Automorphic Forms on Loop Groups

这项提议的目的是研究20世纪60年代末两个重要的数学发展之间的联系，这两个发展都是由加拿大数学家提出的。首先是罗伯特·朗兰兹（Robert Langlands）富有远见的计划，他将谐波分析技术应用于数论的核心问题。这个数论信息的载体是自同构的l函数，这个程序的一个基石是研究这些函数的解析性质。自成立以来，朗兰兹程序在数学中几乎无处不在，它既影响了表示理论、代数几何、弦理论，甚至代数拓扑的重要发展，也受到了这些发展的影响。第二个发展是由Victor Kac和Robert Moody引入的一个无限维对称族，它推广了在他们之前一个世纪的经典李理论。虽然本质上是无限维的，但这些新物体被证明具有许多与有限维物体相同的结构特征。此外，它们还经常与数学的其他领域表现出令人惊讶和深远的联系。这一点在Kac和Moody引入的无限维对象的“最简单”类，即所谓的环群和环代数（或仿射Kac-Moody群和代数）中得到了更大的注意。在有限单群理论和数学物理发展的部分推动下，环代数和群在20世纪80年代与代数组合学的核心问题、顶点算子理论和量子群理论联系在一起。这个建议的目标是扩大朗兰兹程序的范围，不仅考虑有限维群，而且考虑无限维环群。朗兰兹哲学的一个核心方面是，即使是最小的群体上的l函数（以及它们所编码的数论问题）也不应该孤立地研究。相反，人们需要将它们与更大的、通常是完全不同的群体中的类似对象放在一起考虑。综合起来，这些信息提供了对原始问题的有力见解，并且可以考虑的组越多，对原始问题的见解就越精细。爱森斯坦系列结构是一种从小群体向大群体转移的工具。它的对应物，常数项（或者更一般的傅里叶系数）结构从较大的组移动到较小的组。Langlands和Shahidi通过组合这两种结构开发了分析自同构l函数的机制。即首先形成一个爱森斯坦级数，然后利用谐波分析的工具对其进行研究，最后利用常项构造回归到原来的设置。这种结构的效率依赖于爱森斯坦系列可以通过光谱技术进行研究，这在原始设置中是不可用的。在实践中，这种范式在解析数论的许多著名问题中产生了最强有力的已知结果。为了开始使用Langlands-Shahidi方法，我们需要大量更大的群来考虑爱森斯坦级数。在有限维的环境中，这样的群体数量有限，其中大部分的信息已经被收集了。然而，引入无限维群极大地拓宽了该方法的范围。本提案旨在发展一种机制，将这些无限维群纳入朗兰兹程序，从而纳入可用的工具库，以解决解析数论的核心问题。

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.