Reductive Dual Pairs in Exceptional Groups: Cohomology, and Unipotent Representations

摘要李 θ级数与约化对偶对的“经典”理论 在辛群中已经成为表示的标准部分 理论和自守形式，并继续吸引大量的 兴趣和活动。例外群的相似理论 这是几年前开始的，现在正在快速发展。 在以前的NSF赞助项目中，提案人卓有成效地研究了 理论中的一些基本问题，如 无穷小的字符，并开发了新的方法来确定 特殊对偶对的离散谱。在这里，他建议 继续研究剩下的基本问题，并应用于 具体问题的结果，如算术的上同调 流形和幂单表示。他计划将自己的联合收割机 与汉堡和Sarnak的方法，并希望建立密切的 某些非对称齐性空间的离散级数与约化对偶对的谱之间的关系。 李群表示理论是研究 某种对称性。当两个群体被嵌入一个更大的 环境群中的一个有趣的方式，人们可以问， 的大集团将行为限制在两个嵌入式 组这通常会导致深层信息， 在表示论和数论中都有重要作用。 在提议者和其他数学家早期工作的基础上， 该项目旨在解决几个基本问题， 研究这种限制的对称性，并应用结果 对几何和算术的研究， 与所讨论的组相关联的空间。

Abstract Li The ``classical'' theory of theta series and reductive dual pairs in the symplectic group has become a standard part of representation theory and automorphic forms, and continues to attract a great amount of interest and activity. The analogous theory for exceptional groups was begun just a few years ago, and is going through rapid development. In previous NSF sponsored projects, the proposer fruitfully studied some basic questions in the theory, such as the correspondences of infinitesimal characters, and developed new methods to determine the discrete spectrum of exceptional dual pairs. Here he proposes to continue his study of the remaining fundamental problems, and to apply the results to specific problems such as cohomology of arithmetic manifolds and unipotent representations. He plans to combine his own methods with those of Burger and Sarnak, and hopes to establish close relations between discrete series of certain non-symmetric homogeneous spaces and the spectrum of reductive dual pairs. The theory of Lie group representations is concerned with the study of certain kinds of symmetries. When two groups are embedded in a larger ambient group in a interesting way, one could ask how the symmetries of the larger group would behave upon restriction to the two embedded groups. Frequently this leads to deep information which plays important roles in both representation theory and number theory. Building upon earlier work by the proposer and other mathematicians, this project aims to settle several fundamental questions in the study of such restrictions of symmetries, and apply the results to the study of geometry and arithmetic of various interesting spaces associated to the groups in question.