  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.
