Arithmetic and Representation Theory of Reductive Groups

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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.