Pro-<Emphasis Type="Italic">p</Emphasis>-Iwahori Hecke ring and supersingular -representations |
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Authors: | Marie-France Vignéras |
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Institution: | (1) Institut de Mathématiques de Jussieu, Université de Paris 7-Denis Diderot, 2 place Jussieu, 75251 Paris Cedex 05, France |
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Abstract: | The motivation of this paper is the search for a Langlands correspondence modulo p. We show that the pro-p-Iwahori Hecke ring
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
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
to k which is “as null as possible” will be called null. The simple
-modules with a null central character are called supersingular. When G=GL(n), we show that each simple
-module of dimension n containing a character of the affine subring
is supersingular, using the minimal expressions of Haines generalized to
, 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).
An erratum to this article can be found at |
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Keywords: | |
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