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

摘要：Gopal Prasad的主要研究方向是约化李氏代数群——主要研究几何或数论根源的问题。他一直在研究关于某些有趣的子群的问题，例如，算术子群或几何或数论中出现的其他大型（Zariski-dense）子群。确定这些群体的所有正常亚群是很重要的。像局部域和全局域这样有趣的域，有kneser - tits和Margulis-Platonov的猜想，它们提供了对正规子群的描述。对于许多类别的群体来说，这些猜想都得到了肯定的证实。然而，仍然有一些非常有趣的群体，这些猜想是开放的。普拉萨德计划研究这些群体。对于算术群，著名的“同余子群问题”是关于有限指标的正规子群是否存在同余子群的问题。解决这一问题最成功的方法有两个部分：(1)“元核”的计算。(2)同子群核的中心性。Prasad与m.s. raghunathan和A.S.Rapinchuk共同完成了对所有群体的metapletickernel的非常精确的计算。另一方面，对于一些重要的群，同余子群核的中心性仍然是未知的。普拉萨德的目标是调查这些群体，并在已知案例中找到概念上更好的中心性证据。他将继续与A.S. Rapinchuk在这个项目上合作。普拉萨德将研究的另一个课题是约化p进群的表示理论。Prasad一直对不可约可容许表征的分类很感兴趣，他的目标是根据“类型”理论获得分类。在这个方向上，对于一般约化群来说，他与艾伦·梅的联合研究开创了一个开端——他们引入的几何技术——不代表理论已经被证明是非常有用的。普拉萨德计划继续他对可接受陈述分类的研究。许多几何和数论对象的对称集合形成群。为了研究这些几何数论对象，研究它们的对称群是很重要的。普拉萨德一直在研究与这些群体及其子群体相关的问题。这些问题具有数论或几何根源，因此它们的解决方案将在这些领域具有重要的应用。Prasad建议研究kneser - tits和Margulis-Platonov问题，这两个问题提供了对正子群的推测性描述；这两个问题已经为一大群人解决了，但一些具有挑战性的案件仍在审理中。Prasad还建议继续他在著名的同余子群问题上的工作，他和他的合作者Raghunathan和Rapinchuk在这个问题上做出了许多重要的贡献。在另一个方向上，Prasad建议研究约化p进群的表示理论。这些群的表示在各种情况下自然出现，它们的研究是现代数论中朗兰兹程序的重要组成部分。普拉萨德和艾伦·梅在该领域引入的几何方法被证明是非常有用的。Prasad建议改进这些方法，对所有可接受的表征进行完整的分类。

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.