Geometric Representation theory seminar

Organizers: Will Donovan, Penghui Li, Peng Shan, Changjian Su

Time: 10:30 am -- 11:30 am, Friday

Place: 宁斋 Ningzhai W11, Yau Mathematical Sciences Center,  Tsinghua University

official seminar webpage,  which contains the zoom information for online talks

some other related activities this semester

Minicourse by Michael McBreen: Introduction to microlocal sheaves

Workshop on Geometric Representation Theory and Applications July 10-14, 2023, Tsinghua University

• 3/03 Jianrong Li (University of Vienna)

Quantum affine algebras and Grassmannians

In this talk, I will talk about the joint work with Wen Chang, Bing Duan, and Chris Fraser on quantum affine algebras of type A and Grassmannian cluster algebras.

Let g=sl_k and U_q(^g) the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism Phi from the Grothendieck ring R_l^g of a certain subcategory C_l^g of finite dimensional U_q(^g)-modules to a quotient C[Gr(k,n, \sim)] of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux. Using the isomorphism, we defined ch(T) in  C[Gr(k,n, \sim)] for every rectangular semistandard tableau T.

Using the isomorphism and the results of Kang, Kashiwara, Kim, Oh, and Park and the results of Qin, we proved that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form ch(T) for some real (resp. prime real) rectangular semi-standard Young tableau T.

We translated a formula of Arakawa–Suzuki and Lapid-Minguez to the setting of q-characters and obtained an explicit q-character formula for a finite dimensional U_q(^sl_k)-module. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables. We also give a mutation rule for Grassmannian cluster algebras using semi-standard Young tableaux.

• 3/10 Jianrong Li (University of Vienna)

Quantum affine algebras and KLR algebras

Recently, Baumann-Kamnitzer-Knutson introduced a remarkable algebra morphism: \bar{D} from C[N] to the field of rational functions C(a_1, ..., a_n), where N is the unipotent radical of a simply laced complex algebraic group and a_i are simple roots, in their proof of a conjecture of Muthiah about MV basis of C[N]. The algebra C[N] and a larger algebra K_0(C^{\xi}) have monoidal categorifications using representations of quantum affine algebras introduced by Hernandez and Leclerc. We defined an algebra morphism \tilde{D} from K_0(C^{\xi}) to C(a_1, ..., a_n) and proved that when restricts to C[N], \tilde{D} coincides with \bar{D}. Moreover, using \tilde{D} and \bar{D}, we can recover information of q-characters of representations of quantum affine algebras from ungraded characters of modules of KLR algebras and vice versa. This is joint work with Elie Casbi.

• 3/17 Yu Zhao (Kavli IPMU)

Derived blow ups and birational geometry of nested quiver varieties

Given a quiver, Nakajima introduced the quiver variety and the Hecke correspondence, which is a closed subvariety of Cartesian products of quiver varieties. We introduce two nested quiver varieties which are the fiber products of Hecke correspondences along the projection morphisms. We prove that, after blowing up the diagonal, they are isomorphic to a smooth variety which Negut observed for the Jordan quiver. We also prove that the blow up has an derived enhancement in the sense of Hekking.

• 3/24 Yixuan Li (University of California, Berkeley)

Towards a geometric proof of the Donkin's tensor product conjecture

In the modular (char p) representation theory of algebraic reductive groups, the Frobenius twist is a great self-symmetry of the category of representations. Geometrically this self-symmetry is related to the embedding of the affine grassmannian, which is the based loop space of the reductive group, into itself as based loops that repeat themselves for p times. I'll explain an interpretation of the Donkin's tensor product conjecture as a consequence of this geometry and point out some potential ways to turn this into a proof. I'll also explain how to prove the quantum group version of the Donkin's tensor product theorem using this geometry.

• 3/31 Xun Lin (YMSC)

Infinitesimal categorical Torelli

Motivated from the categorical Torelli theorems, we introduce two types of infinitesimal  categorical problems, connecting infinitesimal Torelli problems with a commutative diagram. Our constructions are general, and the main examples in this talk are nontrivial components of derived categories of Fano 3-folds. The infinitesimal categorical Torelli theorems for Fano 3-folds are summarized. I will  talk about the unknown cases, and explain how the infinitesimal categorical Torelli theorems apply to Kuznetsov Fano 3-folds conjecture, and the categorical Torelli problems for hypersurfaces. This is based on joint works with J. Augustinas, Zhiyu Liu, and Shizhuo Zhang.

