Ekaterina Bogdanova, Non-vanishing of quantum geometric Whittaker coefficients
Abstract: We will discuss the functor of geometric Whittaker coefficients in the context of quantum geometric Langlands program. Concretely, we will prove (modulo the spectral decomposition conjecture) that the functor of quantum geometric Whittaker coefficients is conservative on the category of cuspidal automorphic D-modules. The proof will combine generalizations of representation-theoretic and microlocal methods from the preceding works of Faergeman-Raskin and Nadler-Taylor respectively.
Sanath Devalapurkar, Ruminations about Langlands duality with generalized coefficients
Abstract: In this talk, I will discuss some conjectures (and supporting results) surrounding studying (local) geometric Langlands duality – in particular, the derived geometric Satake equivalence -- with coefficients in ring spectra. The latter can be thought of as generalized cohomology theories with a well-behaved "chain level" cup product. One of the most profound realizations of homotopy theory in the past few decades, originating with Quillen and Morava, has been the tight relationship between ring spectra and the theory of 1-dimensional formal groups. Motivated by this, I will explain a conjecture which aims to describe how the spectral side of the derived geometric Satake equivalence depends on the choice of ring spectrum coefficients, which involves the corresponding 1-dimensional formal group in a crucial way. Time permitting, I will explain a construction (coming from this generalized version of geometric Satake) of an associative algebra associated to any 1-dimensional formal group,
which specializes to the usual enveloping algebra of SL_2 for the additive formal group (corresponding to Satake with coefficients in ordinary rings), and to a variant of the quantum group U_q(SL_2) for the multiplicative formal group (corresponding to Satake with coefficients in topological K-theory). This algebra has many interesting properties, which I hope to explain in the talk.
Gurbir Dhillon, On modular representations of affine Lie algebras
Abstract: While the highest weight representation theory of affine Lie algebras in characteristic zero is by now fairly well understood, far less has been known in characteristic p > 0. We will report on some basic results (partly in progress) in this direction, e.g. the Harish--Chandra center of the enveloping algebra at all levels, the linkage principle, an analogue of the Kac--Kazhdan conjecture, and a local de Rham geometric Langlands correspondence. This is based on joint work with Ivan Losev.
Joakim Faergeman, Singular support for G-categories and applications W-algebras
Abstract: We introduce a notion of singular support for categorical representations of a reductive group and give a characterization of the (nilpotent) singular support in terms of the vanishing of the Whittaker models. We apply this theory to reprove a theorem of Losev-Ostrik and Bezrukavnikov-Losev on the classification of finite-dimensional representations of W-algebras.
Tony Feng, Modular functoriality in the Local Langlands correspondence
Abstract: I will talk about some results towards Langlands functoriality in the Fargues-Scholze correspondence between a group and its fixed subgroup under a prime order automorphism, in defining characteristic. In particular, using tools from Smith theory and modular representation theory, I establish some conjectures of Treumann-Venkatesh giving a generalization of the Jacquet functor to such situations. This also has applications to explicitly calculating the Fargues-Scholze parameters of supercuspidal representations.
Jessica Fintzen, An introduction to the representation theory of p-adic groups
Abstract: An explicit understanding of the (category of all smooth, complex) representations of p-adic groups provides not only an important tool for the representation theory of these groups and the construction of an explicit local Langlands correspondence, it also provides inspiration for categorical and geometric version of the Langlands correspondence and allows for global applications.
This mini-course will start by recalling some fundamental notions about representations of p-adic groups and then likely cover the following topics.
(a) It will provide an overview of the construction of supercuspidal representations, which are the building blocks of all representations of p-adic groups.
(b) We will see that the whole category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks, and that each block can be described explicitly in terms of Hecke algebras. We will also see that (under minor tameness assumptions), arbitrary Bernstein blocks are equivalent to depth-zero Bernstein block, which are those that correspond (roughly) to representations of finite groups of Lie type.
(c) In order to do (a) and (b) above, the mini-course will first provide an introduction to the Moy-Prasad filtration and Bruhat-Tits theory.
Dmitry Kubrak, Сan one compute the singular cohomology of BG via derived Satake?
Abstract: Derived geometric Satake equivalence with Z-coefficients remains conjectural in general, but is strongly believed to be true. One of its consequences would be a formula for singular Z-cohomology of BG as an E_3-algebra in terms of E_2-Hochschild cohomology HH_{E_2}(G^v) of the Langlands dual group G^v. Even just computing the individual cohomology groups H^*_sing(BG,Z) is in general a difficult problem, and for some G (e.g. PGL_n) the answer is not known. If we replace G^v in HH_{E_2}(G^v) by the Lie algebra g^v, we get a more familiar formula as (derived) G^v-invariants in "Sym^* g^v", and one can ask whether this still provides a formula for the singular cohomology of BG. I will explain that the answer is no in general, and it fails in particular for Spin_n if n=3,4,5 modulo 8. The proof uses another, p-adic Hodge-theoretic, deformation of RГ_sing(BG,Z/2) which turns out to have the same associated graded and is given explicitly by mod 2 prismatic cohomology. If time permits, I will also discuss some non-trivial questions about the E_3-structure on RГ_sing(BG,Z/2) in the case when groups H^*_sing(BG,Z) are not so interesting.
