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 Fréchet completions thereof. Secondly, a refined form of the Langlands classification for G, which clarifies the relation between the smooth spectrum and the tempered spectrum.