The Gan-Gross-Prasad Conjecture: Archimedean Theory

该奖项支持首席研究员在数论和表征理论方面的研究。数论的历史根源在于对自然数的研究。它是数学最古老的分支之一。在过去的半个世纪里，它已经成为一个不可或缺的工具，在数据传输和处理、通信系统和互联网安全等领域得到了广泛的应用。表征理论产生于人们解代数方程的尝试。这门学科在20世纪发展成熟，至今仍在快速发展。它研究物体的对称性，并应用于数学、物理学、宇宙学和材料科学的几乎所有分支。本项目将探讨这两个领域的一些主要猜想。此外，该奖项将支持PI所在机构的研究生。20世纪60年代，朗兰兹提出了一系列关于表示论和数论的深刻猜想，这些猜想从此成为数论和表示论的指导原则。20世纪90年代，Gross和Prasad在Langlands哲学的指导下，对正交群的表征理论提出了几个猜想。他们与Gan一起，将这些猜想扩展到所有经典群。这些猜想以及它们的精化和算术对应物站在数论、表示论和算术几何的十字路口。该项目的目标是研究这些猜想。具体地，我们研究了以下两个问题：(1)实正交群的局部Gan—Gross—Prasad猜想；(2)由Gan—Gross-Prasad猜想的算术版本产生的算术基本引理的阿基米德对立物。对于第一个问题，我们应用了对应的方法，以及Shahidi局部系数的相对版本，这是我们的主要创新点。该方法还给出了正交群的l包的归纳构造，而正交群本身就是Langlands猜想的重要研究对象。对于第二个问题，我们利用了相关的轨道积分和球面特性的归纳结构。它将问题简化为使用倍积分理论的显式计算。该奖项反映了美国国家科学基金会的法定使命，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。

This award supports the principal investigator's research on Number Theory and Representation Theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool, with diverse applications in areas such as data transmission and processing, communication systems, and internet security. Representation Theory emerges from people’s attempt to solve algebraic equations. The subject matured in the 20th century and is still fast developing. It studies the symmetries of objects and has applications to almost all branches of mathematics, physics, cosmology and material science. This project will explore some major conjectures in these two areas. Additionally, this award will support graduate students in the PI's institution. In the 1960s Langlands formulated a series of profound conjectures relating representation theory and number theory, and these conjectures have become guiding principles for number theorists and representation theorists since then. In the 1990s, Gross and Prasad formulated several conjectures on representation theory of the orthogonal groups, guided by the philosophy of Langlands. Together with Gan, they extended these conjectures to include all classical groups. These conjectures and their refinement and arithmetic counterparts stand at the crossroads of number theory, representation theory, and arithmetic geometry. The goal of the project is to study these conjectures. In more detail, we study the following two problems (1) the local Gan--Gross--Prasad conjecture for real orthogonal groups; (2) the archimedean counterpart of the arithmetic fundamental lemma arising from an arithmetic version of the Gan--Gross—Prasad conjecture. For the first problem we apply the method of theta correspondences, and a relative version of Shahidi’s local coefficients which is our main point of innovation. This method also gives an inductive construction of L-packets for orthogonal groups, which are themselves important objects of study in the conjectures of Langlands. For the second problem, we make use of an inductive structure of the relevant orbital integrals and spherical characters. It reduces the problem to explicit calculations using the theory of doubling zeta integrals.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.