Periodic cyclic homology of reductive $p$-adic groups

Abstract

Let $G$ be a reductive $p$-adic group, $\mathcal{H}(G)$ its Hecke algebra and $\mathcal{S}(G)$ 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.