Periodic cyclic homology of reductive p-adic groups

TL;DR

This paper demonstrates that the Hecke algebra and Schwartz algebra of a reductive p-adic group share the same periodic cyclic homology, offering potential new approaches to the Baum-Connes conjecture.

Contribution

It establishes an equivalence of periodic cyclic homology for key algebras associated with p-adic groups and introduces new comparison and classification results.

Findings

Hecke and Schwartz algebras have identical periodic cyclic homology.

A general comparison theorem for finite type algebras and Fréchet completions.

A refined Langlands classification relating smooth and tempered spectra.

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 might be used to provide 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 Fr\'echet completions thereof. Secondly a refined form of the Langlands classification for G, which clarifies the relation between the smooth spectrum and the tempered spectrum.