中科院李理论研讨会（2022）

时间：2022年11月11，12，18，19，25，26日，12月2，3日

地点：腾讯会议： 759-9643-5802 会议密码：123321

      会议链接：https://meeting.tencent.com/dm/noNRTPSDjxDV

会议主办方：中国科学院数学与系统科学研究院

会议组织者：孙斌勇（中国科学院数学与系统科学，浙江大学）

            包益欣（哈尔滨工业大学（深圳））

            陈阳洋 （江南大学）

            王楚韵（西湖大学）

会议简介：群体现了抽象元素与结构的对称性，而李群则体现了具有拓扑结构的抽象空间的“连续对称性”。这种对称性在代数几何，代数数论甚至物理中也有丰富的应用。李理论作为目前基础数学的主流方向之一，正是研究李群，李代数以及相关理论的一门学科。在本次研讨会中，我们邀请了国内优秀的李理论的专家学者们，为大家介绍李理论的最近的发展。

会议报告人：

白占强（苏州大学）

蔡延安（苏州大学）

陈佳源（上海数学中心）

陈阳洋（江南大学）

董超平（苏州大学）

韩刚（浙江大学）

黄家裕（香港中文大学（深圳））

李宁（南开大学）

栾永志（山东大学）

马家骏（厦门大学）

肖维（深圳大学）

薛华健（汕头大学）

王之程（苏州大学）

张鸿锋（香港中文大学（深圳））

会议日程：（11月11日，11月12日安排）

（11月18日，11月19日安排）

（11月25日，11月26日安排）

（12月2日，12月3日安排）

报告摘要：

11月11日：

Wavefront sets and descent method for finite symplectic groups

报告人：王之程 （苏州大学）

摘要：D. Jiang and L. Zhang proposed a conjecture which related the wavefront sets of irreducible admissible representations and the descent method in the local fields case. Recently, they and D. Liu define the arithmetic wavefront set which based on the local Gan-Gross-Prasad conjecture. In this talk, we study the relationship between wavefront sets, descent method and finite Gan-Gross-Prasad problem for finite symplectic groups. We calculate the multiplicity of an irreducible representation in the generalised Gelfand-Graev representation corresponding to finite fields analogies of the arithmetic wavefront sets. In particular, for cuspidal representations, we give certain multiplicity one theorem and show that the finite fields analogy of arithmetic wavefront sets coincides with the wavefront sets in the sense of G.Lusztig and N. Kawanaka..

Dirac series of rank seven exceptional Lie groups

报告人：董超平（苏州大学）

摘要：This talk will introduce the classification of Dirac series of the three linear real forms of complex E7: E7(-25), E7(-5) and E7(7). We will try to explain how the group structure affects the classification results, in particular the cancellation phenomenon.

11月12日：

               Dixmier Conjucture

报告人：韩刚（浙江大学）

摘要：会介绍一下人们关于Weyl代数A1的Dixmier猜想的一些进展，包括A1与Heisenberg代数和sl_2的泛包络代数之间的关系。

The cohomology of locally analytic representations

报告人：陈阳洋 （江南大学）

摘要：We will report Jan Kohlhaase’s work on cohomology theory of p-aidc reprensentations of p-adic groups

11月18日

 Annihilator varieties of highest weight modules for classical Lie algebras

报告人：白占强(苏州大学)

摘要： Let g be a semisimple complex Lie algebra. For a U(g)-module M, let I(M) =Ann(M) be its annihilator ideal in U(g) and J(M) be the corresponding graded ideal in S(g) = gr(U(g)). The zero set of J(M) in the dual vector space g* of g is called the annihilator variety of M, which is also called the associated variety of I(M). A two-sided ideal in U(g) is called primitive if it is the annihilator of an irreducible representation of g. In 1985, Joseph proved that the associated variety of a primitive ideal is the Zariski closure of a nilpotent orbit in g*. In this talk, we will give a combinatorial characterization of the annihilator varieties of highest weight modules for classical Lie algebras. We find that our method is much faster than Atlas software after some communications with Vogan.

Generic Hecke algebra and theta correspondence over finite fields I : Conservation relation. 

报告人：马家骏（厦门大学）

摘要：Theta correspondence is a basic tool to relate representations of different classical groups, which is expected to be a concrete realization of certain Langlands functorialities. 

