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

Time: 10:30 am - 11:30 am, Friday

Place: B627, Shuangqing Complex Building（双清综合楼B627教室), 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

• 09/11 双清B626, 10:00 am - 11:00 am

Shilin Yu 余世霖 (华东师范大学/厦门大学)

Geometry of special pieces and orbit method

I will report two ongoing works: 

The first one is joint work with Juteau, Levy and Sommers. We confirm the special pieces conjecture of Lusztig in all cases, which claims that the special piece associated to any special nilpotent orbit of a complex semisimple Lie algebra is the quotient of a smooth (symplectic variety) by a finite group action. The case of classical Lie algebras was previously obtained by Kraft-Procesi. Our proof is based on the previous work of Fu-Juteau-Levy-Sommers on the geometry of Slodowy slices of special pieces. 

The second work is by the speaker alone. I will exhibit an unexpected connection between the special pieces conjecture and unitarity of special unipotent representations of real linear reductive Lie groups in the sense of Arthur and Adams-Barbasch-Vogan. As a consequence, a uniform proof of unitarity of all special unipotent representations of all reductive groups is obtained, based on recent preprints of Adams-Mason-Brown-Vogan, Davis-Mason-Brown and my previous joint work with Conan Leung on orbit method. The case of classical groups was previously established by Barbasch-Ma-Sun-Zhu using Howe duality.

• 09/19 Sergey Oblezin (BIMSA)

Whittaker functions in representation theory, geometry and number theory

In my talk I report on an ongoing research project have been running jointly with A. Gerasimov and D. Lebedev. I present the key results and constructions restricting myself to the special case of GL(2,R). I start with introducing GL(2,R)-Whittaker function associated with a spherical principal series representation. Then I explain its role in the local Langlands correspondence and its geometric interpretation in terms of quantum cohomology of complex projective line. In the second part of my talk, I introduce the Hecke-Baxter for spherical principal series representations and its global extension for the GL(2)-Eisenstein series.

• 09/26 Hyun Kyu Kim (KIAS)

Frobenius homomorphisms for SL_n-skein algebras of surfaces

The $SL_n$-skein algebra $\mathscr{S}_q(S)$ of an oriented surface $S$ is a quantization of the $SL_n$-character variety of $S$. It is spanned by isotopy classes of framed links in the thickened surface $S \times (-1,1)$. When the quantum parameter $q$ is specialized at a root of unity $\omega$ (with some condition), there is a so-called Frobenius homomorphism $\mathscr{S}_1(S) \to \mathscr{S}_\omega(S)$, which can be viewed as a generalization of the Frobenius maps for quantum groups. We review its recent developments, and discuss some future directions. The talk is partially based on the joint work (arXiv:2504.08657) with Thang Lê and Zhihao Wang.

• 10/10 Yongchang Zhu 朱永昌 (YMSC)

Genus One Conformal Block of Affine Vertex Operator Algebras

An affine vertex operator algebra (VOA) is a VOA constructed from modules of an affine Kac-Moody algebra. In this talk, we will recall the definition of conformal blocks for affine VOAs on genus-one Riemann surfaces. We will then discuss the structure and key properties of these spaces, which are central to the representation theory of VOAs and its connections to other subjects.

• 10/17 Double Talks

10:00am-11:00am Max Gurevich (Technion)

Weak sphericity, weak Arthur packets, and Springer parameters

Arthur packets are natural combinations of irreducible representations of classical-type p-adic groups that typically arise in an automorphic context. There are ongoing attempts to treat the same notion from a microlocal perspective, that is, through the local character expansions of the packets’ constituents. In joint work with Emile Okada, we highlight the role of weakly spherical Arthur packets for that task, namely those containing a representation admitting invariants under a (not necessarily hyperspecial) maximal compact subgroup. Perhaps surprisingly, weakly spherical representations of symplectic/orthogonal groups turn out to be classified by the Lusztig’s canonical quotients that are attached to adjoint nilpotent orbits. If time permits, I will also discuss a crucial tool enabling such studies: A comparison between the endoscopic theory for p-adic groups and the K-theoretic realization of the Deligne-Langlands correspondence for affine Hecke algebras.

