2025年港大-中科院表示论研讨会

时间：2025年5月19-20日

邀请报告人：

陈佳源（香港大学）     陈全勇（哈尔滨工程大学）

陈晓煜（上海师范大学） 方杰鹏（香港大学）

何旭华（香港大学）     李鹏辉（清华大学）

钱子诚（中科院）       吴凯迪（香港大学）

余  君（北京大学）     杨若涛（中科院）

谢锴涛（香港大学）     张鸿锋（香港大学）

张伟楠（香港大学）

组织者：

陈佳源（香港大学）

何旭华（香港大学）

聂思安（中科院）

地点：

中国科学院数学与系统科学研究院 晨兴110

报告安排

报告题目和摘要

Discrete series and Springer correspondence for type H_4

陈佳源

The classical Springer correspondence constructs Weyl group representations from the geometry of Springer fibers. Lusztig models this theory in the framework of the graded Hecke algebra. From this algebraic viewpoint, one can establish the Springer correspondence for all finite reflection groups. For example, nilpotent orbits can be replaced by certain combinatorial data studied by Heckman-Opdam. In this talk, I shall explain several ingredients to establish the Springer correspondence for the reflection group of type H_4, including the Ext-vanishing properties of discrete series. The main results in this talk are joint with Simeng Huang (Fudan University).

Higher Ext between locally analytic generalized Steinberg with applications to higher L invariants for GL(n)

钱子诚

The definition/study of L invariants via locally analytic representation theory has been initiated by Breuil in the case of GL(2,Qp). The GL(3,Qp) case has been studied extensively by Schraen and Breuil-Ding from different aspects. Motivated by their work and Gehrmann's work on automorphic L invariants, it is clear that understanding certain higher Ext groups between various locally analytic generalized Steinberg representations is crucial to develop the theory of (higher) L invariants for GL(n). A key example of such Ext groups is Ext_G^{n-1}(1,St_G^{an}) with G=PGL(n,K) and St_G^{an} being the locally analytic Steinberg representation of G. We would report on some recent progress towards this direction.

Braid group actions on the Poisson homogeneous spaces arising from quantum symmetric pairs

张伟楠

The fundamental work of De Concini-Kac-Procesi shows that one can recover the dual Poisson Lie group by taking a suitable semi-classical limit on quantum groups. The quantum symmetric pairs are quantization of symmetric pairs, and they involve coideal subalgebras of quantum groups, called i-quantum groups. Recently, Song obtained a class of (dual) Poisson homogeneous spaces by taking suitable semi-classical limits on i-quantum groups. In this talk, using the braid group actions on i-quantum groups, we will construct braid group actions and PBW type basis on these Poisson homogeneous spaces. This is joint with Jinfeng Song (National University of Singapore).

Geometric approach to the i-quantum group of affine type D

陈全勇

We establish a lattice presentation of complete and n-step flag varieties of affine type D and study the structures of Schur algebra and Lusztig algebra associated to the partial flag varieties of affine type D. We show that there is a subalgebra of Lusztig and the quantum groups arising from this subalgebra via stabilization procedures is a coideal subalgebra of quantum group of affine sl type. We construct monomial and canonical bases of the idempotented quantum algebra and establish the positivity properties of the canonical basis, respect to multiplication and the bilinear pairing.  

Classification of GL(n,R)-distinguished unitary representation of GL(n,C)

张鸿锋

In this talk, I will explore distinction problems, with a focus on classifying unitary representations of GL(n,C) that are distinguished by GL(n,R). Specifically, I will present criteria for determining when a unitary representation admits a non-zero GL(n,R)-invariant linear form, and compare the similar results of GL(n) over p-adic field. If time permits, I will further illustrate the explicit construction of such invariant periods and establish their non-vanishing on distinguished minimal K-types. This is joint with Basudev Pattanayak and Kaidi Wu.

Casselman-Jacquet functor and Jacquet functor

吴凯迪

The Jacquet functor plays a fundamental role in representation theory, particularly in the study of branching laws. While the computation of Jacquet modules and derivatives is well-developed in the p-adic setting, significant challenges remain for real reductive groups--both for smooth representations and Harish-Chandra modules. In this talk, we investigate the Jacquet module of Harish-Chandra modules through the lens of the Casselman-Jacquet functor. This functor exhibits several advantageous properties, most notably its geometric realization. By leveraging this framework, we establish a formula describing the associated variety of the Jacquet module. This is based on a joint work in progress with Hongfeng Zhang.

Birkhoff-Bruhat Atlas and Total Positivity

谢锴涛

A Birkhoff-Bruhat atlas locally models a stratified space by open Kazhdan-Lusztig varieties on a flag variety. In this talk, we will explore some applications of Birkhoff-Bruhat atlases to the study of total positivity. This is based on a joint work in progress with Huanchen Bao and Xuhua He.

Cocenters of Hecke algebras/categories

何旭华

On the Gaiotto conjecture

杨若涛

The Gaiotto conjecture says that the category of finite dimensional representations (resp, category O) of the quantum supergroup is closely related to a certain twisted sheaf category on the affine Grassmannian (resp, affine flags). It is the quantum extension of (a particular case of) the relative Langlands conjecture. In this talk, we will explain the background and the statement of the Gaiotto conjecture and recent progress on this conjecture. If time permits, we will also discuss the strategy of proofs as well as remaining questions. It is based on ongoing and in-preparation works joint with M.Finkelberg and R. Travkin.

Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

方杰鹏

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the canonical bases and the fundamental classes. This is based on a joint work with Yixin Lan.

Relative Serre duality for Hecke categories

李鹏辉

In a joint work with Quoc P. Ho, We prove a conjecture of Gorsky, Hogancamp, Mellit, and Nakagane in the Weyl group case. Namely, we show that the left and right adjoints of the parabolic induction functor between the associated Hecke categories of Soergel bimodules differ by the relative full twist.

Positivity of Fourier transform of zonal spherical functions

余君

Given a semisimple real linear group, a zonal spherical function is matrix coefficient associated to the unique spherical vector with value 1 at identity element in a unitary spherical principal series, which are important object in representation theory and harmonic analysis. Each zonal spherical function is a positive definite function. Hence, its Fourier transform along a maximal split torus is everywhere non-negative by a classical theorem of Salomon Bochner. In this talk we report a result in a recent joint work: the Fourier transform along a maximal split torus of any zonal spherical function takes positive value everywhere.

The boundness of Lusztig’s a-function for Coxeter groups of finite rank

陈晓煜

Lusztig defined the a-function for a Coxeter group in 1984, and proposed the famous conjecture P1-P15 in 2004, which will hold for equal parameter case once the positivity of Kazhdan-Lusztig polynomials and the boundness of of a-function hold. The boundness conjecture of a-function for finite rank Coxeter groups is one of the four open problems on Hecke algebras, and is of great interest and still open in most cases. We prove that: (1) Each term in the expansion of product of standard bases of Hecke algebra gives rise to a set of reflecting hyperplanes that pairwisely intersect in the interior of Tits cone (intersecting subset), (2) The cardinality of intersecting subsets is bounded. As a consequence, we prove that a-function is bounded for any Coxeter group of finite rank.