On the K0 of a p-adic group

This article deals with various topics related with Grothendieck groups, invariant distributions, parabolic and compact inductions... for a p-adic group G. The main result is a description of the K0 of the Hecke algebra ℋ of G in terms of discrete series of Levi subgroups, which has an interesting behavior with regard to parabolic restriction and induction. A similar description – but no more compatible with these parabolic functors – is obtained for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\overline{\mathcal{H}}$\end{document}=ℋ/[ℋ,ℋ] and the Hattori rank map gets an easy description in this dictionary.¶We follow a beautiful idea of J. Bernstein consisting in comparing two natural filtrations on these objects, one of combinatorial nature and one of topological nature. The combinatorial filtrations are related to the structure of Levi subgroupsin G and have counterparts concerning many classical objects of interest as the Grothendieck group of finite length G-modules R(G), the set Ωsr of regular semi-simple conjugacy classes, and the variety Θ(G) of infinitesimal characters. These filtrations will turn out to be “compatible”, in a sense to be specified, with regard to all the classical operations or morphisms between these objects.