Mathematical Sciences: Automorphic L-functions and Interwining Operators

数学科学：自守 L 函数和交织算子

• 批准号: 9622585
• 负责人: Freydoon Shahidi
• 金额: $ 8.1万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622585&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Automorphic; functions; Interwining

Shahidi In the next three years, the investigator will study the following problems in Automorphic Forms and Representation Theory. As his first problem, the investigator wants to show that the poles of the standard intertwining operators for parabolically induced representations of a quasisplit group over a local field are among those of certain Langlands L-functions. This will have important consequences in the theory of Eisenstein series and global liftings. In particular, he plans to prove a definitive reducibility criterion for representations induced from irreducible quasi-tempered generic representations of Levi subgroups of these groups in terms of L-functions. This is a consequence of another problem to be studied by him which extends a result of Vogan to p-adic groups. It states that the standard modules whose Langlands quotients are generic are irreducible. These are parts of a joint project with W. Casselman. As his second and third problems, he will continue his work on the tempered spectrum of classical groups (with D. Goldberg) and their residual spectrum (with H. Kim). He also plans to prove the equality of certain coefficients defined by two different methods (Rankin-Selberg and Langlands-Shahidi) as well as the study of different approaches to understanding poles of L-functions using the second method. He will also try to establish certain identities satisfied by normalized intertwining operators, extending his results from the tempered case to the non-tempered ones. Finally, he plans to further study the symmetric cube L-function of a cusp form on GL(2) and its twists with arbitrary cusp forms with the hope of better understanding the symmetric cube lift from GL(2) to GL(4). The research falls in the general mathematical area of the Langlands program.The Langlands program is part of Number Theory, which is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving forc e in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.

在接下来的三年里，研究者将研究自同构形式和表征理论中的以下问题。作为他的第一个问题，研究者想要证明局部场上准分裂群的抛物诱导表示的标准缠绕算子的极点是某些朗兰兹l函数的极点。这将对爱森斯坦级数和全局提升理论产生重要影响。特别地，他计划证明由这些群的Levi子群的不可约拟调质一般表示在l -函数中导出的表示的确定性可约准则。这是他要研究的另一个问题的结果，这个问题将Vogan的结果扩展到p进群。说明朗兰商是泛型的标准模是不可约的。这些是与W. Casselman合作项目的一部分。作为他的第二个和第三个问题，他将继续研究经典群的调和谱（与D. Goldberg）和它们的残差谱（与H. Kim）。他还计划证明由两种不同方法（Rankin-Selberg和Langlands-Shahidi）定义的某些系数的相等性，并研究使用第二种方法来理解l函数极点的不同方法。他还将尝试建立由归一化缠结算子满足的恒等式，将他的结果从调和情况推广到非调和情况。最后，他计划进一步研究GL(2)上的尖形对称立方l函数及其任意尖形的扭转，以期更好地理解从GL(2)到GL(4)的对称立方提升。这项研究属于朗兰兹纲领的一般数学领域。朗兰兹程序是数论的一部分，数论研究的是整数的性质，是数学中最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分中创造新的数学提供了一种推动力。朗兰兹纲领是一种将数论与微积分联系起来的普遍哲学；它体现了研究整数的现代方法。现代数论是非常技术性和深奥的，但它在理论计算机科学和编码理论等领域有着惊人的应用。