• 4/07 (3:30 pm-4:30 pm) Xiaomeng Xu (BICMR) 

Crystals from the Stokes phenomenon 

This talk first gives an introduction to the Stokes phenomenon of meromorphic linear system of ordinary differential equations. It then explains how crystal bases in the representation theory naturally arise from the Stokes phenomenon.

• 4/14 Penghui Li (YMSC)

Functions on commuting stack via Langlands duality

 We explain how to calculate the dg algebra of global functions on commuting stacks using tools from Betti Geometric Langlands. Our main technical results include: a semi-orthogonal decomposition of the cocenter of the affine Hecke category; and the calculation of endomorphisms of a Whittaker sheaf in a diagram organizing parabolic induction of character sheaves. This is a joint work with David Nadler and Zhiwei Yun.

• 4/21 Yalong Cao (RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program)

From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials 

I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.

• 4/28 Wenbin Yan (YMSC)

Modularity for W-algebras and affine Springer fibers

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibers. Furthermore, we can also match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras. We will also explain how to extend these relations to representations of W-algebras. This is based on joint work with Peng Shan and Dan Xie.

• 5/05 劳动节 Holiday

• 5/12 Jonathan Gruber (National University of Singapore)

Linkage and translation for tensor products

Let G be a simple algebraic group over an algebraically closed field of characteristic p>0. The decomposition into blocks of the category of finite-dimensional rational G-modules is described by two classical results of H.H. Andersen and J.C. Jantzen: The linkage principle and the translation principle. We will start by recalling these results and explaining why they are a-priori not well suited for studying tensor products of G-modules. Then we introduce a tensor ideal of 'singular G-modules' and give a linkage principle and a translation principle for tensor products in the corresponding quotient category. This also gives rise to a decomposition of the quotient category as an external tensor product of its principal block with the Verlinde category of G.

• 5/19 Michael McBreen (Chinese University of Hong Kong)

Microlocal Sheaves on Affine Slodowy Slices 

I will describe certain moduli of wild Higgs bundles on the line, and explain why they are affine analogues of Slodowy slices. I will then describe an equivalence between microlocal sheaves on a particular such space and a block of representations of the small quantum group. Joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Zhiwei Yun.

• 5/26 Yixin Lan (Tsinghua University)

Lusztig's perverse sheaves for quivers and integrable highest weight modules

Lusztig has introduced semisimple perverse sheaves for quivers and the induction and restriction functors to categorify the positive part of the quantum groups and provoided the existence of the canonical basis. Even though one can use an algebraic construction to obtain the canonical basis of irreducible integrable highest weight modules, how to realize the integrable highest weight modules and their canonical bases via Lusztig’s sheaves is still an important problem. We generalize Lusztig’s theory to N-framed quivers and define certain localizations of Lusztig’s perverse sheaves to realize (tensor products of) irreducible integrable highest weight modules. As a byproduct, we give a proof of the Yang-Baxter equation by using the coassociativity of Lusztig’s restriction functor. This is a joint work with Jiepeng Fang and Jie Xiao.

• 6/02 [POSTPONED] Bart Vlaar (BIMSA)

Cylindrical structures for Drinfeld-Jimbo quantum groups and the origin of trigonometric K-matrices

The study of R-matrices, matrix solutions of the spectral (parameter-dependent) Yang-Baxter equation, was a major motivation for the discovery of quantum groups. The quasitriangular structure of these bialgebras is the origin of large classes of R-matrices. The Yang-Baxter equation has a "twisted type-B/cylindrical" counterpart: the reflection equation. Its matrix solutions, known as K-matrices, have been studied since the 1980s. Is there an analogous origin for these solutions? 

To answer this, in joint works with Andrea Appel we develop a general framework, in terms of braided tensor categories with additional structures. Concretely, take any Letzter-Kolb quantum symmetric pair: a Drinfeld-Jimbo quantum group (quantized enveloping algebra of a Kac-Moody algebra) together with a suitable subalgebra (also known as i-quantum group). Further to works by Bao & Wang and Balagovic & Kolb, a twisted intertwiner of the subalgebra satisfies a reflection equation, acts on (integrable) category O modules and endows this braided tensor category with a twisted cylinder braiding. For affine quantum groups one can develop the parallel with R-matrices much further and account for large classes of so-called trigonometric K-matrices.

• 6/09 Du Pei (University of Southern Denmark)

Brane and DAHA Representations

In this talk, we will use "brane quantization" to study the representation theory of Double affine Hecke algebras.

