数学学科2023系列学术报告之十二

　　报告题目：Hecke endomorphism algebras: stratification, finite groups of Lie type, and i-quantum groups

　　报告人：杜杰，新南威尔士大学教授

　　时间：2023年8月24日(星期四）10:00-11:00

　　地点：1-301

　　摘要：Hecke endomorphism algebras are a natural generalization of q-Schur algebras from symmetric groups to arbitrary Coxeter groups. They appear naturally in the study of representations of finite groups of Lie type, especially finite general linear groups whose representations are connected with those of quantum general linear groups.Generally speaking, the structure and representations of Hecke endomorphism algebras are difficult to understand if the associated Coxeter group is not the symmetric group. Over twenty years ago, B. Parshall, L. Scott and the speaker investigated some rough stratification structure for those associated with Weyl groups. We conjectured the existence of a finer stratification by Kazhdan–Lusztig two-sided cells for an enlarged endomorphism algebra. I will report on the progression of ideas in our successful efforts to prove a (very slightly modified) version of the conjecture. Inspired by an Ext1 vanishing condition uncovered in a local case, we use exact categories to formulate by analogy a tractable global version, not mentioning localization and often requiring less vanishing. After constructing many relevant exact category settings, we are eventually able to prove this exact category Ext1 vanishing in one of them that contains all the filtered objects we need.

　　报告人简介：

　　杜杰，澳大利亚新南威尔士大学教授，在Weyl群的胞腔分解、代数群，q-Schur代数及其表示、在Ringle-Hall代数及量子群和量子超群等方面取得了一系列原创性的成果，目前已经在国际一流杂志发表论文80余篇，合作完成专著《Finite dimensional algebras and quantum groups》和《A double Hall algebra approach to quantum affine Schur-Weyl theory》,分别在美国数学会和伦敦数学会出版。