The refined Arthur--Langlands conjectures beyond the supercuspidal case

本项目研究朗兰兹计划的各个方面--数学的一个基本领域，将数论、几何和分析联系起来，并系统地检验代数方程的解的内在对称性。通过将这些对称性与几何联系起来，该程序创建了一种获取算术信息的强大方法。该项目特别关注以算法可计算的方式获得此类信息。对数字算术的深刻而明确的了解对于过去几十年的许多技术进步至关重要，我们今天认为这些进步是理所当然的，包括加密和数字安全。这项工作的一部分包括对年轻科学家的指导，更准确地说，该项目旨在加深我们对一般连通和不连通还原群的改进的Arthur-Langland猜想的理解，特别是局部和全局包的内部结构以及它们所满足的特征恒等式。对不连通群的研究虽然本身很有意义，但对连通群的研究也是必不可少的。实数上互不连通的群的特殊情况将作为第一步。在获得离散参数以外的精化猜想方面，一个重要的作用是通过作用于抛物线诱导表示的自缠绕算子，该项目寻求获得这些猜想的自然规格化和缠绕关系的精确表述。该项目还将为本地和全球酉群构造非通用的Arthur包，并获得全球酉群的全离散自同构谱的描述。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates facets of the Langlands program -- a fundamental area of Mathematics bridging number theory, geometry, and analysis, and examining in a systematic way the symmetries inherent in solutions of algebraic equations. By relating these symmetries to geometry, the program creates a powerful way of obtaining arithmetic information. The project is particularly concerned with obtaining such information in an algorithmically computable way. Deep and explicit knowledge of the arithmetic of numbers is essential for many technological advances over the last decades that we take for granted today, including encryption and digital security. Part of this work involves mentoring of young scientists.More precisely, the project aims at deepening our understanding of the refined Arthur-Langlands conjectures for general connected and disconnected reductive groups, in particular the internal structure of local and global packets and the character identities satisfied by them. The study of disconnected groups, while of interest in its own right, is also essential for the study of connected groups. The special case of disconnected groups over the real numbers will serve as a first step. An important role in obtaining the refined conjectures beyond the case of discrete parameters is played by self-intertwining operators acting on parabolically induced representations, and the project seeks to obtain natural normalizations of these and a precise statement of the intertwining relation. The project will also construct non-generic Arthur packets for local and global unitary groups, and obtain a description of the full discrete automorphic spectrum of global unitary groups.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.