Geometric Representation Theory Seminar

Yau Mathematical Science Center, Tsinghua University

2024 Fall

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

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

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

Opers ー what they are and what they are good for

I will introduce the space of (q-)opers on a projective line for a simple simply-conncted Lie group G. I will explain how this space is related to previously known results in geometry, physics, and integrable systems.


Genus 2 Macdonald functions

We present a system of 3 integrable difference equations in 3 variables. We prove that automorphisms associated to this system form a genus 2 mapping class group. We study eigenfunctions of this system and call them genus 2 Macdonald functions. Finally, we work on genus 3 and higher genus generalization of this system.


Slices in the loop spaces of symmetric varieties

Let X be a symmetric variety. J. Mars and T. Springer constructed conical transversal slices to the closure of Borel orbits on X and used them to show that the IC-complexes for the orbit closures are pointwise pure. This is an important geometric ingredient in their work providing a more geometric approach to the results of Lusztig-Vogan. In the talk, I will discuss a generalization of Mars-Springer's construction of transversal slices to the setting of the loop space LX of X where we consider closures of spherical orbits on LX. I will also explain its applications to the formality conjecture in the relative Langlands duality. If time permits, I will discuss similar constructions for Iwahori orbits. This is a joint work with Tsao-Hsien Chen.


       Calogero-Moser spaces vs unipotent representations

Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group W (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from W . Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with W (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this talk is to gather these observations, state precise conjectures and provide general facts and examples supporting these conjectures.


Monoidal Jantzen Filtrations and quantization of Grothendieck rings

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal categories with generic braidings.

It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring. This is a joint work with Ryo Fujita.


Genus two Double Affine Hecke Algebra and its Classical Limit. 

Double Affine Hecke Algebras were originally introduced by I.Cherednik and used in his 1995 proof of Macdonald conjecture from algebraic combinatorics. These algebras come equipped with a large automorphism group SL(2,Z) which has geometric origin, namely it is the modular group of a torus. It was subsequently shown that spherical Double Affine Hecke Algebras realize universal flat deformations of the quantum chracter variety of a torus and their existence is closely related to the fact that classical SL(n,C)-character varieties admit symplectic resolution of singularities via the Hilbert Scheme Hilb_n(\mathbb C*\times\mathbb C*). 

In 2019 G. Belamy and T. Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare-Birkhoff-Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century".  (joint with Sh. Shakirov)

Wonderful compactification over an arbitrary base scheme 

Wonderful compactifications of adjoint reductive groups over an algebraically closed field play an important role in algebraic geometry and representation theory. In this talk, we will construct an equivariant compactification for adjoint reductive groups over arbitrary base schemes, which parameterize classical wonderful compactifications of De Concini and Procesi as geometric fibers. Our construction is based on a variant of the Artin–Weil method of birational group laws. In particular, our construction gives a new intrinsic construction of wonderful compactifications. If time permits, we will also discuss several applications of our compactification in the study of torsors under reductive group schemes.


Symmetric functions via motivic Chern classes

The Macdonald polynomials are very important symmetric functions, generalizing the Schur functions, Hall-Littlewood polynomials, etc. I will report a joint work in progress with B. Ion and L. Mihalcea, in which we define a symmetric function via the motivic Chern classes of the Schubert cells in the affine Grassmannian. These polynomials depends on two variables q and t, and enjoy very similar specializations as the Macdonald polynomials.


Geometric representation of Hecke categories

Hecke categories are the geometrization/categorification of Hecke algebras and play a key role in geometric representation theory. We shall survey some recent progress regarding Hecke categories and their applications in number theory, combinatorics, Lie theory, and topology. Part of this talk is based on joint work with Quoc P. Ho, David Nadler, and Zhiwei Yun.


4d mirror symmetry for class-S theories

I will explain a (conjectural) correspondence between the representation theory of a vertex operator algebra (VOA) associated to a genus g Riemann surface with n-punctures and the geometry of Hitchin moduli space / character variety of the same Riemann surface. They are coming from Higgs / Coulomb branches 0f 4d class-S theories respectively. We propose that there exists a bijection between modular invariant modules of the VOA and fixed manifolds of Hitchin moduli spaces. The modularity is also related to dimensions of fixed manifolds. Moreover, all these data can be read from the mixed Hodge polynomials of the character variety. I will explain this correspondence in detail and provide examples.


Character varieties, Coulomb branches, and duality 

I will describe a project with Peng Shan which aims to relate character varieties of surfaces to K-theoretic Coulomb branches in the sense of Braverman, Finkelberg, and Nakajima. Focusing on the case where the surface is a once-punctured torus, I will show how our results provide a rigorous framework for studying the dualities of certain quantum field theories.