Finitude pour les representations lisses de groupes p-adiques

Abstract

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are noetherian whenever R is ; a question left open since Bernstein's fundamental work for R=C. In a first step, we prove that this noetherian property would follow from a generalization of the so-called Bernstein's second adjointness property between parabolic functors for complex representations. Then, to attack this second adjointness, we introduce and study "parahoric functors" between representations of groups of integral points of smooth integral models of G and of their "Levi" subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. According to recent announcements by Kim and Yu, the same strategy should also work for "tame groups", using Yu's generic characters.