The Lafforgue variety and its deformation theory — Kostas Psaromiligkos (Université Clermont Auvergne)

The category of smooth admissible representations of a p-adic reductive group G admits a concrete splitting by the Bernstein decomposition theorem into blocks indexed by the connected components of the Bernstein variety.

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 the smooth irreducible representations of a p-adic reductive group. The construction will proceed by constructing the Lafforgue variety for Hecke algebras of Bernstein components.

I will also show the Lafforgue variety deforms flatly under regular deformations of the parameters of an affine Hecke algebra, when the center of the algebra is regular. This is related to a conjecture by Aubert, Baum and Plymen.