Pro-p-Iwahori Hecke algebra 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 H(1) of a split reductive p-adic group G over a local field F of finite residue field Fq with q elements, admits an Iwahori-Matsumoto presentation and a Bernstein Z-basis, and we determine its centre. We prove that the ring H(1) is finitely generated as a module over its centre. These results are proved in [12] 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 H(1) to k which is “as null as possible” will be called null. The simple H k -modules with a null central character are called supersingular. When G = GL(n), we show that each simple H k -module of dimension n containing a character of the affine subring H aff is supersingular. The proof uses the minimal expressions of Haines generalized to H(1), and that the number of such modules is equal to the number of irreducible k-representations of the Weil group WF of dimension n (when the action of an uniformizer pF in the Hecke algebra side and of the determinant of a Frobenius FrF in the Galois side are fixed), i.e. the number Nn(q) of unitary irreducible polynomials in Fq[X] of degree n. One knows that the converse is true by explicit computations when n = 2 [11], and when n = 3 (Rachel Ollivier).