Square-integrable automorphic forms, local Langlands correspondence and Gross-Prasad conjecture

PI建议研究由朗兰兹计划产生的表示理论和自同构形理论中的各种基本问题，以及它们在算术兴趣问题中的应用。多年来，PI一直致力于亚瑟猜想所预言的平方可积自同构形的构造和分类，特别是在例外群的背景下。他希望在接下来的三年内完成这项研究。他还打算研究某些古典和特殊群体的当地朗兰兹书信。在另一个方向上，PI希望建立Gross-Prasad猜想的某些情况，该猜想关于将正交或酉群的表示限制为更小的表示，该猜想适用于L函数的特定值。最后，PI建议在theta对应理论的背景下建立正则化的Siegel-Weil公式的第二项恒等式。朗兰兹程序是现代数论研究的一个组成部分。它最初的目标是理解在数论和表象理论中自然产生的某些群。近年来，它已经超越了传统的界限，与代数几何和数学物理等领域建立了联系。它已经在现实生活中找到了意想不到的应用。事实上，作为朗兰兹计划的第一个主要结果之一的雅克-朗兰兹通信已经被用来构造所谓的Ramanujan图。这些图是高度连通的，是有效的网络。事实证明，它们在通信理论中非常有用。希望本文所研究的问题能够帮助挖掘出更多这类应用。

The PI proposes to study various basic questions in representation theory and the theory of automorphic forms arising from the Langlands program, and their applications to questions of arithmetic interest. For a number of years, the PI has been pursuing the construction and classification of square-integrable automorphic forms as predicted by Arthur's conjecture, especially in the context of the exceptional groups. The PI hopes to complete this study in the next 3-year period.He also intends to study the local Langlands correspondence for certain classical and exceptional groups. In another direction, the PI hopes to establish certain cases of the Gross-Prasad conjecture regarding the restriction of representations of an orthogonal or unitary group to a smaller one, which has applications to special values of L-functions. Finally, the PI proposes to establish the second term identity for the regularized Siegel-Weil formula in the context of the theory of theta correspondences.The Langlands program is an integral part of modern number theoretic research. Its initial goal is to understand certain groups which arise naturally in number theory and representation theory. In recent years, it has expanded beyond its traditional boundaries to connect with areas such as algebraic geometry and mathematical physics. It has already found unexpected applications in real life. Indeed, the Jacquet-Langlands correspondence, which is one of the first major results in the Langlands program, has been exploited to give a construction of the so-called Ramanujan graphs.These graphs are highly connected and serve as efficient networks.They have turned out to be very useful in communications theory. It is hoped that the questions investigated in this proposal can help to unearth more applications of this type.