| Résume | 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 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 comparison theorems between derived categories of modules of graded
Hecke algebras and derived categories of equivariant sheaves.
We can apply that in the local Langlands program, conjecturally for all
reductive p-adic groups and certainly for some well-known groups. From that we
obtain a comparison between the derived category of finite length smooth G-representation
and derived categories of equivariant constructible sheaves on complex varieties related to
L-parameters. |
