Workshop

The goal of this talk is to formalize the spectral side of the geometric Langlands equivalence. We define the stack LocSys of de Rham local systems and prove its basic properties. Beilinson and Drinfeld's "best hope" for the geometric Langlands equivalence uses quasi-coherent sheaves on LocSys, but that category is too small for the equivalence. We therefore enlarge QCoh(LocSys) to the category of ind-coherent sheaves on LocSys with nilpotent singular support.

Motivated by Lusztig's definition of character sheaves on a reductive Lie algebra, we propose a definition of character sheaves on the loop Lie algebra. The construction can be viewed as a p-adic analogue of the orbit method, and the resulting sheaves are closely related to characters of supercuspidal representations of J.K.Yu. Further, the construction suggests a definition of wildly ramified Hitchin moduli spaces suitable for the study of ramified geometric Langlands. This is joint work with Bao Chau Ngo.

I will some (partly conjectural) ideas relating the following topics: (i) Smith-Treumann localization for sheaves on affine flag varieties; (ii) Kazhdan-Lusztig cells and $p$-cells in the affine Weyl group; and (iii) cohomology for (higher) Frobenius kernels. This talk is based on joint work in progress with R. Bezrukavnikov, P. Boixeda Alvarez, and S. Riche.