On a geometric parametrization of smooth irreducible p-adic representations

 Let Irr(G) denote the set of smooth irreducible complex representations of a reductive p-adic group G. The Bernstein decomposition theorem provides us with a natural partition of \Pi(G) into Bernstein components, and the irreducible representations on each component are in bijection with those of a Hecke algebra.

In this talk, I will explain how to construct the Lafforgue variety Laf(G), an infinite disjoint union of affine schemes equipped with an open dense subscheme whose geometric points parametrize \Pi(G). I will also show it comes equipped with a finite morphism to the Bernstein variety.

Time allowing, I will also report on related work in progress studying how the Lafforgue variety deforms when changing the parameter of an affine Hecke algebra. This is related to a conjecture by Aubert, Baum and Plymen.