Unipotent Representations and Automorphic Forms

单能表示和自同构形式

• 批准号: 0901104
• 负责人: Dan Barbasch
• 金额: $ 34.19万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901104&HistoricalAwards=false
• 关键词: Unipotent; Representations; Automorphic; Forms

AbstractBarbaschThere are two major themes in this proposal, the determination of the unitary dual, and the structure of unipotent representations. Barbasch will continue and extend his previous work to find necessary and sufficient conditions for unitarity for classes of Langlands parameters such as principal series in real and p-adic groups. The aim is to find proofs that are as uniform and conceptual as possible, and express the answers in a closed form. Unipotent representations are (conjectured to be) the building blocks of the unitary dual. Their character theory plays an important role in the theory of automorphic forms. Barbasch will study the character theory of unipotent representations in the real case, with an emphasis on relations to endoscopic groups, particularly twisted ones. Realizations of unipotent representations as local factors of automorphic forms, and in the context of the dual pairs correspondence are also envisioned. This research will provide a deeper understanding of representation theory of reductive Lie groups, in particular unitary representations, which play an essential role in the applications of representation theory to automorphic forms, mathematical physics analysis and geometry. This research will generate problems suitable for Ph.D. theses as well as combinatorial problems for undergraduate research. Together with several faculty members at Cornell, Barbasch will run the ``Sophus Lie Days at Cornell'' an instructional conference designed to expose undergraduates and graduate students in mathematics and other sciences to the theory of Lie groups and their representations, one of the aims being to get them interested and involved in research at an early stage. The results of this research will be disseminated through research papers, talks at seminars and conferences, and various web sites.

Barbasch这一提议有两个主要主题，一个是酉对偶的确定，另一个是幂等表示的结构。Barbasch将继续和推广他以前的工作，以找到实群和p-ady群中的主级数等朗兰兹参数类具有唯一性的充要条件。其目的是找到尽可能统一和概念性的证据，并以封闭的形式表达答案。幂等表示是(猜想是)酉对偶的积木。他们的特征理论在自同构形理论中占有重要地位。Barbasch将研究真实情况下的幂等表示的特征标理论，重点是与内窥镜群，特别是扭曲群的关系。还设想了在对偶对对应的情况下，将幂等表示实现为自同构形式的局部因子。这一研究将加深对约化李群的表示理论，特别是么正表示的理解，它在表示理论在自同构形式、数学物理分析和几何中的应用中起着至关重要的作用。这项研究将产生适合于博士论文的问题，以及适合本科生研究的组合问题。Barbasch将与康奈尔大学的几名教职员工一起举办“康奈尔大学的Sophus Lie Day”教学会议，旨在让数学和其他科学的本科生和研究生了解李群理论及其表示，目的之一是让他们对早期阶段的研究感兴趣并参与到研究中来。这项研究的结果将通过研究论文、研讨会和会议上的演讲以及各种网站进行传播。