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

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

Place: Lecture Hall B626, Shuangqing Complex Building（双清综合楼B626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

• 02/21 Robert McRae (YMSC)

Tensor structure on the Kazhdan-Lusztig category of affine $\mathfrak{sl}_2$ at admissible levels

For a simple Lie algebra $\mathfrak{g}$ and a level $k$, the Kazhdan-Lusztig category $KL_k(\mathfrak{g})$ is the category of finite-length modules for the affine Lie algebra of $\mathfrak{g}$ at level $k$ whose composition factors have highest weights which are dominant integral for the subalgebra $\mathfrak{g}$. In this talk, I will discuss joint work with Jinwei Yang showing that $KL_k(\mathfrak{sl}_2)$ is a non-rigid braided tensor category when $k=-2+\frac{p}{q}$ is admissible, and that there is an exact and essentially surjective (but not quite full or faithful) tensor functor from $KL_k(\mathfrak{sl}_2)$ to the non-semisimple category of finite dimensional weight modules for Lusztig's big quantum group of $\mathfrak{sl}_2$ at the root of unity $e^{\pi i q/p}$. I will also discuss prospects for extending such results to higher rank $\mathfrak{g}$.

• 03/07 Huanhuan Yu 余欢欢 (BICMR)

A refinement of the Coherence Conjecture of  Pappas-Rapoport

The Coherence Conjecture of Pappas-Rapoport, proven by X. Zhu, establishes a relationship between the geometry of different affine Schubert varieties, notably providing dimension equalities for the sections of line bundles on (unions of) affine Schubert varieties in different affine partial flag varieties. In this talk, I will present a refinement of this conjecture, demonstrating that these spaces of global sections are isomorphic as representations of certain group. I will also discuss applications of this result, particularly to affine Demazure modules, accompanied by concrete examples. This is joint work with Jiuzu Hong.

• 03/14 Konstantin Jakob (TU Darmstadt)

Counting absolutely indecomposable G-bundles

About 10 years ago, Schiffmann proved that the number of absolutely indecomposable vector bundles on a curve over a finite field (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave another proof of this result in a slightly different formulation, but neither proof generalizes in an obvious way to G-bundles for other reductive groups G. In joint work with Zhiwei Yun, we generalize the above results to G-bundles. Namely, we express the number of absolutely indecomposable G-bundles on a curve X over a finite field in terms of the cohomology of the moduli stack of stable parabolic G-Higgs bundles on X.

• 03/21 Tomasz Przezdziecki (University of Edinburgh)

q-Characters for quantum symmetric pairs

It is well known that quantum affine algebras admit three distinct presentations (Kac-Moody, new Drinfeld and RTT). Relatively recently, the same has been shown to hold for a broad family of quantum affine symmetric pairs. In particular, a Drinfeld-type presentation, due to Lu-Wang, is a new and exciting development. The focus of my talk will be the relationship between the usual Drinfeld presentation of quantum affine algebras and the Lu-Wang presentation of their coideal subalgebras. Remarkably, both presentations exhibit large commutative subalgebras, which are of particular interest to representation theory. More specifically, I will present several results concerning the properties of the generators of these commutative subalgebras, including their behaviour under inclusion and coproduct, as well as their spectra on finite-dimensional representations. These results will then be used to define an analogue of the q-character homomorphism for quantum symmetric pairs. I will compute the q-characters of evaluation modules, and discuss applications to categorification.

• 03/28 Jianrong Li 李建荣 (University of Vienna)

Cluster structures on spinor helicity and momentum twistor varieties

We study cluster structures on spinor helicity and momentum twistor varieties which describe the kinematic spaces of massless scattering. Both varieties are certain partial flag varieties. We exhibit embeddings of the corresponding cluster algebras into cluster algebras of sufficiently large Grassmannians and show how the former is obtained by freezing certain cluster variables from the latter. This is joint work with Lara Bossinger.

