Speaker: Yakov Varshavsky (Hebrew University)

Series Title: Harmonic analysis on p-adic groups and sheaves on loop groups.

Abstract:

Harmonic analysis on reductive p-adic groups (such as the group GL(n,Q_p) of invertible matrices with entries in the p-adic field) is a central part of Langlands program.

It follows from general topological properties that characters of irreducible representations form a topological basis in the space of invariant generalized functions on the group. Langlands' theory of endoscopy studies subtle properties of these characters stemming from algebro-geometric and arithmetic nature of the group. In particular, Langlands conjectured that some specific linear combinations of characters satisfy the so called E-stability property, which relates the values of the class function on two elements conjugate over the algebraic closure of the p-adic field.

After a general introduction to this circle of ideas I will present an approach to these conjecture based on the idea that the above linear combinations should come from geometry : they are expected to be closely related to perverse equivariant sheaves on the loop group. While perverse sheaves have become a standard tool in geometric representation theory, the present setting involves "truly infinite dimensional" spaces, the definition and basic properties of such sheaves in this setting were only worked out recently.

The above picture will be illustrated by specific results covering two particular cases: cuspidal Deligne-Lusztig representations and unipotent representations of GL(n,F). The results are partly motivated by conjectures proposed by Lusztig.

Most original results to be presented in the lectures are from joint works  with Bezrukavnikov, Bouthier and Kazhdan.