Spherical varieties in the Langlands program

这个项目的目的是调查球形品种和朗兰兹计划之间的关系的各个方面。它的动机是最近的progratures有关的期间自守形式的球面子群欧拉产品的泛函所产生的Plancherel公式的球面品种，并表示支持Plancherel措施的亚瑟参数。该项目的一个主要部分是致力于创建跟踪公式理论工具，在本地和全球范围内，有必要把正确的设置，澄清一些方面，并证明他们的特定实例的结构。其他部分包括扩展以前的调和分析结果，并在特定情况下发展一个相对迹公式，以分析自守形式的相关周期。L-函数是数论各个分支中非常重要的对象，并且有强有力的图解和结果（在朗兰兹纲领中和周围）将大多数有趣的L-函数类型与称为自守的L-函数联系起来。自守L-函数是通过构造整体调和分析来研究的，即函数在李群的商上通过离散算术子群的显式积分，但这些构造在使用了几十年后仍然是神秘的。它们中的许多涉及球面（“大”）子群，并且已经开始出现一种通用理论，该理论通过球面簇的局部调和分析将全局调和分析与欧拉乘积（以及因此的L-函数）联系起来。根据这一理论，目前主要是理论性的，局部朗兰兹猜想承认推广到局部域上的球面簇的调和分析，其中一个“对偶群”附在球面簇上并描述由它区分的表示;满足某些假设的自守形式的球面周期是非常明确的欧拉周期，因此，与L-函数或其特殊值有关。本项目的目的是改进这些公式，并研究其证明方法，主要是利用迹公式理论技术。

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.