Multiplicities and Period Integrals for Spherical Varieties

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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.