RUI: Mathematical Sciences: Spherical Characters on P-adic Coset Spaces and the Relative Trace Formula

RUI：数学科学：P-进陪集空间上的球面特征和相对迹公式

• 批准号: 9623125
• 负责人: Cary Rader
• 金额: $ 4.8万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623125&HistoricalAwards=false
• 关键词: RUI; Mathematical; Sciences; Spherical; Characters

Rader This award provides funding for a project in the Langlands program. The Relative Trace Formula (as initiated by Jacquet) is a new tool for studying automorphic L-functions and functoriality in the Langland's program. Rader and S. Rallis have been collaborating to prove some of the local results they think necessary to make the Relative Trace Formula a theorem instead of just a collection of examples. They have shown that many of the results known for usual character and central orbital integrals generalize to the case of spherical characters and double coset integrals on certain p-adic symmetric spaces. They plan to expand their research to include spherical subgroups such in the sense of M. Brion. Although it will take many years to complete such a project, it seems to be the natural domain for relative trace formulae, because (probably) one dimensional representations of H occur in irreducible smooth infinite dimensional representations of G with finite multiplicity. Rader also plans to complete a collaborative effort with Marie-France Vigneras investigating the geometric Zelevinskii involution for nilpotent orbits in certain Lie algebras. This should have applications in constructing modular representations of Hecke algebras, after Kazhdan-Lusztig. The Langlands program is part of number theory. Number theory 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 force in creating new mathematics in other diverse parts of the discipline. The Langland's 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.

该奖项为朗兰兹计划中的一个项目提供资金。相对迹公式（由Jacquet提出）是研究Langland程序中自同构l函数和泛函的一种新工具。Rader和S. Rallis一直在合作证明一些他们认为必要的局部结果，使相对迹公式成为一个定理，而不仅仅是一个例子的集合。他们证明了许多已知的关于通常特征和中心轨道积分的结果可以推广到某些p进对称空间上的球形特征和二重协集积分的情况。他们计划扩大他们的研究范围，包括像M. Brion这样的球形亚群。虽然完成这样一个项目需要很多年的时间，但它似乎是相对迹公式的自然领域，因为（可能）H的一维表示出现在具有有限多重性的G的不可约光滑无限维表示中。Rader还计划完成与Marie-France Vigneras的合作，研究某些李代数中幂零轨道的几何Zelevinskii对合。在Kazhdan-Lusztig之后，这应该在构造Hecke代数的模表示中有应用。朗兰兹程序是数论的一部分。数论是对整数性质的研究，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动力。朗兰程序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。