Abstract: In the 60s, Gel'fand defined a Bessel function associated with irreducible generic representations of GL(n, F_q). The Bessel function plays an important role in the representation theory of GL(n, F_q), and served as a key ingredient in the proof of the finite field analog of Jacquet's conjecture. However, the computation of explicit values of the Bessel function is difficult. In this talk, I'll explain a method based on gamma factors to compute values of the Bessel function. I will also explain a relation between some values of the Bessel function and Katz's generalized Kloosterman sheaves. 

Abstract: I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan,

arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists, Kontsevich, Langlands, Teschner, and Gaiotto-Witten. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun_G of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). 

I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.

Abstract: A compact hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. An elliptic fibration of a K3 surface correspondingly generalizes to the so-called Lagrangian fibration of a compact hyper-Kähler manifold. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, I will propose a construction for the dual Lagrangian fibration of all currently known examples of compact hyper-Kähler manifolds, and try to justify this construction.

Abstract: Classical holomorphic modular forms are number-theoretic objects that have been intensely studied.  The split exceptional group G_2 does not support a theory of holomorphic modular forms, but it does possess so-called quaternionic modular forms.  These are a special class of automorphic forms that appear to behave similarly to holomorphic modular forms.  In the talk, I will describe a theory of modular forms of half-integral weight on G_2 and other exceptional groups.  In particular, we prove the existence of a modular form of weight 1/2 on G_2 whose Fourier coefficients are related to the 2-torsion in the narrow class groups of totally real cubic fields.  This is joint work with Spencer Leslie. 

It has long been expected, due to work and conjectures of Beilinson--Drinfeld and Frenkel--Gaitsgory, that one has a localization theorem for representations of affine Lie algebras at critical level with unramified central characters as certain D-modules on the affine Grassmannian. We have proven this in a joint work with Sam Raskin. After reviewing the motivation and context for this conjecture, we will describe some new methods used in the proof. 

Abstract: For a class of quotient stacks V/G for G a reductive group and V a symmetric G-representation, we propose a definition of the intersection K-theory of the quotient V//G, which is a Q-vector space denoted IK(V//G).  There is a Chern character from IK(V//G) to intersection cohomology IH(V//G) and IK(V//G) satisfies Kirwan surjectivity. We construct IK(V//G) as a direct summand of the Grothendieck group (tensor with Q) of a noncommutative resolution of singularities of V//G constructed by Spenko-Van den Bergh. We explain extensions of some of these results for stacks with good moduli spaces. We discuss an application of this construction to a PBW-type theorem for (Kontsevich-Soibelman) K-theoretic Hall algebras of quivers with potential.  

Abstract: Let D be the sheaf of differential operators on a complex manifold M. If we turn on the Planck parameter, the ring D becomes a deformation quantization of the cotangent bundle of M. There is an analogue to this operation on the Betti side. Namely, we can "turn on the Planck parameter” for constructible sheaves. The resulting concept is called sheaf quantization. This concept (originally introduced by Tamarkin) has proven useful in many areas. In this talk, I will give an introduction to the concept, including applications to symplectic topology, WKB analysis, RH correspondence, etc. 

Abstract: I will talk about the problem of classifying local systems of geometric origin on algebraic varieties over complex numbers, from the point of view of arithmetic.

Conjecture: For a smooth algebraic variety S over a finitely generated field F, a semi-simple Q_l-local system on S_{\bar{F}} is of geometric origin if and only if it extends to a local system on S_{F'} for a finite extension F'\supset F

My main goal will be to provide motivation for this conjecture arising from the properties of the p-adic Riemann-Hilbert correspondence and the Fontaine-Mazur conjecture, and survey known partial results.

Abstract: We construct a new family of irreducible representations of \mathcal{U}_q(\mathfrak{g}_\mathbb{R}) and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of \mathcal{U}_q(\mathfrak{g}_\mathbb{R}) act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations introduced by [Frenkel-Ip] which correspond to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type A_n acting on L^2(\mathbb{R}^n) with minimal functional dimension, and establish the properties of its central characters and universal \mathcal{R} operator. Finally we will explain a positive version of the evaluation module of the affine quantum group \mathcal{U}_q(\widehat{\mathfrak{sl}}_{n+1}) modeled over this minimal positive representation of type A_n.

Abstract: We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth partial compactification of Z by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

Abstract: We discuss topologically twisted five-dimensional SU(2) supersymmetric Yang-Mills theory on M4 x S1, where M4 is a smooth closed four-manifold. We provide two different path integral derivations of certain correlation functions, which can be identified with the K-theoretic version of the Donaldson invariants. In particular we derive their wall-crossing formula, first in the so-called U-plane integral approach and in the perspective of instanton counting. The result reproduces and generalizes the work of Gottsche, Nakajima, and Yoshioka in 2006.

Abstract: Descendent classes on moduli spaces of sheaves are defined via the Chern characters of the universal sheaf.

I will present several conjectures and results concerning Viraoro constraints for integrals of the descendent classes.

Abstract: We prove a gluing theorem on the automorphic side of the geometric Langlands correspondence: roughly speaking, we show that the difference between DMod(BunG) and its full subcategory of tempered objects is compensated by the categories of tempered objects in DMod(BunM) for all standard Levi subgroups M \subset G. This theorem is designed to match exactly with the spectral gluing theorem, an analogous result occurring on the other side of the geometric Langlands conjecture, and is supposed to be an important step in the proof of the conjecture. This is joint work with Dario Beraldo.