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.

Rader 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 发起）是研究 Langland 程序中自同构 L 函数和函子性的新工具。 Rader 和 S. Rallis 一直在合作证明一些他们认为使相对迹公式成为定理而不仅仅是示例集合所必需的局部结果。他们表明，许多已知的常见特征和中心轨道积分结果可推广到某些 p 进对称空间上的球面特征和双陪集积分的情况。他们计划扩大研究范围，将 M. Brion 意义上的球形亚群纳入其中。尽管完成这样一个项目需要很多年，但它似乎是相对踪公式的自然领域，因为（可能）H 的一维表示出现在具有有限重数的 G 的不可约平滑无限维表示中。 Rader 还计划与 Marie-France Vigneras 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，这应该可以应用于构造 Hecke 代数的模表示。 朗兰兹纲领是数论的一部分。数论是对整数性质的研究，是数学最古老的分支。从一开始，数论中的问题就为该学科其他不同领域创造新数学提供了驱动力。朗兰纲领是一种将数论与微积分联系起来的一般哲学； 它体现了研究整数的现代方法。 现代数论技术性很强，也很深奥，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。