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

摘要：Gopal Prasad的主要关注点是对约化Lie和代数群的研究--主要是关于几何或数论起源的问题。他一直在研究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稠密)子群。确定这些群的所有正规子群是很重要的。像局部域和全局域一样，也有Kenser-Tits猜想和Marguis-Platonov猜想，它们提供了正规子群的描述。这些猜想对于许多类别的群体都是肯定的。然而，仍然有一些非常有趣的群体，这些猜想是开放的。Prasad计划研究这些群。对于算术群，著名的“同余子群问题”是关于有限指标的任何正规子群是否包含同余子群的问题。解决这一问题最成功的方法有两个：(1)“元核”的计算。(2)同余子群核的中心性。在Prasad与M.S.Raghunathan和A.S.Rapinchuk的联合工作中，对所有组的亚五核进行了非常精确的计算。另一方面，对于某些重要的群类，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更好的中心性证据。他将继续与A.S.Rapinchuk在这个项目上的合作。Prasad将致力于的另一个主题是约化p-进群的表示理论。普拉萨德一直对可约可接纳表征的分类感兴趣，他的目标是根据“类型”理论获得分类。在他与艾伦·莫伊的合作中，他在这个方向上为一般的约化群做了一个开始--他们在表示理论中引入的几何技巧被证明是非常有用的。普拉萨德计划继续他的研究，对可接受的演讲进行分类。许多几何和数论对象的对称性的集合组成一个群。为了研究这些几何和数论对象，研究它们的对称性群是很重要的。普拉萨德一直在研究与这些群体及其子群体相关的问题。这些问题既有理论上的渊源，也有几何上的渊源，因此它们的解将在这些领域有重要的应用。Prasad建议研究对正规子群提供猜想描述的Kenser-Tits和Marguis-Platonov问题；这两个问题对于大类群都已得到解决，但仍有一些具有挑战性的情况有待解决。Prasad还建议继续他在著名的同余子群问题上的工作，在这个问题上，他和他的合作者Raghunathan和Rapinchuk已经做出了许多重要贡献。在另一个方向上，Prasad提议致力于约化p-进群的表示理论。这些群的表示是在不同的背景下自然产生的，它们的研究是现代数论朗兰兹计划的重要组成部分。Prasad和Allen Moy在该地区介绍的几何方法已被证明是非常有用的。Prasad建议改进这些方法，以给出所有可接受的陈述的完整分类。

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.