In this talk, we consider the theta correspondence of type I dual pairs over a finite field F_q. Aubert-Michel-Rouquier and Pan established an explicit formula for theta correspondence between unipotent representations using the character theory of finite groups of Lie type. We will discuss an alternative method to attack the problem: considering the correspondence of generic Hecke algebras modules. We will explain how the Fourier transform enters the picture and how

the normalization of Hecke algebra leads to the conservation relation for "theta cuspidal"-representations. If time permits, I will also discuss how the correspondence of the correspondence between theta cuspidal representation is pinned by a similar computation. This is joint work with Jialiang Zou and Congling Qiu.

Generic Hecke algebra and theta correspondence over finite fields II : Correspondence of general representations.

报告人：马家骏（厦门大学）

摘要：In this talk, we continue the discussion on the Hecke algebras approach to theta correspondence. We will first explain the classification of certain Harish-Chandra series representations. Then we give an explicit basis for certain Hecke algebra modules. In the end, we deduce the main result via Tits deformation. This is joint work with Jialiang Zou and Congling Qiu.

Full theta lifting for type II dual pairs

报告人：薛华健（汕头大学）

摘要：In this talk, we will introduce the full theta lifting for type II reductive dual pairs with a focus on the equal rank case and the almost equal rank case.

11月19日

Equivalent definitions of generalized Gan-Gross-Prasad relevant pairs for p-adic general linear groups.

报告人：陈佳源（上海数学中心）

摘要: Gan-Gross-Prasad (2020) proposes a notion of relevant pairs, governing the branching law of Arthur type representations. I will explain some equivalent formulations for generalizing the notion to all irreducible representations of p-adic general linear groups. We shall focus on the viewpoint from Kashiwara’s crystal operators.

Cohomology of locally analytic representation II

报告人：陈阳洋 （江南大学）

摘要：We will continue on the report of cohomology of locally analytic representation, now focus on the main results about this theory.

11月25日

Representations for \mathbb{Z}-graded Lie superalgebras of growth 1.

报告人：蔡延安（苏州大学）

摘要：We will introduce recent progress on simple weight modules with finite dimensional weight spaces for \mathbb{Z}-graded Lie superalgebras of growth 1.

Jantzen coefficients and generalized Harish-Chandra modules

报告人：肖维 （深圳大学）

摘要：In this talk, we will give some recent results about generalized Harish-Chandra modules in representation theory of Lie groups and Lie algebras, focusing on highest weight modules. In particular, we introduce the Jantzen coefficient which is a useful tool in the research about highest weight modules.

12月2日

  Compute the multiplicities of K-types via intertwining operators

报告人：张鸿锋（香港中文大学）

摘要: The intertwining operator play an important role in the determination of unitary dual of real groups. Firstly, we review Barbasch’s paper “The unitary spherical spectrum for split classical groups” and explain how to determine the unitarity of Hermitian modules. Secondly, we show how to get the multiplicities of some K-types of the irreducible Langlands subquotient by computing the intertwining operators.

On some results of transfer in local theta correspondence

报告人：李宁（南开大学）

摘要：The notion of transfer is a way to relate representation theory of different real forms of a complex semisimple Lie group. It is realized by the composition of the Zuckerman functor and the forgetful functor. In this talk, I will explicitly describe the transfer of 𝜃𝜃(1) of Sp(2n,R) in the case of non-stable range via the study of K-types. As a corollary, we will prove certain compatibility of local theta correspondence and the transfer functor.

Lie Algebra e_{7 ½} and Dynkin Index

报告人：栾永志 （山东大学）

摘要：The Lie algebra e_{7 ½} comes from Pierre Deligne’s work on the exceptional series of Lie groups. Using the triality algebra, J. Landsberg and L. Manivel construct this Lie algebra in 2006. We study the construction of this Lie algebra, and branching rules of e_{7 ½} to the special linear Lie algebra sl(3,C) and complex exceptional Lie algebra g2. We use the computer algebra system SageMath to carry out our calculations. We find 24 behaves like the `dual Coxeter number’ of e_{7 ½}.

On a conjecture of Salamanca-Riba and Vogan for U(p,q)

报告人：黄家裕（香港中文大学（深圳））

摘要：在1998年, Salamanca-Riba 和 Vogan 在《数学年报》提出一个猜想。倘若猜想成立，则能把实约化群酉表示分类的问题，简化到只考虑最低K型为unitarily-small的 (g,K)-模上。基于初稿（arxiv:2210.08684），我将证明猜想于 U(p,q) 成立。