• 04/01-04/29 minicourse by Henry Liu 刘华昕 (Kavli IPMU)

• 04/11 Mengxue Yang 杨梦雪 (Kavli IPMU)

Conformal limit on Cayley components 

In 2014, Gaiotto conjectured that there is a biholomorphism between Hitchin components and spaces of opers on a punctured sphere via a scaling limit called the $\hbar$-conformal limit. On a compact Riemann surface of $g \ge 2$, this biholomorphism has been proven in 2016. Motivated by the study of higher Teichm\"uller spaces, we may view the Hitchin components as a part of a larger family of special components called Cayley components. I will talk about the Cayley components and propose their conformal limit to be the generalized notion of opers of Collier—Sanders.

• 04/25 Henry Liu 刘华昕 (Kavli IPMU)

Vertex coalgebras and quantum loop algebras

Joyce recently gave a geometric construction of a vertex coalgebra structure on the cohomology of appropriate moduli stacks of linear objects like quiver representations or coherent sheaves. In appropriate settings, Latyntsev showed that this vertex coalgebra is compatible with cohomological Hall algebras, forming a vertex bialgebra. I will explain how these results generalize to (critical) K-theory, yielding multiplicative refinements of vertex bialgebras. This is relevant to wall-crossing problems -- the original focus of Joyce's work -- for K-theoretic enumerative invariants.

• 05/09 Yixin Lan 兰亦心 (AMSS, CAS)

Canonical bases, Littlewood-Richardson rule and branching rule

In this talk, I will realize the subquotient based modules of certain tensor products or restricted modules via Lusztig's perverse sheaves on  multi-framed quivers, and provide a construction of their canonical bases. After applying characteristic cycle maps,   these constructions give a relation between fusion or braching coefficients and  Borel-Moore homology groups for certain locally closed subsets of Nakajima's quiver varieties considered by Malkin and Nakajima.

• 5/15 10:00 am - 11:00 am, Shuangqing 双清 B725

Rui Xiong 熊锐 (University of Ottawa)

An ADE Classification of Hodge–Tate Hyperplanes in Grassmannians 

The ADE classification appears in many contexts where algebraic or geometric structures exhibit finiteness or simplicity, such as simple Lie algebras, quivers of finite type, and finite subgroups of SU(2). In joint work in progress with Sergey Galkin, Naichung Conan Leung, and Changzheng Li, we discover another unexpected instance: Hodge–Tate hyperplane sections of Grassmannians, which parametrize k-dimensional subspaces of an n-dimensional vector space, are also classified bijectively by ADE Dynkin diagrams. We will further discuss possible connections with algebraic geometry and representation theory.

• 05/16 Jie Du (University of New South Wales)

Finite dimensional algebras and quantum groups

Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum gl_n (i.e., the quantum enveloping algebra Uq(gl_n) of the Lie algebra gl_n) and its associated matrix representation of the regular module of Uq(gl_n). This beautiful work shows that the structure of the quantum linear group is hidden in the structure of Hecke algebras. The work has been generalized (either geometrically or algebraically) to quantum affine gl_n, quantum super gl_{m|n}, and recently, to some i-quantum groups of type AIII. (All were good PhD projects.) In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using Hecke-Clifford superalgebras and their associated q-Schur superalgberas.

• 5/22 10:30 am - 11:30 am, Shuangqing 双清 B626

Yaolong Shen 沈耀龙 (University of Ottawa)

Quantum supersymmetric pairs and iSchur duality

Let g be a semisimple Lie algebra and \theta be an involution of g. The quantization (U_q(g),U^i) of the symmetric pair (g,g^\theta) was systematically developed by Letzter where U_q(g) is the Drinfeld-Jimbo quantum group and U^i is a coideal subalgebra of it. We usually refer to U^i as the iquantum group. Over the last decade, many fundamental constructions in quantum groups have been generalized to iquantum groups by Wang and his collaborators. In this talk, we will discuss the construction of U^i in the context of Lie superalgebras. Moreover, we will demonstrate an iSchur duality between one specific family and the q-Brauer algebras. This duality can be viewed as a quantization of the classical duality between the orthosymplectic Lie superalgebra and the Brauer algebra. This is joint work with Weiqiang Wang.

