Representation Theory and Automorphic Forms

表示论和自守形式

• 批准号: 0300172
• 负责人: Birgit Speh
• 金额: $ 67.11万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300172&HistoricalAwards=false
• 关键词: Representation; Theory; Automorphic; Forms

AbstractSpeh/BarbaschThis proposal is concerned with the study of automorphic forms,representations of reductive groups and their applications. Dan Barbasch, together with various coworkers, is proposing tocontinue investigations of the unitary dual of real and p-adicgroups. In particular he will study necessary conditions forunitarity. He will also continue the study of unipotentrepresentations, in particular their associated cycles. Joint with various coworkers he will study occurences of representations in thedual reductive pairs correspondence, and for p-adic groups he willstudy the Bernstein center. Birgit Speh, together withvarious coworkers, will continue the study of cohomology of locally symmetricspaces. In particular she will study representation theoreticdescriptions of modular symbols. She will also work on proving uniform convergence of terms in the Arthur-Selberg trace formula. It is expected that the results of this proposal will contribute significantly to the understanding of the geometry and topology of locally symmetric spaces. Graduate and undergraduate students as well as postdoctoral faculty are expected to be involved in studying problems generated by this research. Many problems in number theory and mathematical physics are concernedwith functions that are solutions to differential equations which havecertain symmetry properties. These properties are expressed inmathematics as saying that the solutions form a unitary representationof a reductive group. A major part of this proposal is concerned with thedetermination of the building blocks which are calledunitary irreducible representations. The aforementioned functionsrelevant to number theory are called automorphic functions and haveexpansions in terms of unitary irreducible representations. Theseexpansions can be thought of as generalizations of classical Fourierseries. The problem of convergence of these expansions is importantfor applications to number theory, and forms an important componentof this proposal.

Speh/Barbash这一建议涉及到自守形式、约化群的表示及其应用的研究。Dan Barbasch和他的同事们提议继续研究真实的群和p-群的酉对偶。他将特别研究酉性的必要条件。他还将继续研究单极子的介绍，特别是其相关的周期。他将与多位同事一起研究对偶约化对对应中表示的出现，对于p进群，他将研究伯恩斯坦中心。Birgit Speh和他的同事们将继续研究局部拓扑空间的上同调。特别是她将研究表示理论的模块化符号的描述。她还将致力于证明亚瑟-塞尔伯格迹公式中的一致收敛项。预计这一建议的结果将有助于显着的局部对称空间的几何和拓扑的理解。研究生和本科生以及博士后教师预计将参与研究本研究产生的问题。 数论和数学物理中的许多问题都与具有一定对称性的微分方程的解函数有关。这些性质在数学上表示为，解形成一个约化群的酉表示。这一建议的主要部分是关于确定的积木是所谓的unitary不可约表示。上述与数论相关的函数称为自守函数，并具有酉不可约表示的展开式。这些展开式可以看作是经典傅里叶级数的推广。 这些展开式的收敛问题对于数论的应用是重要的，并且形成了这个建议的一个重要组成部分。