Periodic cyclic homology of reductive p-adic groups

Abstract

Let G be a reductive p-adic group, H (G) its Hecke algebra and S (G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This provides an alternative proof of the Baum-Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly, a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Frechet completions thereof. Secondly, a refined form of the Langlands classification for G, which clarifies the relation between the smooth spectrum and the tempered spectrum.