Arithmetic and Representation Theory of Reductive Groups over Local and Global Fields

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Abstract for NSF-award number DMS-0400640 of PrasadAbstract: Gopal Prasad's primary focus has been study of reductive Lieand algebraic groups--mostly on problems which have either geometric ornumber theoretic origin. He has been investigating questions about certaininteresting subgroups, for example, arithmetic subgroups or other large(Zariski-dense) subgroups which arise in geometry or number theory. Itwould be important to determine all normal subgroups of these groups. Overinteresting fields like local and global fields, there are conjectures ofKneser-Tits and Margulis-Platonov which provide a description of normalsubgroups. These conjectures have been settled in the affirmative for manyclass of groups. However, there still remain some very interesting groupsfor which these conjectures are open. Prasad plans to study these groups.For arithmetic groups, the famous "congruence subgroup problem" is thequestion whether any normal subgroup of finite index contains a congruencesubgroup. The most successful approach to settling this problem has twoparts: (1) Computation of the "metapectic kernel". (2) Centrality of thecongruence subgroup kernel. A very precise computation of the metaplectickernel for all groups has been done in Prasad's joint work withM.S.Raghunathan and A.S.Rapinchuk. On the other hand, the centrality ofthe congruence subgroup kernel is still unknown for some important classof groups. Prasad's goal is to investigate these groups and also find aconceptually better proof of the centrality in the known cases. He willcontinue his collaboration with A.S. Rapinchuk on this project. Anothertopic on which Prasad will work on is the representation theory ofreductive p-adic groups. Prasad has been interested in classification ofirreducible admissible representations where his goal is to obtain aclassification in terms of theory of "types". A begining in thisdirection, for general reductive groups, was made in his joint work withAllen Moy--the geometric techniques which they introduced inrepresentation theory have turned out to be very useful. Prasad plans tocontinue his research towards classification of admissiblerepresentations. The set of symmetries of many geometric and number theoretic objects forma group. For studying these geometric and number theoretic objects, it isimportant to study their groups of symmetries. Prasad has been studyingproblems related to these groups and their subgroups. These problems havenumber theoretic or geometric origins and therefore their solution willhave important applications to these areas. Prasad proposes to work on theKneser-Tits and Margulis-Platonov problems which provide conjecturaldescription of normal subgroups; both the problems have been settled for alarge class of groups but some challenging cases remain open. Prasad alsoproposes to continue his work on the famous congruence subgroup problemwhere he and his collaborators Raghunathan and Rapinchuk have made manyimportant contributions. In another direction, Prasad proposes to work onthe representation theory of reductive p-adic groups. Representations ofthese groups arise naturally in various contexts and their study is animportant component of the Langlands program in modern number theory. Thegeometric methods which Prasad and Allen Moy introduced in the area haveturned out to be very useful. Prasad proposes to refine these methods togive a complete classification of all admissible representations.