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 $\mathfrak{g}$.


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.