Xuhua He, HKU

Positivity of Canonical Bases

Lusztig's theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. In symmetric types, it is known that the structure constants for multiplication in the negative part $U^-$, as well as for the action of Chevalley generators $E_i$ and $F_i$ on a single simple module, all belong to $\mathbb{N}[v, v^{-1}]$.

Lusztig conjectured that this strong positivity holds for the multiplication within the modified quantum group and the action on the tensor product of modules. In this talk, I will present recent joint work with Jiepeng Fang towards this conjecture.

A key innovation in our approach is the "thickening philosophy", an algebraic technique inspired by geometric ideas from total positivity, building on my earlier work with Huanchen Bao. This method embeds a suitable approximation of the tensor product into the negative part $\tilde{U}^-$ of a larger quantum group, constructed via a framed quiver. This allows us to inherit the desired positivity directly from the well-established positivity of the canonical basis of $\tilde{U}^-$. This approach demonstrates how the large Kac-Moody groups can provide a powerful framework for elucidating the structure and representations of quantum groups even for the finite and affine types.

Yujie Xu (Columbia U)

On Hecke algebras for p-adic groups and applications to the Langlands correspondence

I will talk about several results on Hecke algebras attached to Bernstein blocks of arbitrary reductive p-adic groups, and their applications to the local Langlands program. One such application is an explicit understanding of the (classical) arithmetic Local Langlands correspondence with explicit L-packets. Another such application involves various categorical "upgrades", for example, an equivalence between module categories of Hecke algebras arising from both the automorphic and the spectral sides. If time permits, I will talk about a categorical local Langlands correspondence featuring certain coherent Springer sheaves on moduli spaces of Langlands parameters, and prove a fully faithful functor from the automorphic side to the spectral side. 

Allen Moy (HKUST emeritus)

Construction of essentially compact invariant idempotent distributions on a p-adic reductive Lie algebra

Suppose G is a p-adic reductive group.  The Bernstein center is the analogue for G of the center of the universal enveloping algebra of a reductive Lie group.  For example, the Bernstein center acts by scalars on any irreducible representation, and as such is of great importance in understanding the representations of G.

One realization of the Bernstein center of G is as the algebra of essentially compact G-invariant distributions on G.  An essentially compact distribution is one whose convolution with any locally constant compactly supported function is again compactly supported.  An example is the delta function at an element of G.  Here, only central elements yield a G-invariant distribution.

One can also consider the algebra of essentially compact G-invariant distributions on the Lie-algebra Lie(G).  For a semisimple element Y in the dual (Lie(G))^{*}, we attach a canonical embedding of the Bruhat-Tits building B(C_{G}(Y)) of the centralizer C_{G}(Y) of Y into the Bruhat-Tits building B(G).  This embedding is compatible with the MP-filtrations on the two groups.  To the G-orbit of this embedding, we consider a Euler-Poincaré sum over the facets of a refinement of the simplicial structure on B(G).  We conjecture this Euler-Poincaré sum is a G-invariant essentially compact distribution on Lie(G) and this construction provides a basis of G-invariant idempotent distributions with support in the depth 0^{+} (topologically nilpotent) elements of the Lie algebra.  

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.