Arithmetic and Representation Theory of Reductive Groups

还原群的算术与表示论

• 批准号: 0100429
• 负责人: Gopal Prasad
• 金额: $ 10.56万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100429&HistoricalAwards=false
• 关键词: Arithmetic; Representation; Theory; Reductive; Groups

Reductive groups often arise as groups of symmetries of arithmetic and geometric structures. This connection makes their study very interesting and it is also a major source of important questions. For example, for the celebrated Langlands Program (in Number Theory), a complete classification of unitary representations of reductive groups over local fields is needed.I propose to investigate arithmetic, group theoretic and geometric properties of reductive groups using various tools. Among the powerful tools I have used in the past is the geometry of a nice space known as the "building" of the group provided by the Bruhat-Tits theory. This theory, together with a detailed understanding of the structure of reductive groups, and their cohomology (these are certain subtle geometric invariants of the groups), has helped to settle many important questions about these groups. In my own work on rigidity of certain "large" subgroups known as "lattices", and also in my study of arithmetic questions about these groups, including the congruence subgroup problem, these techniques played crucial role. In my joint work with Allen Moy, the Bruhat-Tits theory of reductive groups over local fields was used to settle some questions about their representations and also to classify admissible representations of depth zero. Subsequently, several other mathematicians used our frame-work and techniques to find solutions of many interesting problems in the representation theory and harmonic analysis. I will use some of the geometric techniques mentioned above to find a classification of irreducible admissible representations of reductive groups over local fields. I will also investigate the congruence subgroup problem which remains unresolved for certain (anisotropic) groups. The latter would require understanding their normal subgroups first. I am working on this question with Andrei Rapinchuk and Yoav Segev. I also plan to write a book on the congruence subgroup problem in collaboration with Andrei Rapinchuk.

约化群经常作为算术和几何结构的对称群出现。 这种联系使他们的研究非常有趣，也是重要问题的主要来源。 例如，著名的朗兰兹计划（数论），一个完整的分类的酉表示的约化群在当地fields.I建议调查的算术，群论和几何性质的约化群使用各种工具。 在我过去使用过的强大工具中，有一个很好的空间的几何学，它被称为布鲁哈特-山雀理论所提供的群体的“建筑”。 这个理论，加上对约化群的结构及其上同调（这些是群的某些微妙的几何不变量）的详细理解，帮助解决了关于这些群的许多重要问题。在我自己的工作刚性的某些“大”分组被称为“格”，也在我的研究算术问题，这些团体，包括同余分组问题，这些技术发挥了至关重要的作用。在我与艾伦莫伊的合作中，局部域上约化群的布鲁哈特-蒂茨理论被用来解决关于它们的表示的一些问题，并对深度为零的容许表示进行分类。 随后，其他几位数学家使用我们的框架和技术，找到了许多有趣的问题，在表示论和调和分析的解决方案。 我将使用上面提到的一些几何技巧来找到局部域上约化群的不可约容许表示的分类。我还将调查的同余子群问题，仍然悬而未决的某些（各向异性）群体。 后者首先需要了解它们的正规子群。 我正在与Andrei Rapinchuk和Yoav Segev研究这个问题。 我还计划与Andrei Rapinchuk合作写一本关于同余子群问题的书。