Spherical varieties in the Langlands program

朗兰兹计划中的球形品种

• 批准号: 1101471
• 负责人: Ioannis Sakellaridis
• 金额: $ 11.64万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101471&HistoricalAwards=false
• 关键词: Spherical; varieties; Langlands; program

The objective of this project is to investigate various aspects of the relationship between spherical varieties and the Langlands program. It is motivated by recent conjectures relating periods of automorphic forms over spherical subgroups to Euler products of functionals arising from the Plancherel formula of a spherical variety, and expressing the support of Plancherel measure in terms of Arthur parameters. A major part of the project is devoted to creating trace formula-theoretic tools, both locally and globally, necessary to put the conjectures in the correct setting, to clarify some aspects and to prove particular instances of them. Other parts include extending previous harmonic-analytic results and developing a relative trace formula in specific cases in order to analyze the pertinent periods of automorphic forms.L-functions are very central objects in various branches of number theory, and there are strong conjectures and results (in and around the Langlands program) relating most of the interesting types of L-functions to those L-functions which are called automorphic. Automorphic L-functions are studied by constructions of global harmonic analysis, that is explicit integrals of functions on the quotient of a Lie group by a discrete, arithmetic subgroup, but these constructions remain mysterious after many decades of use. A lot of them involve spherical ("large") subgroups, and a general theory has started to emerge which connects global harmonic analysis to Euler products (and hence L-functions) via local harmonic analysis on spherical varieties. According to this theory, which for now is mostly conjectural, the local Langlands conjecture admits a generalization to the harmonic analysis of a spherical variety over a local field, where a "dual group" is attached to the spherical variety and describes the representations distinguished by it; and spherical periods of automorphic forms, satisfying certain assumptions, are eulerian in a very explicit way and, hence, related to L-functions or special values of those. This project aims at improving these conjectures and investigating ways for their proof, mainly by trace formula-theoretic techniques.

该项目的目的是研究球形品种与兰兰兹计划之间关系的各个方面。这是由于最近的猜想是由球形亚组与球形子组的欧拉产物相关的猜测，这是由球形品种的plancherel公式引起的功能的欧拉产物，并根据Arthuragorters表示了对Plancherel度量的支持。该项目的主要部分致力于在本地和全球范围内创建痕量公式理论工具，以将猜想放在正确的环境中，以阐明某些方面并证明其特定实例。其他部分包括扩展先前的谐波分析结果并在特定情况下开发相对的微量公式，以分析自动形式的相关时期。L功能在数量理论的各个分支中都是非常核心的对象，并且在Langlands计划和周围有强烈的猜想和结果（Langlands计划和周围），这些功能与那些与自动函数相关的大多数有趣类型，这些官能与自动函数相关联。通过全球谐波分析的结构来研究自动型L功能，这是通过离散的，算术亚组在谎言组上的功能的明确积分，但是经过数十年的使用后，这些结构仍然是神秘的。它们中的许多涉及球形（“大”）亚组，并且已经开始出现一般理论，该理论通过局部谐波分析将全球谐波分析与欧拉产品（以及l功能）联系起来。根据这一理论，目前主要是猜想的，当地的兰兰兹猜想承认对球形品种在局部领域的谐波分析的概括，在该领域，“双重组”附在球形品种上，并描述了其与众不同的表示。自动形式的球形时期（满足某些假设）是以非常明确的方式是欧拉（Eulerian）的，因此与这些假设或与l功能或特殊值有关。该项目旨在改善这些猜想并研究其证明的方法，主要是通过痕量公式理论技术。