Unipotent Representations and Automorphic Forms

【摘要】barbasch在这一提议中有两个主要的主题，即酉对偶的确定和单性表征的结构。Barbasch将继续并扩展他之前的工作，以找到实群和p进群中的主级数等朗兰参数类的一致性的充分必要条件。其目的是找到尽可能统一和概念化的证明，并以封闭的形式表达答案。单性表示（推测为）是酉对偶的组成部分。他们的性格理论在自同构形式理论中占有重要地位。Barbasch将在真实案例中研究无能表征的特征理论，重点是与内窥镜群体的关系，特别是扭曲的群体。作为自同构形式的局部因素的单能表示的实现，以及在对偶对应的背景下也被设想。本研究将使人们对约化李群的表示理论，特别是酉表示有更深入的了解，这对于表示理论在自同构形式、数学物理分析和几何中的应用具有重要意义。本研究将产生适合博士论文的问题，以及适合本科生研究的组合问题。Barbasch将与康奈尔大学的几位教职员工一起举办“康奈尔大学的Sophus Lie Days”，这是一个教学会议，旨在让数学和其他科学专业的本科生和研究生接触李群理论及其表示，目的之一是让他们在早期阶段就对研究产生兴趣并参与其中。这项研究的结果将通过研究论文、研讨会和会议的演讲以及各种网站传播。

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.