报告题目：Unifying highest weight modular representation theories

报告人：林宗柱 教授，美国堪萨斯州立大学

报告时间：6月22日（周四），上午10:00-11:30

报告地点：东区管理科研楼1418教室

摘要：

Given a generalized Cartan matrix, there are many different algebras and groups one can attach to, the classical Kac-Moody Lie algebra and the corresponding Kac-Moody groups as well as quantum groups (superquantum groups) with one or many parameters appearing in the literature. One of the themes in representation theories of these algebras is to compute the decomposition numbers of irreducible modules in the universal highest weight modules. More generally, one want to study the Kazhdan-Lusztig theory, in particular, one wants to compute the Kazhdan-Lusztig polynomials. In this series of lectures, we will define Lusztig's modified quantum groups, called U-dot system, for each of these algebras. It turns out that the highest weight representations of these algebras are dependent only on the U-dot systems. We will prove that the U-dot systems are isomorphic up to base change. Thus one can compare different highest weight representation theories of different types of quantum groups/algebras by comparing their U-dot systems. Therefore their decomposition numbers as well as the Kazhdan-Lusztig polynomials can be transported freely among different quantum groups or algebraic groups as well as their modular representation theories. Lusztig's various conjectures in the path of proving Lusztig's character formula conjecture for algebraic groups in positive characteristic case using representations of quantum groups, representations of affine Kac-Moody Lie algebras is, in fact, to compare the U-dot systems. This is a joint work with Zhaobin Fan and Yiqiang Li.

报告人简介：林宗柱，美国堪萨斯州立大学终身教授，博士生导师，现任三峡数学研究中心主任，曾任美国科学基金会NSF项目主任和《中国科学：数学》编委。主要从事表示论、代数群以及量子群等方面的研究，论文发表在 Invent. Math.，Adv. Math., Trans. Amer. Math. Soc., CMP 和J. Algebra 等重要学术期刊上，标志性成果包括林-Nakano定理，出版学术著作五部。