Abstract. Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GLn(F ) and its derived subgroup SLn(F ). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL2(Qp) and smooth mod-p representations of SL2(Qp) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SLn(F ), and irreducible, n-dimensional projective Galois representations.

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Abstract. Let p be an odd prime number, and F a nonarchimedean local field of residual characteristic p. We classify the smooth, irreducible, admissible genuine mod-p representations of the twofold metaplectic cover Sp 2n (F ) of Sp 2n (F ) in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart-Vignéras as well as new technical results to adapt Herzig's method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations generalizing the complete irreducibility of principal series representations in the rank 1 case, as well as the fact that irreducibility is preserved by parabolic induction from the cover of the Siegel Levi subgroup.

Let p ≥ 5 be a prime number, G a split connected reductive group defined over a p-adic field, and I1 a choice of pro-p-Iwahori subgroup. Let C be an algebraically closed field of characteristic p and H the pro-p-Iwahori-Hecke algebra over C associated to I1. In this note, we compute the action of H on H 1 (I1, C) and H top (I1, C) when the root system of G is irreducible. We also give some partial results in the general case.

Suppose that G is a connected reductive group over a finite extension F/Qp, and that C is a field of characteristic p. We prove that the group G(F ) admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over C.