.MathSciNetMATHCrossRefGoogle Scholar
  27. [L6]
    G. Lusztig, Green functions and character sheaves, Annals of Math. 131 (1990), 355–408.MathSciNetMATHCrossRefGoogle Scholar
  28. [L7]
    G. Lusztig, A unipotent support for irreducible representations, Adv. in Math. 94 (1992), 139–179.MathSciNetMATHCrossRefGoogle Scholar
  29. [L8]
    G. Lusztig, Remarks on computing irreducible characters, J. of the Amer. Math. Soc. 5 (1992), 971–986.MathSciNetMATHCrossRefGoogle Scholar
  30. [LS]
    G. Lusztig et N. Spaltenstein, Induced unipotent classes, J. of London Math. Soc. (2) 19 (1979), 41–52.MathSciNetMATHCrossRefGoogle Scholar
  31. [R]
    R.W. Richardson, Conjugacy classes in parabolic subgroups, Bull. Lond. Math. Soc. 6 (1974), 21–24.MATHCrossRefGoogle Scholar
  32. [S]
    T. Shoji, Character sheaves and almost characters of reductive groups I, II, Adv. in Math 111, No 2 (1995), 244–313, 314–354.MathSciNetMATHCrossRefGoogle Scholar
  33. [V]
    M.F. Vignéras, An elementary introduction to the local trace formula of J. Arthur. The case of finite groups, Jubiläum band DMV (1991), B.G. Teubner Stuttgart.Google Scholar
  34. [W]
    J.-L. Waldspurger, Quelques questions sur les intégrales or bitales et les algèbres de Hecke, Bull. Soc. Math. France 124 (1996), 1–34.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Anne-Marie Aubert
    • 1
  1. 1.Département de Mathématiques et d’InformatiqueEcole Normale SupérieureParisFrance

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