• 05/23 Yu Li 李昱 (University of Toronto)

Polynomial integrable systems from cluster structures

We present a general framework for constructing polynomial integrable systems on linearizations of Poisson varieties that admit log-canonical systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures. As examples, we consider an arbitrary standard complex semisimple Poisson Lie group $G$ with the Berenstein-Fomin-Zelevinsky cluster structure; nilpotent Lie subgroups of $G$ associated to elements of the Weyl group of $G$, identified with Schubert cells in the flag variety of $G$ and equipped with the standard cluster structure (first defined by Geiss-Leclerc-Schr\"oer when $G$ is simply-laced); and the restriction of the Gekhtman-Shapiro-Vainshtein generalized cluster structure on the Drinfeld double of the Poisson Lie group ${\rm GL}(n, \mathbb C)$ to its dual Poisson Lie group ${\rm GL}(n, \mathbb C)^*$. In each of the three cases, we show that every extended cluster in the respective cluster structure gives rise to at least one polynomial integrable system on the respective Lie algebra with respect to the linearization of the Poisson structure at the identity element. For some of the polynomial integrable systems thus obtained, we give Lie theoretic interpretations of their Hamiltonians, and we further show that their Hamiltonian flows are complete. This is joint work with Yanpeng Li and Jiang-Hua Lu.

• 05/30 Fan Qin 覃帆 (Beijing Normal University)

PBW type bases for quantum cluster algebras

Quantum cluster algebras associated with double Bott-Samelson cells have PBW type bases, which consist of monomials of fundamental variables. By extending them to the infinite rank setting, we recover the cluster structure on the virtual Grothendieck rings of representations of quantum affine algebras (i.e., derived Hall algebras). This allows us to compute the fundamental variables via braid group actions.

• 06/06 10:00 am - 11: 00 am, 双清C641

Changlong Zhong 钟昌龙 (State University of New York at Albany)

Perterson algebra and generalized cohomology of affine Grassmannians

Peterson algebras are defined as Cohomology and K-theory of affine Grassmannians. They have rich combinatorial structure, and are closely related to quantum cohomology and quantum K-theory of flag varieties. This was firstly studied and conjectured by Peterson and proved by him, Lam-Shimozono, and Syu Kato. These algebras can be embedded into the cohomology/K-theory of the affine flag variety and coincide with certain centralizer. They also have Hopf algebra structure. In this talk I will talk about generalization of these construction from cohomology/K-theory to general oriented cohomology theory by using the Kostant-Kumar method. This generalization has the advantage of unifying all previous work. Part of this work is jointly with Rui Xiong and Kirill Zainoulline.

• 06/06 Anders Buch (Rutgers University)

Pieri formulas for the quantum K-theory of cominuscule Grassmannians 

The quantum K-theory ring QK(X) of a flag variety X=G/P encodes the K-theoretic Gromov-Witten invariants of X, defined as arithmetic genera of Gromov-Witten varieties parametrizing curves meeting fixed Schubert varieties. A Pieri formula means a formula for multiplication with a set of generators of the ring QK(X). Such a formula makes it possible to compute efficiently in this ring. I will speak about a Pieri formula for QK(X) when X is a cominuscule Grassmannian, that is, an ordinary Grassmannian, a maximal orthogonal Grassmannian, or a Lagrangian Grassmannian. This formula has a simple statement in terms of order ideals in a partially ordered set that encodes the degree distance between opposite Schubert varieties. This is joint work with P.-E. Chaput and N. Perrin.

