Number Theory Seminar: Describing the cohomology of congruence subgroups via categorical Langlands

In this talk I will present a conjectural description of the compactly supported cohomology of locally symmetric spaces arising as congruence quotients in terms of the coherent cohomology of stacks of global Galois representations with coefficients in certain sheaves constructed via categorical local Langlands correspondences. This is joint work with Xinwen Zhu in the Hermitian (i.e. Shimura variety) case, and with Dougal Davis and Kari Vilonen in the general case.

My hope in the talk is to illustrate the conjecture by considering explicit examples in which everything can be computed (granting various expected properties of the relevant Galois representations and their cohomology).