Automorphic Forms on Loop Groups

循环群上的自守形式

• 批准号: RGPIN-2014-04622
• 负责人: Patnaik, Manish
• 金额: $ 1.17万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587918
• 关键词: Automorphic; Forms; Loop; Groups

The aim of this proposal is to study the link between two important mathematical developments from the late 1960s, both by Canadian mathematicians. The first of these was the visionary program of Robert Langlands to apply the techniques of harmonic analysis to the central questions in number theory. The carriers of this number theoretic information are automorphic L-functions, and a cornerstone of this program has been the study of the analytic properties of these functions. Since its inception, the Langlands program has grown to become nearly ubiquitous in mathematics, having both influenced and been influenced by important developments in representation theory, algebraic geometry, string theory, and even algebraic topology. The second of these developments was the introduction by Victor Kac and Robert Moody of a family of infinite-dimensional symmetries generalizing the classical Lie theory of the century preceding them. Although infinite-dimensional in nature, these new objects were shown to possess many of the same structural features as their finite-dimensional counterparts. Moreover, they often also exhibited surprising and far reaching connections with other areas of mathematics. Nowhere has this been more noticed than in the ‘simplest’ class of infinite-dimensional objects introduced by Kac and Moody, the so called loop groups and loop algebras (or affine Kac-Moody groups and algebras). Spurred in parts by the theory of finite simple groups and also by developments in mathematical physics, loop algebras and groups were connected in the 1980s with the central questions in algebraic combinatorics, the theory of vertex operators, and the theory of quantum groups. The goal of this proposal is to enlarge the range of the Langlands program by considering it not just for finite-dimensional groups, but also for infinite-dimensional loop groups. A central facet of the Langlands philosophy is that L-functions (and the number theoretical questions which they encode) on even the smallest of groups should not be studied in isolation. Rather, one needs to consider them alongside analogous objects on larger and often quite different groups. Put together, this information gives powerful insights into the original question, and the more groups one can consider, the more refined the insight into the original problem. A vehicle for moving from smaller to larger groups is the Eisenstein series construction. It's counterpart, the constant term (or more generally the Fourier coefficient) construction moves from larger to smaller groups. Langlands and Shahidi have developed the machinery to analyze automorphic L-functions by composing these two constructions. Namely, they first form an Eisenstein series, then study it using tools from harmonic analysis, and finally return to the original setting using the constant term construction. The efficiency of this construction relies on the fact the Eisenstein series is amenable to study via spectral techniques, not available in the original setting. In practice, this paradigm has produced the strongest known results in a variety of celebrated questions in analytic number theory. To get the Langlands-Shahidi method started, one needs a rich supply of larger groups on which to consider Eisenstein series. There are a limited number of such groups within a finite-dimensional context, the information from most of which have already been gleaned. However, introducing infinite-dimensional groups into the picture vastly broadens the scope of the method. This proposal aims to develop the machinery to incorporate these infinite-dimensional groups into the Langlands program, and hence into the arsenal of tools available attack the central questions of analytic number theory.

该提案的目的是研究加拿大数学家的1960年代后期的两个重要数学发展之间的联系。其中第一个是罗伯特·兰兰兹（Robert Langlands）的远见计划，将谐波分析技术应用于数字理论中的中心问题。该数字理论信息的载体是自动形态L功能，该程序的基石是对这些功能的分析特性的研究。自成立以来，Langlands计划已经发展到数学上几乎无处不在，受到代表理论，代数几何学，弦乐理论甚至代数拓扑的重要发展的影响和影响。 这些事态发展中的第二个是Victor Kac和Robert Moody的介绍，该家族的无限维度对称性概括了本世纪之前的古典谎言理论。尽管本质上是无限尺寸的，但这些新物体被证明具有许多与有限维度对应物相同的结构特征。此外，他们经常也暴露了惊喜，并与其他数学领域建立了很大的联系。在KAC和Moody引入的“最简单”类别的无限二维对象中，这一所谓的循环群和循环代数（或Aggine Kac-Moody群和代数）所引入的“最简单”的无限二维对象。在1980年代，循环代数和群体的发展理论以及数学物理学的发展以及数学物理学的发展与部分相关的部分与代数组合学，顶点操作者的理论和量子群的理论相连。 该提案的目的是通过考虑有限维组，而且针对无限维循环组来扩大兰兰兹计划的范围。 Langlands哲学的一个主要方面是，即使在最小的群体上，L功能（以及它们编码的数量理论问题）也不应孤立地研究。相反，人们需要在较大且通常完全不同的群体上与类似物体一起考虑它们。结合在一起，这些信息对原始问题提供了有力的见解，并且可以考虑的群体越多，对原始问题的见解就越完善。 Eisenstein系列的结构是从较小组移动到较大组的车辆。它是对应物，恒定的术语（或更一般的傅立叶核心）构造从较大的组移动到较小的组。 Langlands和Shahidi已经开发了通过组成这两种结构来分析自动型L功能的机械。也就是说，它们首先形成爱森斯坦系列，然后使用谐波分析中的工具对其进行研究，最后使用恒定的术语结构返回原始设置。这种结构的效率取决于爱森斯坦系列的事实，可以通过光谱技术进行研究，而原始环境中不可用。实际上，该范式在分析数理论的各种著名问题中产生了众所周知的结果。 为了启动Langlands-Shahidi方法，人们需要大量的较大群体来考虑Eisenstein系列。在有限维的环境中，此类组数量有限，其中大多数的信息已经被清洁。但是，将无限二维组引入图片大大扩大了该方法的范围。该提案旨在开发机械，将这些无限尺寸群体纳入兰兰兹计划，从而融入可用工具的武器库中，攻击了分析数理论的中心问题。