• 6/20 (4pm-5pm, Conference Room 3, Jin Chun Yuan West Building) 

Peter McNamara (University of Melbourne)

KLR algebras and their representations

KLR algebras are a family of algebras introduced approximately fifteen years ago as part of the programme of categorifying quantum groups. We will discuss various representation-theoretic properties of these algebras. Along the way, we see an interesting mix of geometry, combinatorics and homological algebra.

• 7/05 Ruotao Yang (Skoltech)

On the Gaiotto conjecture I: statement

A conjecture of Davide Gaiotto predicts a geometrization of the category of representations of the quantum supergroup. This geometrization is given by the category of twisted D-modules on the affine Grassmannian with a certain equivariant condition. In this talk, we will introduce the precise statement of the Gaiotto conjecture and the recent progress. This is based on the joint works with R. Travkin, and also the works of A. Braverman, M. Finkelberg, V. Ginzburg and R.Travkin.

• 7/07 Ruotao Yang (Skoltech)

On the Gaiotto conjecture II: sketch of proof

In this talk, our goal is to introduce the tools used in the proof and sketch the proof of the Gaiotto conjecture for GL(M|N). 

• 7/10-7/14 Workshop

• 7/24 (Jin Chun West Building, 1st conference room, 4-5 pm)

Vivek Shende (Syddansk Universitet and UC Berkeley)

Microsheaves from Hitchin fibers

For geometric reasons, one expects that any natural procedure `quantizing the Hitchin system', i.e. carrying fibers of the Hitchin fibration to sheaves on Bun_G, should produce Hecke eigensheaves.  I will explain how recent developments around microsheaves and Fukaya categories give rise to such a procedure, and discuss what further developments would be needed to check the `Hecke eigen' property.  

• 7/26 (Jin Chun West Building, 1st conference room, 4-5 pm)

Vivek Shende (Syddansk Universitet and UC Berkeley)

Skein-valued curve counting

The skein relations of quantum topology arise originally from properties of R-matrices for quantum groups, or equivalent representation-theoretic data.  In the present talk I will explain how the same skein relations arise from the geometry of counting holomorphic curves with boundary in Calabi-Yau 3-folds.  Time permitting, I will explain some algebraic questions which arise in this context, around holonomicity for skein-valued recursion relations, and the construction of skein-valued cluster algebras.  

• 7/28 (Jin Chun West Building, 1st conference room, 4-5 pm)

Anna Lachowska (EPFL)

The derived center of the small quantum group

Let u=u_q(g) denote the small quantum group associated to a semisimple complex Lie algebra g and a root of unity q. The Hochschild cohomology  $HH^*(u)$, also known as the derived center of $u$, has a rich structure, its graded dimensions are unknown in general. I will discuss a natural action of the Lie algebra $g$ and a projective modular group action on $HH^*u$. As an illustration, I will present the $g = sl_2$ example. 

• 8/2 (Jin Chun West Building, 1st conference room, 2-3 pm)

Tahsin Saffat (UC Berkeley)

Eisenstein Series for P^1 with three points

I’ll explain a conjecture about the space of Eisenstein series for the function field of P^1 with three points of tame ramification as a trimodule for the Hecke algebra. It can be thought of as a generalization of the fact that the space of automorphic functions for P^1 with two points is the regular bimodule for the Hecke algebra.

• 8/4 (Jin Chun West Building, 1st conference room, 4-5 pm)

[POSTPONED] Zhiwei Yun (MIT)  

An irregular Deligne-Simpson problem and Cherednik algebras

The Deligne-Simpson problem asks for a criterion of the existence of connections on an algebraic curve with prescribed singularities at punctures. We give a solution to a generalization of this problem to G-connections on P^1 with a regular singularity and an irregular singularity (satisfying a condition called isoclinic). Here G can be any complex reductive group. Perhaps surprisingly, the solution can be expressed in terms of rational Cherednik algebras. This is joint work with Konstantin Jakob.

• 8/7 (Jin Chun West Building, 1st conference room, 2-3 pm)

Jianqiao Xia (Harvard)

Hecke categories and positive depth representations of loop group 

Inspired by the theory of unrefined minimal K-types developed by Moy-Prasad, we study some associated Hecke categories and prove that they are monoidally equivalent to the affine Hecke categories of some smaller groups. This equivalence relates the study of certain positive depth representations of the loop group to the study of depth zero representations, thus having potential applications to the categorical version of wildly ramified local Langlands correspondence. 

The equivalence can be seen as a categorical version of a Hecke algebra isomorphism proved by Ju-Lee Kim.

seminar arxiv: 2022 fall, official page