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

局部和全局域上还原群的算术和表示论

• 批准号: 0400640
• 负责人: Gopal Prasad
• 金额: $ 13.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400640&HistoricalAwards=false
• 关键词: Arithmetic; Representation; Theory; Reductive; Groups

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.

摘要为NSF奖号DMS-0400640的普拉萨德摘要：戈帕尔普拉萨德的主要重点一直是研究约化李和代数群-主要是对问题，无论是几何ornumber理论的起源。他一直在调查的问题certainly有趣的小组，例如，算术分组或其他大（Zagliki密集）分组出现在几何或数论。这将是重要的，以确定所有正常的子群，这些群体。像局部域和全局域一样，有Kneser-Tits和Margulis-Platonov的结构，它们提供了正规子群的描述。对于许多类别的群体来说，这些假设已经得到了肯定的解决。然而，仍然有一些非常有趣的群体，这些课程是开放的。 对于算术群，著名的“同余子群问题”是指任意有限指数的正规子群是否包含一个同余子群的问题。解决这一问题最成功的方法有两个部分：（1）计算“元几何核”。(2)一致子群核的中心性。 普拉萨德与M.S.Raghunathan和A. S. Rapinchuk的联合工作中对所有群的元格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，并在已知的案例中找到更好的中心性证据。他将继续与A.S.合作。Rapinchuk在这个项目上。另一个主题上的普拉萨德将工作是代表性理论的还原p进群。Prasad一直对不可约容许表示的分类感兴趣，他的目标是根据“类型”理论获得分类。一个开始在这个方向上，一般还原群，是在他的联合工作与艾伦莫伊-几何技术，他们介绍了在代表性理论已被证明是非常有用的。普拉萨德计划继续他的研究，对可接受的陈述分类。许多几何和数论对象的对称性的集合形成一个群。为了研究这些几何和数论对象，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亚群体有关的问题。这些问题有许多的理论或几何根源，因此它们的解在这些领域有重要的应用。Prasad提出的工作theNeser-Tits和Margulis-Platonov问题，提供了正规子群的数学描述;这两个问题已经解决了一大类群，但一些具有挑战性的情况仍然开放。普拉萨德还建议继续他的工作，著名的同余子群问题，他和他的合作者Raghunathan和Rapinchuk作出了许多重要贡献。在另一个方向上，普拉萨德提出工作onthe表示理论的约化p-adic集团。这些群的表示在各种背景下自然出现，它们的研究是现代数论中朗兰兹纲领的重要组成部分。Prasad和艾伦Moy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，对所有可接受的表示进行完整的分类。