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

Time: 3:30 pm -- 4:30 pm, Friday

Place: Lecture Hall B627, Shuangqing Complex Building（双清综合楼B627报告厅), Yau Mathematical Sciences Center,  Tsinghua University

双清综合楼地址：北京市海淀区逸清南路西延6号院1号，双清公寓马路对面、清华附小（双清校区）西侧。

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

minicourses: 

Stephen Doty (Loyola University Chicago)

A finite dimensional algebra approach to quantum groups

Thomas Haines (University of Maryland)

The geometric Satake correspondence

Gurbir Dhillon (Yale University)

Geometric representations of vertex algebras

• Weinan Zhang (University of Hong Kong), Jingzhai 静斋 304, 08/30, 08/31, and 09/01, 10:00 am - 11:30 am

Quantum symmetric pairs and $\imath$quantum groups

The classical theory of symmetric pairs concerns Lie algebras with involutions. The theory of quantum symmetric pairs, systematically developed by Letzter around 2000, is a quantization of classical symmetric pairs. The $\imath$quantum groups, arising from quantum symmetric pairs, are certain coideal subalgebra of quantum groups. The $\imath$quantum groups can be viewed as a natural generalization of quantum groups, and many results for quantum groups have been generalized to $\imath$quantum groups.

In the first lecture, we will review some basics of symmetric pairs and introduce quantum symmetric pairs and $\imath$quantum groups. We will construct the quasi K-matrix associated to a quantum symmetric pair, which will be a key ingredient in the later construction of relative braid group symmetries.

In the second lecture, we will construct the relative braid group symmetries (associated to the relative Weyl group of the underlying symmetric pair) on $\imath$quantum groups and on their modules. These symmetries generalize Lusztig's braid group symmetries on quantum groups.

In the third lecture, we will construct the Drinfeld type presentation for (quasi-split) affine $\imath$quantum groups, which generalizes the Drinfeld (loop) presentation for affine quantum groups. The relative braid group symmetries will play an important role in this construction. The Drinfeld type presentation for affine $\imath$quantum groups will lead to a Drinfeld type presentation for twisted Yangians.

• Jiajun Ma (Xiamen University), Jingzhai 静斋 304, 09/04, 10:00 am--11:30 am

Generic Hecke algebra modules in theta correspondence over finite fields

In this talk, we consider the theta correspondence of type I dual pairs over a finite. Aubert, Michel, and Rouquier established an explicit formula for theta correspondence between unipotent representations of unitary groups and made a conjecture for the symplectic group-even orthogonal group dual pair. Shu-Yen Pan recently proved the conjecture. These works are based on Srinivasan's formula for the uniform projection of the Weil representation.

Joint with Congling Qiu and Jialang Zou, we found an alternative approach to solve the problem by analyzing the relevant Hecke algebra bimodules. Joint with Zhiwei Yun, we geometrized the whole picture. Consequently, we obtained a  relation between the Springer correspondence and theta correspondence. 

• Zhiwei Yun (MIT) online colloquium talk,  09/22, Fri., 10:00-11:00 am, 

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.

• 09/29 Holiday 中秋节

• 10/20 Bart Vlaar (BIMSA)

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

The study of spectral R-matrices, matrix solutions of the (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 spectral R-matrices. The Yang-Baxter equation has a type-B/cylindrical counterpart: the reflection equation. Its matrix solutions, spectral K-matrices, have been studied since the 1980s. Do they have a similar origin?

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 together with a suitable coideal subalgebra (also known as i-quantum group). Building on works by Bao & Wang and Balagovic & Kolb, we show that a twisted intertwiner of the subalgebra satisfies a twisted reflection equation, acts on (integrable) category O modules, and endows this braided tensor category with a twisted cylinder braiding. In the case of affine quantum groups one can then indeed answer the above question, explaining large classes of so-called trigonometric K-matrices.

