Finitude pour les representations lisses de groupes p-adiques

Jean-Francois Dat (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

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.

Mots clés: groupes p-adiques; algebres de Hecke

Domaines: Théorie des représentations [math.RT]; Théorie des nombres [math.NT]; Anneaux et algèbres [math.RA]

Jean-Francois Dat. Finitude pour les representations lisses de groupes p-adiques. 2006. ⟨hal-00085832⟩