Pro-p-Iwahori Hecke ring and supersingular -representations

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).