• 10/27 Shoma Sugimoto (YMSC)

On the Feigin-Tipunin's construction

The induced G-module of a VOA with a B-action again has a VOA structure (Feigin-Tipunin's construction). In the original preprint of FT and my papers, we used this technique to construct the (1,p)-VOA from a lattice VOA with a good B-action and proved some basic results. As a matter of fact, it is the good B-module structure, not the VOA structure, that plays an essential role in the proof of these results. Therefore, It is expected that we can construct and study interesting VOAs from other VOAs with similar B-actions using the FT construction. In this talk, I will outline the FT construction and my results, and then discuss future works relating the FT construction to plumbed 3-manifolds.

• 11/10 Tianqing Zhu (YMSC)

Quantum difference equations for affine type A quiver varieties

The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. Its construction relies on the K-theoretic stable envelope and it is conjectured that the construction is related to the quantum affine algebra of the corresponding quiver type. 

In this talk, I will focus on the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ and give an analog of the qde. We explicitly construct the qde and give the explicit formula for the case of instanton moduli space and Hilbert scheme of A_n-singularities. We also discuss its connection to the Dubrovin connection of the quantum cohomology with the example of the instanton moduli space.

• 11/17 Qixian Zhao (YMSC)

Non-integral Kazhdan-Lusztig algorithm and an application to Whittaker modules

Let g be a complex semisimple finite dimensional Lie algebra, and consider a category of representations of g where a Kazhdan-Lusztig algorithm exists for integral regular infinitesimal characters. In this talk, we will discuss a potential approach for extending the integral algorithm to arbitrary non-integral regular infinitesimal characters, using intertwining functors. We will then apply this approach to Whittaker modules and demonstrate the non-integral algorithm there using an explicit example.

• 11/24 Huanhuan Yu (BICMR)

Fusion product and Global modules

Let g be a finite dimensional simple Lie algebra. In this talk, I will start with the definition of fusion product of g[t]-modules, and then introduce several approaches to study it, such as the global modules and Borel-Weil type theorem. In the meantime, I will talk about some partial results on the fusion product and applications including my work on twisted global Demazure modules joint with Jiuzu Hong.

• 12/01 (8:30 am-9:30 am) Chengze Duan (University of Maryland-College Park)

Good position braids, transversal slices and affine Springer fibers

Let G be a reductive group over an algebraically closed field and W be its Weyl group. Using Coxeter elements, Steinberg constructed cross-sections of the adjoint quotient of G which also yield transversal slices of regular unipotent classes. In 2012, He and Lusztig constructed transversal slices using minimal length elements in elliptic conjugacy classes in W, which yield transversal slices of basic unipotent classes. In this talk, we generalize minimal length elements to good position braids in the associated braid monoid of W and use these elements to construct transversal slices of all unipotent classes in G. We shall see these new elements also appear in many other aspects of representation theory, such as affine Springer fibers and the partial order on unipotent classes, etc.

• 12/08 Caroline Namanya (Makerere University) online talk

Pure braids and group actions

The first part of the talk will be about a new and simplified presentation of the classical pure braid group. Motivated by twist functors from Algebraic geometry, the generators are given by the squares of longest elements over connected subgraphs, and the relations are either commutators or certain length 5 palindromic relations.

In the second part of the talk, I will summarise a construction of derived autoequivalences associated to an algebraic flopping contraction. These functors are constructed naturally using bimodule cones, and these cones are locally two-sided tilting complexes. The autoequivalences combine to give an action of the fundamental group of an associated infinite hyperplane arrangement on the derived category.

• 12/15 Conference at Sanya

• 12/29 Gurbir Dhillon (Yale)

The FLE and the W-algebra

The FLE is a basic assertion in the quantum geometric Langlands program, proposed by Gaitsgory-Lurie, which provides a deformation of the geometric Satake equivalence to all Kac-Moody levels. We will report on a proof via the representation theory of the affine W-algebra, which is joint work in progress with Gaitsgory.

• 01/26 Raphael Rouquier (UCLA)

Higher representation theory of gl(1|1) 

The notion of representations of Lie algebras on categories ("2-representations") has proven useful in representation theory. I will discuss joint work with Andrew Manion for the case of the super Lie algebra gl(1|1). A motivation is the reconstruction of Heegaard-Floer theory, a 4-dimensional topological field theory, and its extension down to dimension 1.

seminar arxiv: 2023 Spring, 2022 Fall, official page