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.

Shahidi在接下来的三年中，研究人员将研究以下形式和代表理论中的以下问题。作为他的第一个问题，研究人员希望证明，标准的交织操作员在当地领域的Quasisplit组的抛物面诱导的代表性是某些Langlands L功能。 这将在爱森斯坦系列和全球举起的理论中产生重要的后果。特别是，他计划证明，根据l功能，这些群体的Levi子组的不可还原准通用的通用表示产生的表示标准。这是他要研究的另一个问题的结果，该问题将Vogan的结果扩展到P-Adic群体。它指出，兰格兰商通用的标准模块是不可还原的。这些是与W. Casselman联合项目的一部分。作为他的第二和第三个问题，他将继续在经典群体（与D. Goldberg）和他们的残留谱（与H. Kim一起）上继续工作。他还计划证明由两种不同方法（Rankin-Selberg和Langlands-Shahidi）定义的某些系数的平等，以及对使用第二种方法理解L功能的不同方法的研究。他还将试图建立通过标准化的交织运营商满足的某些身份，从而将其结果从纠正的案例扩展到非脾气暴躁的案例。最后，他计划进一步研究GL（2）上尖尖的对称立方体L功能，其曲折具有任意的尖端形式，希望更好地了解从GL（2）到GL（4）的对称立方体提升。 这项研究属于兰兰兹计划的一般数学领域。兰兰兹计划是数字理论的一部分，这是对整数属性的研究，并且是数学最古老的分支。从一开始，数字理论的问题就为在学科的其他不同部分创建新数学方面提供了驱动器。 Langlands计划是一种将数字理论与微积分联系起来的一般理念。 它体现了整个数字研究的现代方法。 现代数字理论非常技术性和深刻，但在理论计算机科学和编码理论等领域中具有惊人的应用。