Geometric Representation Theory Seminar

Yau Mathematical Science Center, Tsinghua University

2025 Spring

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

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 $\mathfrak{g}$.


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.


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.


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.


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.



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.


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.