Extending the Langlands-Shahidi Program to the non-generic spectrum and related problems

将 Langlands-Shahidi 计划扩展到非通用频谱及相关问题

• 批准号: 0400958
• 负责人: David Goldberg
• 金额: $ 12万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400958&HistoricalAwards=false
• 关键词: Extending; Langlands; Shahidi; Program; non

Abstract for award DMS-0400958 of GoldbergThe primary goal of this project is to extend the Langlands-Shahidi method to non-generic cusp forms of certain classical groups. This is joint work with S. Friedberg. Our plan is to use Bessel models to play the role Whittaker models play in the generic case. This work should lead in the direction of establishing the automorphic transfer of cusp forms from classical to general linear groups, generalizing the generic case established by Cogdell - Kim - Piatetski-Shapiro - Shahidi. It should also allow for the completion of the work of Jiang - Soudry on the local Langlands correspondence for special orthogonal groups of odd dimension. Even establishing the transfer in low rank cases would give significant results. For example, we anticipate results on Hilbert-Modular forms with trivial central character from the most primitive example. In the course of this project we expect to address the local theory for the Archimedean case, as well as refining our earlier results in the non-Archimedean case. Additional goals of this project are to understand the poles of intertwining operators for maximal parabolic subgroups of quasi-split reductive groups over non-Archimedean fields (joint with F. Shahidi), construction of types, in the sense of Bushnell - Kutzko for the non-cuspidal spectrum of classical groups over non-Archimedean fields (joint with P. Kutzko and S. Stevens), and understanding reducibility of parabolically induced representations of Spin groups over non-Archimedean fields via Knapp - Stein theory. Among the most powerful results in mathematics are those that display a connection between seemingly disparate ideas. These connections tend to improve our understanding of both ideas, and are interesting both as a matter of philosophy and application. For example, the power of Calculus lies not in our understanding of differential and integral calculus separately, but in the connection between the two via the Fundamental Theorem of Calculus. This is what lends the subject to the myriad applications which have impacted upon society, but also makes the subject a great achievement in human thought. This project is designed to address certain problems within what is known as the Langlands Program, which seeks to unify three major areas within mathematics. Namely, Number Theory, Harmonic Analysis, and Algebraic Geometry. Our work concentrates on one aspect of the conjectural connection between the first two areas. Tremendous progress has been made in recent years, and we hope to contribute to furthering our understanding of these subjects. This project will have broader impact not only through its contribution to mathematics (and more specifically number theory), but through raising the level of research at a public university with a diverse student population.

摘要奖DMS-0400958 Goldberg该项目的主要目标是将Langlands-Shahidi方法扩展到某些经典群的非通用尖点形式。这是与S的合作。弗里德伯格我们的计划是使用贝塞尔模型发挥作用的惠特克模型发挥在一般情况下。这项工作应导致的方向，建立自守转移的尖点形式从经典的一般线性群，推广通用的情况下建立的科格德尔-金-皮亚特斯基-夏皮罗-沙希迪。它还应允许完成工作的江-苏德里对当地朗兰兹对应特殊正交群的奇数维。即使在低级别的情况下建立转移也会产生显著的结果。例如，我们预期结果希尔伯特模形式与平凡的中心字符从最原始的例子。在这个项目的过程中，我们希望解决阿基米德的情况下，以及完善我们的早期结果在非阿基米德的情况下的本地理论。这个项目的其他目标是了解非阿基米德域上拟分裂约化群的极大抛物子群的交织算子的极点（与F。Shahidi），Bushnell-Kutzko意义下的类型构造，用于非阿基米德域上经典群的非尖点谱（与P. Kutzko和S. Stevens），并通过克纳普-斯坦理论理解非阿基米德场上自旋群的抛物线诱导表示的约化。数学中最有力的结果之一是那些显示出看似不同的想法之间联系的结果。这些联系有助于我们更好地理解这两个概念，无论是从哲学角度还是从应用角度来看，都很有趣。例如，微积分的力量不在于我们分别理解微积分和微积分，而在于通过微积分基本定理将两者联系起来。这是什么借给这个问题的无数应用程序，影响了社会，但也使这个问题在人类思想的一个伟大成就。该项目旨在解决所谓的朗兰兹计划中的某些问题，该计划旨在统一数学中的三个主要领域。即数论、调和分析和代数几何。我们的工作集中在前两个领域之间的自然联系的一个方面。近年来取得了巨大进展，我们希望为进一步了解这些问题作出贡献。该项目将产生更广泛的影响，不仅通过其对数学（更具体地说，数论）的贡献，而且通过提高公立大学的研究水平与多样化的学生群体。