FINITENESS FOR SMOOTH REPRESENTATIONS OF p-ADIC GROUPS

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; this question arose naturally with Bernstein's fundamental work for R = C, in which case lie proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a. generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein 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 C 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. The same strategy should apply to 'tame' groups, using Yu's smooth models and generic characters.