Automorphic Forms on Loop Groups

这项建议的目的是研究20世纪60年代末加拿大数学家的两项重要数学发展之间的联系。其中第一个是罗伯特·朗兰兹的富有远见的计划，他将调和分析技术应用于数论中的中心问题。这种数论信息的载体是自同构的L函数，而本程序的一个基石就是研究这些函数的解析性质。自从朗兰兹程序开始以来，它已经在数学中变得几乎无处不在，既影响了表示论、代数几何、弦理论，甚至代数拓扑学的重要发展。**这些发展的第二个是维克多·卡克和罗伯特·穆迪引入了一族无限维对称，推广了他们上个世纪的经典李氏理论。虽然这些新物体本质上是无限维的，但它们具有许多与有限维对应物体相同的结构特征。此外，它们还经常表现出与其他数学领域令人惊讶和深远的联系。这一点在Kac和Moody引入的“最简单”的无限维对象类中最为明显，即所谓的循环群和循环代数(或仿射Kac-Moody群和代数)。在有限单群理论和数学物理发展的部分刺激下，圈代数和群在20世纪80年代与代数组合学、顶点算符理论和量子群理论的中心问题联系在一起。**这项建议的目标是扩大朗兰兹程序的范围，不仅考虑有限维群，而且考虑无限维循环群。朗兰兹哲学的一个核心方面是，不应该孤立地研究L对最小群的函数(以及它们所编码的数论问题)。相反，人们需要将它们与更大且往往相当不同的群体中的类似物体一起考虑。总而言之，这些信息提供了对原始问题的强大洞察力，一个人可以考虑的群体越多，对原始问题的洞察就越精致。**从较小的群体转移到更大的群体的工具是艾森斯坦系列结构。与之对应的是，常数项(或者更广泛地说，傅立叶系数)结构从较大的组移动到较小的组。朗兰兹和沙希迪通过组成这两个结构，发展了分析自同构L函数的机制。也就是说，他们首先形成一个爱森斯坦级数，然后使用调和分析的工具进行研究，最后使用常数项构造返回到原始环境。这种结构的效率依赖于这样一个事实，即爱森斯坦系列可以通过光谱技术进行研究，而这在原始环境中是不可用的。在实践中，这种范式在解析数论中的各种著名问题上产生了已知的最强结果。**为了启动朗兰兹-沙希迪方法，人们需要有大量更大的基团来考虑艾森斯坦级数。在有限维的背景下，存在有限数量的这样的群，其中大多数信息已经被收集。然而，在图像中引入无限维群极大地拓宽了该方法的范围。这项提议旨在开发一种机制，将这些无限维群纳入朗兰兹计划，从而纳入解决解析数论中心问题的可用工具的武器库。

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.