  Let G be any reductive p-adic group. We discuss several conjectures, some of
them new, that involve the representation theory and the geometry of G.
  At the heart of these conjectures are statements about the geometric
structure of Bernstein components for G, both at the level of the space of
irreducible representations and at the level of the associated Hecke algebras.
We relate this to two well-known conjectures: the local Langlands
correspondence and the Baum--Connes conjecture for G. In particular, we present
a strategy to reduce the local Langlands correspondence for irreducible
G-representations to the local Langlands correspondence for supercuspidal
representations of Levi subgroups.
