Graded Hecke algebras and equivariant sheaves in the local Langlands program

Maarten Solleveld
Radboud Universiteit Nijmegen, The Netherlands

Date(s) : 02/05/2022   iCal
14 h 00 min - 15 h 00 min

It has been conjectured that the local Langlands correspondence for a reductive p-adic group G (itself also partly conjectural) can be categorified. Then it should relate the category of complex smooth G-representations with a category of equivariant sheaves on a variety of Langlands parameters for G. If it exists, such a categorification will probably arise via Hecke algebras.

In this talk we will discuss several steps in this direction. Our main players will be graded Hecke algebras, which appear both on the p-adic side and on the Galois side of the local Langlands program. We will see that graded Hecke algebras can not only be constructed in terms of generators and relations, but also geometrically, as endomorphism algebras of certain equivariant constructible sheaves. That leads to to geometric constructions of irreducible Hecke algebra modules and to comparison theorems between derived categories of equivariant sheaves and derived module categories of graded Hecke algebras.

We can apply these Hecke algebras techniques in the local Langlands program, conjecturally for all reductive p-adic groups and certainly for some well-known groups. That provides a description of derived categories of equivariant constructible sheaves on suitable varieties of Langlands parameters.


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