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)

11:10am-12:10am Liang Xiao 肖梁 (PKU)

• 10/24  Double Talks

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

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)

• 11/07 Aron Heleodoro (SIMIS)

• 11/14 Tanmay Deshpande (TIFR, Mumbai)

• 11/21

• 11/28 Xuanzhong Dai (RIMS)

• 12/05

• 12/12 Sian Nie 聂思安 (AMSS, CAS)

• 12/19

• 12/26

• 01/02

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