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Fibration de Hitchin et endoscopie

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Abstract

We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and MacPherson. For unitary groups, this global purity statement has been used, in a joint work with G. Laumon, to prove the fundamental lemma over a local fields of equal characteristics.

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Références

  1. Beauville, A., Laszlo, Y.: Un lemme de descente. C. R. Acad. Sci., Paris 320, 335–340 (1995)

    MATH  MathSciNet  Google Scholar 

  2. Beauville, A., Narasimhan, M., Ramanan, S.: Spectral curves and generalized theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)

    MATH  MathSciNet  Google Scholar 

  3. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1982)

    MathSciNet  Google Scholar 

  4. Biswas, I., Ramanan, S.: Infinitesimal study of Hitchin pairs. J. Lond. Math. Soc. 49, 219–231 (1994)

    MATH  Google Scholar 

  5. Bosch, S., Lutkebohmert, W., Raynaud, M.: Neron models. Ergeb. Math. 21, 1–325 (1990)

    MathSciNet  Google Scholar 

  6. Deligne, P.: La conjecture de Weil II. Publ. Math., Inst. Hautes Étud. Sci. 52, 137–252 (1980)

    MATH  MathSciNet  Google Scholar 

  7. Faltings, G.: Stable G-bundles and projective connections. J. Algebr. Geom. 2, 507–568 (1993)

    MATH  MathSciNet  Google Scholar 

  8. Donagi, R., Gaitsgory, D.: The gerb of Higgs bundles. Transform. Groups 7, 109–153 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goresky, M., Kottwitz, R., MacPherson, R.: Homology of affine Springer fiber in the unramified case. Duke Math. J. 121, 509–561 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique : IV. Étude locale des schémas et de morphismes de schémas, chapîtres III et IV. Publ. Math., Inst. Hautes Étud. Sci. 28, 5–255 (1966) et 32, 5–361 (1967)

    Google Scholar 

  11. Hitchin, N.: Stable bundles and integrable connections. Duke Math. J. 54, 91–114 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kostant, B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)

    MATH  MathSciNet  Google Scholar 

  13. Kottwitz, R.: Isocristal with additionnal structures. Compos. Math. 56, 201–220 (1985)

    MATH  MathSciNet  Google Scholar 

  14. Kottwitz, R.: Stable trace formula: cuspidal tempered terms. Duke Math. J. 51, 611–650 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kottwitz, R.: Stable trace formula: elliptic singular terms. Math. Ann. 275, 365–399 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kottwitz, R.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 2, 373–444 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kottwitz, R.: Transfert factors for Lie algebra. Represent. Theory 3, 127–138 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Langlands, R.: Les débuts d’une formule des traces stables. Publ. Univ. Paris 7, vol. 13 (1983)

    Google Scholar 

  19. Laumon, G.: Fibres de Springer et Jacobiennes compactifiées. Prépublication arXiv

  20. Laumon, G., Ngô, B.C.: Le lemme fondamental pour les groupes unitaires. Prépublication arXiv

  21. Rapoport, M.: A guide to the reduction of Shimura varieties. Astérisque 298, 271–358 (2005)

    MATH  MathSciNet  Google Scholar 

  22. Serre, J.-P.: Cohomologie galoisienne. Lect. Notes Math., vol. 5. Springer 1973

  23. Springer, T.: Reductive groups. In: Automorphic Forms, Representations, and L-Functions. Proc. Symp. Pure Math. vol. 33-1, pp. 3–27. Am. Math. Soc. 1979

  24. Veldkamp, F.D.: The center of the universal enveloping algebra of a Lie algebra in characteristic p. Ann. Sci. Éc. Norm. Supér. 5, 217–240 (1972)

    MATH  MathSciNet  Google Scholar 

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  1. Département de Mathématiques, UMR 8628, Université Paris-Sud, 91405, Orsay, France

    Bao Châu Ngô

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  1. Bao Châu Ngô
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Correspondence to Bao Châu Ngô.

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Ngô, B. Fibration de Hitchin et endoscopie. Invent. math. 164, 399–453 (2006). https://doi.org/10.1007/s00222-005-0483-7

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  • DOI: https://doi.org/10.1007/s00222-005-0483-7

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