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

辛群中的西塔级数和约化对偶的“经典”理论已成为表示论和自同构形式的标准部分，并继续吸引着大量的兴趣和活动。针对特殊群体的类似理论仅在几年前才开始，并正在迅速发展。在以往的国家自然科学基金资助项目中，作者对无穷小特征标对应等理论中的一些基本问题进行了卓有成效的研究，并提出了确定例外对偶离散谱的新方法。在这里，他建议继续研究剩下的基本问题，并将结果应用于特定的问题，如算术流形和幂等表示的上同调。他计划将自己的方法与Burger和Sarnak的方法相结合，并希望在某些非对称齐次空间的离散级数和约化对偶谱之间建立密切的联系。李群表示理论主要研究某些类型的对称性。当两个群以一种有趣的方式嵌入到一个较大的环境群中时，人们可能会问，当限制到两个嵌入的群时，较大群的对称性将如何表现。这往往会导致深层信息，这在表象理论和数论中都扮演着重要的角色。在提出者和其他数学家早期工作的基础上，这个项目旨在解决这种对称性限制研究中的几个基本问题，并将结果应用于与所讨论的群相关的各种有趣空间的几何和算法的研究。

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.