• 06/13 Yang Yang 杨阳 (East China Normal University)

Geometric Construction of Quantum affine Schur algebras

The quantum affine Schur algebras are generalizations of affine Hecke algebras. They naturally appear in quantum Schur-Weyl duality. In this talk, I will firstly provide a geometric construction of a series of generalized quantum affine Schur algebras of any type via equivariant K-groups of generalized Steinberg varieties. As applications, we obtain a Schur algebra analogue of the local geometric Langlands reciprocity of any type and provide an equivariant K-theoretic realization of quasi-split i-quantum groups of affine type AIII. Secondly, I will provide a geometric construction of  affine Schur algebras of  type C with three parameters via equivariant K-groups of generalized exotic Steinberg varieties and provide a K-theoretic realization of three parameters i-quantum groups of affine type AIII. If time permits, I will discuss how to give an equivariant K-groups realization of two parameters affine Schur algebras of type BCFG based on a recent work of Jonas Antor. This is joint work with Li Luo and Zheming Xu.

• 6/23 10:00-11:00 am, Shuangqing 双清 B725 

Zhiwei Yun 恽之玮 (MIT) 

Character sheaves on the loop Lie algebra 

Motivated by Lusztig's definition of character sheaves on a reductive Lie algebra, we propose a definition of character sheaves on the loop Lie algebra. They are closely related to characters of supercuspidal representations of J.K.Yu, and geometrize Lie algebra analogs of characters of arbitrary depth. This is joint work with Bao Chau Ngo.

• 06/27

10:30am-11:30am (双清 Shuangqing B626) Sam DeHority (Yale)

Quiver folding and cohomological Hall modules

We investigate modules for cohomological Hall algebras which arise from (anti-)involutions of the underlying quiver, and find that the cohomology of moduli stacks of objects with classical type structure groups (e.g. for orthosymplectic quivers) gives a module which satisfies an interesting axiom (the twisted Yetter-Drinfeld condition). We also discuss examples with applications to moduli of classical type bundles on surfaces and the AGT correspondence.

14:00pm-15:00pm (双清 Shuangqing B626) Ben Webster (University of Waterloo and Perimeter Institute )

Coulomb branches without tears (or affine Grassmannians)

I’ll try to summarize what is known about constructing about Coulomb branches without using any affine Grassmannians or convolution algebras (though a little bit of geometry might sneak in through the back door) .

15:30pm-16:30pm (online, and watch together in 双清 Shuangqing B626)  Alexander Samokhin (Bielefeld)

Semiorthogonal decompositions for algebraic groups

This talk is about a joint work with Wilberd van der Kallen available at arXiv:2407.13653. 

Given a split reductive Chevalley group scheme G over the integers and a parabolic subgroup scheme P of G, we construct G-linear semiorthogonal decompositions of the bounded derived category of noetherian representations of P with each semiorthogonal component being equivalent to the bounded derived category of noetherian representations of G. The ensuing semiorthogonal decompositions are compatible with the Bruhat order on cosets of the Weyl group of P in the Weyl group of G. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag schemes G/P over the integers.

We will explain the statement of the main theorem, its relation to classical results on K-theory of flag varieties, and mention the main ingredients used in the proof. Time permitting, we will discuss further connections of our results to geometric representation theory.

• 7/15 14:00-15:00 pm, Shuangqing 双清C548

Weiqiang Wang (University of Virginia)

Drinfeld presentations of twisted Yangians and applications 

As a generalization of Yangians introduced by Drinfeld in 1986, twisted Yangians are deformation of twisted current algebras. In recent joint work with Kang Lu and Weinan Zhang, we have obtained Drinfeld type current presentations of twisted Yangians of type AI and beyond through Gauss decomposition and degeneration. Further parabolic and shifted generalization of these presentations in type AI and AII have led to connections to Slodowy slices and finite W-algebras of type BCD, in joint work with Lu, Peng, Tappeiner and Topley. In this talk, we will explain some of these works.

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