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Mathematische Annalen

, Volume 331, Issue 3, pp 523–556 | Cite as

Pro-p-Iwahori Hecke ring and supersingular Open image in new window -representations

  • Marie-France Vignéras
Article

Abstract.

The motivation of this paper is the search for a Langlands correspondence modulo p. We show that the pro-p-Iwahori Hecke ring Open image in new window of a split reductive p-adic group G over a local field F of finite residue field F q with q elements, admits an Iwahori-Matsumoto presentation and a Bernstein Z-basis, and we determine its centre. We prove that the ring Open image in new window is finitely generated as a module over its centre. These results are proved in [11] only for the Iwahori Hecke ring. Let p be the prime number dividing q and let k be an algebraically closed field of characteristic p. A character from the centre of Open image in new window to k which is “as null as possible” will be called null. The simple Open image in new window -modules with a null central character are called supersingular. When G=GL(n), we show that each simple Open image in new window -module of dimension n containing a character of the affine subring Open image in new window is supersingular, using the minimal expressions of Haines generalized to Open image in new window , and that the number of such modules is equal to the number of irreducible k-representations of the Weil group W F of dimension n (when the action of an uniformizer p F in the Hecke algebra side and of the determinant of a Frobenius Fr F in the Galois side are fixed), i.e. the number N n (q) of unitary irreducible polynomials in F q [X] of degree n. One knows that the converse is true by explicit computations when n=2 [10], and when n=3 (Rachel Ollivier).

Keywords

Prime Number Local Field Explicit Computation Central Character Irreducible Polynomial 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Marie-France Vignéras
    • 1
  1. 1.Institut de Mathématiques de JussieuUniversité de Paris 7-Denis DiderotParis Cedex 05France

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