Tsao-Hsien Chen, Minnesota

Bernstein centers of p-adic groups and perverse sheaves

I will explain a description of the  Bernstein center of a split p-adic reductive group as a limit of parahoric Hecke algebras, extending the previous work of Bezrukavnikov-Kazhdan-Varshavsky on explicit descriptions of depth-r Bernstein projectors. Then I will explain how such a  description allows us to apply the theory of perverse sheaves (e.g. character sheaves on reductive groups, graded Lie algebras, parahorics or Gaitsgory's central sheaves) to study  Bernstein centers. I will discuss some examples and applications, including a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-r Langlands parameters. Time permitting, I will mention connections with Stable Center Conjecture in Local Langlands correspondence.

The talk is based on joint works with Sarbartha Bhattacharya, Charlotte Chan, Stephen Debacker and Cheng Chiang Tsai.

Joakim Faergeman, Yale

Motivic realization of rigid local systems on curves via geometric Langlands

A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. Simpson conjectured that for a reductive group G, rigid G-local systems with suitable conditions at infinity are motivic. This was proven for curves when G = GL_n by Katz who classified such rigid local systems. In this talk, we prove Simpson's conjecture for curves for an arbitrary reductive group G. Our proof goes through the (tamely ramified)  geometric Langlands program in characteristic zero. If time permits, we state a generalization of Simpson's conjecture to an arbitrary smooth projective variety.

Sanath Devalapurkar, Chicago

Koszul duality for ring spectra

Koszul duality relates (graded variants of) B-constructible sheaves of complex vector spaces on the flag variety G/B with B^-constructible sheaves of complex vector spaces on G^/B^, where G^ is the Langlands dual group. This swaps equivariance on one side with monodromicity on the other side. What happens when we replace complex vector spaces by the category of k-modules for a more general commutative ring k? What if k is a commutative ring *spectrum*?

In this talk, I will explain a conjecture about an analogue/generalization of this result which relates the category of B-constructible sheaves of k-modules on G/B with the category of monodromic "F-D-modules" on G^/B^. When k is an ordinary commutative ring, "F-D-modules" are just usual D-modules, and when k is complex K-theory, "F-D-modules" are q-D-modules. (For general k, I will give an explicit definition of this category of "F-D-modules" at least for  G^ = SL_2; I don't know how to define it for general G^, unless k is an ordinary commutative ring or complex K-theory, and in the latter case the definition is via prismatization.)

Pam Gu, Michigan

On values of Bessel functions for generic principal series representations of finite groups

For a connected split reductive group G over a finite field, an irreducible generic representation of G admits a distinguished matrix coefficient known as the Bessel function. While such functions have been extensively studied for G=GL_n, much less is known for other groups. In this talk, I will present new results establishing a connection between Bessel functions and Kloosterman sheaves constructed by Heinloth-Ngô-Yun for a broad class of groups. Specifically, for generic principal series representations of G, we show a relation between values of the associated Bessel functions at certain Weyl group elements corresponding to a maximal cominuscule parabolic subgroup of G and the trace of Frobenius acting on the geometric stalk of the minuscule representation of a corresponding Kloosterman sheaf. This is joint work with Robert Cass and Elad Zelingher.