Xinchun Ma, Cherednik algebras and Hilbert schemes
Abstract: For every nonnegative integer m, we construct a coherent sheaf F_{m,n} on the Hilbert scheme of n points on the plane. The K-theory class of F_{m,n} computes the bigraded character of the (y-ified) weighted sum of minimally supported modules (in the sense of Etingof–Gorsky–Losev) for the type-A rational Cherednik algebra at slope m/n. This computation is conjectured to recover the Euler characteristic of the (y-ified) Khovanov–Rozansky homology of the (m, n)-torus link, which has been confirmed when m is coprime to n. The sheaf F_{m,n} arises as the associated graded of a D-module on sl_n. When m is divisible by n, F_{m,n} is a twist of the Procesi bundle.
Hiraku Nakajima, S-dual of Hamiltonian G spaces and relative Langlands duality
Abstract: In the first talk, I will explain a mathematically rigorous definition of Coulomb branches of 3d SUSY gauge theories given jointly with Braverman and Finkelberg, and then explain a definition of S-dual Hamiltonian spaces, as its variant. In the second and third talks, I will give examples of S-duals, which are related to bow varieties and their variants. In the meantime, I will explain the relevance of S-dual in relative Langlands by Ben-Zvi, Sakellaridis, and Venkatesh.
Taeuk Nam, Tamely Ramified Geometric Langlands
Abstract: The unramified Geometric Langlands Correspondence is a categorical equivalence between D-mod(Bun_G) and IndCoh_Nilp(LocSys_G^\vee). The analogue of ramification in the geometric setting is level structure; in other words, instead of Bun_G, we instead consider the stack of principal G-bundles with H-level structure - that is, a reduction of structure group of the G-bundle restricted to a formal disk from G(O) to a subgroup H of G(O) - which we denote Bun_G^H. This talk will be about a tamely ramified version of the GLC, where we take H to be the Iwahori subgroup. We will discuss recent progress on this project, as well as what remains to be done.
Sam Raskin, Geometric Langlands for projective curves
Abstract: The geometric Langlands equivalence asserts that (suitable) sheaves on the space of G-bundles on X and on the space of \check{G}-local systems are equivalent categories. I will provide some background on the conjecture(s) and highlight some aspects of its proof.
Kenta Suzuki, Fargues' categorical conjecture for elliptic parameters for SL(n)
Abstract: (partially joint work with Eunsu Hur) Fargues and Scholze give a geometric construction of L-parameters attached to smooth irreducible representations of p-adic groups and furthermore predict an enhancement to a category equivalence. I will explain two approaches to prove Fargues and Scholze's functor is an equivalence on elliptic parameters for SL(n). The first approach is character-theoretic, using Fu's recent result on the stability of Fargues-Scholze's L-packets. The second approach follows Gaitsgory and Raskin's proof of the geometric Langlands conjectures for groups with disconnected center, using the 2-Fourier-Mukai transform. As a consequence, we prove Fargues and Scholze's construction gives a bijection between supercuspidal representations of SL(n) and (enhanced) elliptic L-parameters.
Jeremy Taylor, Universal monodromic Hecke categories
Abstract: The universal monodromic affine Hecke category is a family of categories over the dual torus. It consists of sheaves on the enhanced affine flag variety with arbitrary monodromy along the torus orbits. I will discuss the universal monodromic big tilting sheaf and the proof of tame local Betti Langlands (an extension of Bezrukavnikov's equivalence) by localization in the semi-simple directions. This is joint work with Gurbir Dhillon.
David Yang, On the classification of G((t))-categories
Abstract: The local Langlands conjecture gives a classification of representations of G(F), for F a local field. There is a parallel geometric statement, the local geometric Langlands conjecture, which can be stated as a classification of categories (as opposed to vector spaces) acted on by the loop group G((t)). We will explain that this classification at least holds when certain idempotent elements of the Hecke algebra are split, which was automatic in the context of representations. This provides a geometric analogue of recent progress on arithmetic local Langlands for large residue field characteristic. This is joint work with Gurbir Dhillon and Yakov Varshavsky.
Zhiwei Yun, Hitchin moduli spaces and wildly ramified geometric Langlands
Abstract: For a homogeneous element in the loop Lie algebra, we introduce analogs of the Slodowy slice for a nilpotent orbit and its Springer resolution using moduli of Higgs G-bundles with irregular singularities. These spaces fit into a "non-abelian Hodge package". I will then explain their role in formulating special cases of the wildly ramified geometric Langlands conjecture, and present evidence for the conjecture. This is joint work with Bezrukavnikov, Boixeda-Alvarez and McBreen.