Spherical varieties in the Langlands program

该项目的目标是调查球形品种与朗兰兹计划之间关系的各个方面。它的动机是最近的猜想将球子群上自同构形的周期与源于球簇的Plancerel公式的泛函的Euler积联系在一起，并用Arthur参数表示Plcherel测度的支撑性。该项目的主要部分致力于创建本地和全球的踪迹公式理论工具，这些工具对于将猜想放在正确的背景下、澄清某些方面并证明它们的特定实例是必要的。L-函数是数论各个分支中非常核心的对象，在朗兰兹程序内和周围都有很强的猜想和结果，将大多数有趣的L-函数类型与那些被称为自同构的L函数联系在一起。自同构的L函数是通过整体调和分析的构造来研究的，即李群的商上的函数通过离散的算术子群的显式积分，但经过几十年的使用，这些构造仍然是神秘的。它们中的许多都涉及球面(“大”)子群，并且已经开始出现一种一般理论，它通过球面簇上的局部调和分析将全局调和分析与欧拉积(以及L函数)联系起来。根据这一目前主要是猜测的理论，局部朗兰兹猜想承认了局部场上球面簇的调和分析的推广，其中球面簇上附加了一个“对偶群”，并描述了由它区分出来的表示；满足某些假设的自同构形的球面周期以非常明确的方式是欧拉的，因此与L函数或这些函数的特定值有关。本项目旨在改进这些猜想，并探索证明它们的方法，主要是利用踪迹公式理论技术。

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.