The refined Arthur--Langlands conjectures beyond the supercuspidal case

这个项目调查方面的朗兰兹计划-数学的一个基本领域桥接数论，几何和分析，并在一个系统的方式检查固有的对称性的解决方案的代数方程。通过将这些对称性与几何学联系起来，该程序创建了一种获得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。在过去几十年中，我们认为理所当然的许多技术进步，包括加密和数字安全，都离不开对数字算术的深刻而明确的了解。更确切地说，该项目旨在加深我们对一般连通和不连通约化群的精化Arthur-Langlands代数的理解，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。对分离群体的研究，虽然本身就很有意义，但对连接群体的研究也很重要。真实的数上不连通群的特殊情况将作为第一步。一个重要的作用，在获得超越离散参数的情况下，细化的approachtures是发挥作用的自缠绕运营商的抛物线诱导表示，该项目旨在获得自然规范化的这些和一个精确的声明的缠绕关系。该项目还将为本地和全球酉群构建非通用的亚瑟包，并获得全球酉群的完整离散自守谱的描述。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.