Multiplicities and Period Integrals for Spherical Varieties

表示论和自守形式是数学的两个重要分支，与许多其他学科有联系，包括物理学和计算机科学。约化群是一类特殊的拓扑群，具有丰富的对称性。这些对称性是理解我们宇宙中物体内在结构的指导方针。对约化群的研究可以追溯到世纪后期。其中最重要的两个领域是约化群的表示理论和约化群上的自守形式（一种特殊的具有超对称性的函数）。本专题旨在了解约化群的表示对球面子群的限制，并了解自守形式的周期积分。本专题旨在研究球面簇的局部重数和整体周期积分，以及它们与L-函数和算术几何的联系。局部地证明了一般球面簇的重数公式和局部迹公式。另一个目标是证明一些相对迹公式的轨道积分之间的比较，以及一些轨道积分的导数和一些高度配对之间的比较。局部理论中的主要方法是约化群上的调和分析。在全局上，目标是研究周期积分和自守L-函数之间的各种关系。另一个目标是理解一般线性群的平方可积自守形式空间中的非调和项。在全球理论中使用的方法是残差法，相对痕量公式，以及一些来自超越内窥镜理论的想法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Representation theory and automorphic forms are two important branches of mathematics that has connections to many other subjects, including physics and computer science. Reductive group is a special kind of topological groups with abundant symmetries. These symmetries are the guidelines to understanding the intrinsic structures of objects in our universe. The study of reductive groups dates back to the late 19th century. Two of the most important areas are the representation theory of reductive groups and automorphic forms (which is a special kind of functions with extra symmetry) on reductive groups. This project aims to understand the restriction of representations of reductive groups to a spherical subgroup, and to understand the period integrals of automorphic forms.This project is to study the local multiplicities and global period integrals of spherical varieties, as well as their connections to L-functions and arithmetic geometry. Locally the goal is to prove the multiplicity formula and local trace formula for general spherical varieties. Another goal is to prove comparisons between orbital integrals of some relative trace formulas, as well as comparisons between the derivative of some orbital integrals and some height pairings. The main method used in the local theory is harmonic analysis on reductive groups. Globally the goal is to study various relations between period integrals and automorphic L-functions. Another goal is to understand the nontempered terms in the space of square-integrable automorphic forms for the general linear groups in terms of orbital integrals. The methods used in the global theory are the residue method, the relative trace formula, and some ideas from the theory of beyond endoscopy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.