Representation Theory and Automorphic Forms

本文主要研究自同构型、约化群的表示及其应用。Dan Barbasch和他的几位同事提议继续研究实群和p-adic群的酉对偶。特别是，他将研究团结的必要条件。他还将继续研究单位代表制，特别是其相关的周期。他将与不同的同事一起研究对偶还原对对应中表示的出现，对于p-进群，他将研究Bernstein中心。Birgit Speh将与多位同事一起继续研究局部对称空间的上同调。特别是，她将学习模符号的表示理论描述。她还将致力于证明Arthur-Selberg迹公式中项的一致收敛。预计这一建议的结果将大大有助于理解局部对称空间的几何和拓扑。研究生和本科生以及博士后教师将参与研究这项研究产生的问题。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解的函数有关。这些性质在数学上被表示为解形成一个约化群的么正表示。这项提案的一个主要部分涉及到构成要素的确定，这些要素被称为酉不可约表示。上述与数论有关的函数称为自同构函数，它们具有酉不可约表示形式的展开式。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是数论应用的重要问题，也是这一建议的重要组成部分。

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.