We will begin by recalling the periodic cyclic homology of the Hecke algebra of GL(n). The Langlands parameters occur naturally at this point [joint work with J Brodzki].
This has led us to the formulation of a conjecture, according to which there are simple geometric structures underlying the representation theory of p-adic groups.
We will illustrate this conjecture with examples, including GL(n) and the exceptional group G_2.
We will attempt to relate our conjecture to the Langlands-Deligne-Lusztig parameters (s,u,\rho).
[joint work with Anne-Marie Aubert and Paul Baum].