Arithmetic and Representation Theory of Reductive Groups

约化群通常以算术和几何结构的对称群的形式出现。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，对于著名的朗兰兹计划(在数论中)，需要对局部域上的约化群的酉表示进行完全分类。我建议使用各种工具来研究约化群的算术、群论和几何性质。在我过去使用的强大工具中，有一个很好的空间的几何学，被称为Bruhat-Tits理论提供的群的“建筑”。这一理论，加上对还原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。在我自己关于被称为“格”的某些“大”子群的刚性的工作中，以及在我研究关于这些群的算术问题，包括同余子群问题时，这些技巧起到了至关重要的作用。在我与Allen Moy的合作中，我们使用了局部域上的Bruhat-Tits约化群理论来解决关于它们的表示的一些问题，并对深度为零的可容许表示进行了分类。随后，其他几位数学家使用我们的框架和技术，在表示理论和调和分析中找到了许多有趣问题的解决方案。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可容许表示的分类。我还将研究某些(各向异性)群仍未解决的同余子群问题。后者需要首先了解它们的正规子群。我正在与Andrei Rapinchuk和Yoav Segev一起研究这个问题。我还计划与Andrei Rapinchuk合作写一本关于同余子群问题的书。

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.