Mathematical Sciences: Representations of Semisimple Groups over Finite Fields and Quantum Groups

数学科学：有限域和量子群上的半单群的表示

• 批准号: 9207285
• 负责人: George Lusztig
• 金额: $ 37.2万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9207285&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representations; Semisimple; Groups

This research is concerned with the representations of semisimple groups over finite fields and quantum groups. The principal investigator will study characters of finite dimensional irreducible representations of quantum groups at roots of 1. He will also study character sheaves and their relation with the irreducible characters of finite groups of Lie type. Finally, he will study canonical bases of enveloping algebras. One of the postdoctoral associates will work on Lang's conjecture in positive characteristic and Chow groups of some moduli spaces of curves. Another postdoctoral associate will work on a proof of the Lefschetz Trace Formula for algebraic stacks and the conjecture of Weil on Tamagawa Numbers over function fields. Another postdoctoral associate will study both the applications of analytic tools to the understanding of quantum groups and the use of quantum groups to approach questions of analytic interest. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.

本文主要研究有限域和量子群上的半单群的表示。主要研究人员将研究量子群在1的根处的有限维不可约表示的特征，他还将研究特征标层及其与李型有限群的不可约特征的关系。最后，他将学习包络代数的标准基。其中一位博士后助理将在曲线的一些模空间的正特征和Chow群中研究Lang猜想。另一位博士后助理将致力于证明代数堆栈的Lefschetz迹公式和函数域上关于Tamagawa数的Weil猜想。另一名博士后助理将学习分析工具在理解量子群中的应用，以及使用量子群来处理感兴趣的分析问题。量子群对数学家和物理学家来说都是一个新的研究领域。在数学方面，它结合了代数、分析和几何这三个最古老的纯数学领域，但它对致力于共形量子场论的物理学家非常感兴趣。