I will begin with Langlands duality, the origin (in the local Langlands conjecture) of the Langlands-Deligne-Lusztig parameters (t,u,\rho), and the concept of an L-packet.
I will then move on to affine Hecke algebras H(W,q). Each parameter t is the central character of an induced H(W,q)-module.
I'll define the extended quotient, and describe a (conjectural) geometric structure for the set of simple H(W,q)-modules. This conjecture was inspired by the theorem of Baum & Nistor for the periodic cyclic homology of H(W,q).
This will be illustrated in some detail by the affine Hecke algebra attached to the exceptional group G_2.
[Joint work with Anne-Marie Aubert and Paul Baum]