11:10am-12:10am Liang Xiao 肖梁 (PKU)

Higher Chow group version of theta lifts

In the foundational work of Kudla and Millson, they introduced a version of theta lifts which gives rise to generating series with values in Chow cycles on orthogonal of unitary Shimura varieties. A key ingredient in proving the modularity of this generating series is Borcherds' work on singular theta lifts.  In this talk, we explain a framework, which hopes to construct a similar generating series with values in higher Chow groups, in which a (conjectural) version of Borcherds' result on singular theta lifts from Sp4 naturally appears. This is largely motivated by potential applications to  Beilinson conjecture. This is an ongoing series of joint works with Haocheng Fan, Wenxuan Qi, Linli Shi, Peihang Wu, and Yichao Zhang.

• 10/24  Double Talks

10:00am-11:00am Yun Gao 郜云 (York University)

从向量叉乘谈起

简单介绍一些重要的李代数如有限单的李代数，Witt代数，Heisenburg代数，伽利略共形代数，仿射Kac-Moody代数，高维仿射李代数和相交矩阵李代数以及一些应用。

11:10am-12:10am Kari Vilonen (University of Melbourne)

              Unitary representations of Lie groups and Hodge theory.

The determination of the unitary dual of a Lie group is a longstanding problem. In this talk I will explain how the unitarity of a representation of a real reductive group can be read off from its Hodge filtration establishing a conjecture made by Wilfried Schmid and myself a while back. This is part of a larger set of conjectures which postulate that global sections of mixed Hodge modules on flag manifolds give rise to mixed Hodge structures. In joint work with Dougal Davis we have proved a substantial part of these conjectures and in particular the Hodge theoretic unitarity criterion. Hodge theory also allows us to treat other notions of representation theory, such as lowest K-types, geometrically. Along the way we prove general results about mixed Hodge modules which are of interest on their own.

• 10/31 Gaston Burrull (BICMR)

Affine Bruhat order, Kazhdan-Lusztig invariance, and unexpected dualities

I will present experimental discoveries on the Bruhat order of affine Weyl groups, revealing intriguing combinatorial structure. In joint work with Libedinsky and Villegas, we classified thick dominant Bruhat intervals in type A2 tilde, showing that each poset is determined by the isometry class of a certain polygon, providing a strong bridge between Bruhat order and Euclidean geometry. 

These results suggest that the Lusztig-Dyer combinatorial invariance conjecture for Kazhdan-Lusztig polynomials may hold for surprisingly simple reasons. We conjecture that all nontrivial isomorphisms of affine Bruhat intervals appear only as global poset isomorphisms or piecewise local translations, and all the information of a Bruhat interval is captured by its dihedral subintervals. 

I also observed the remarkable phenomenon that some intervals are isomorphic to the dual of others, as if a non-existent phantom longest element exists in affine Weyl groups.

• 11/07 Aron Heleodoro (SIMIS)

Newton Decomposition of Loop Group and Affine Character Sheaves 

In this talk I will explain how one can decompose the loop group associated to a connected reductive group G into parts known as Newton strata. These remarkable strata are invariant under the conjugation of LG and by passing to Levi subgroups one can reduce questions on an arbitrary stratum to questions on basic strata, which are more manageable. I will then explain how one can use these strata to define and study a very sought-after category of character sheaves for loop groups (e.g. p-adic groups or Kac—Moody groups). This is based in joint ongoing work with Xuhua He and Xinwen Zhu.

• 11/14 (3:30 pm online) Tanmay Deshpande (TIFR, Mumbai)

A Verlinde formula for twisted conformal blocks 

Given a finite group G acting on a simple Lie algebra and a positive integral level, I will define the notion of twisted conformal blocks and describe an analogue of the Verlinde formula to compute their dimensions. Our approach is based on studying the relationship of twisted conformal blocks with the notion of G-crossed modular fusion categories and a categorical Verlinde formula in this setting. The main goal of the talk will be to describe this relationship. This is based on joint work with S. Mukhopadhyay.

• 11/21

• 11/28 Xuanzhong Dai 戴烜中 (RIMS)

• 12/05 Qixian Zhao 赵启弦 (YMSC)

• 12/12 Sian Nie 聂思安 (AMSS, CAS)

• 12/19

• 12/26 Yau Wing Li (Melbourne)

• 01/02 Raphaël Rouquier (UCLA)

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