Multiple Dirichlet Series with Applications to Automorphic Representation Theory

在这个项目中，首席研究员Brubaker和他的合作者试图理解亚可分形式，即自同构形式到可分裂的约化代数群的某些覆盖的推广。具体地说，所提出的研究集中于构造具有良好分析性质(例如函数方程和亚纯延拓)的多个复变量的Dirichlet级数，并将这些级数与亚可解群上的Eisenstein级数的Fourier-Whittaker系数联系起来。由于次可解覆盖的构造与数域中的互易定律密切相关，所以Dirichlet级数及其极剩余数包含由幂剩余符号建立的特征标所扭曲的自同构型族。然后将狄里克莱级数的解析性质转化为自同构形式的算术应用，包括自同构L函数的非零化结果。这些MDS的分析性质的构造和随后的证明使用了组合表示理论中的新技术，而这项工作的另一个主要目的是更深入地理解这一表示理论和亚可解形式之间的联系。在这个过程中，这种结构有望进一步阐明组合表示理论与自同构形式的傅立叶-惠特克系数的特例之间的关系。历史上，数论中的问题一直围绕着多项式方程的整数解，即所谓的“丢番图方程”，这些方程可以简单地表述，但往往非常难以证明。曾经有一次，这些问题似乎是纯粹的思想实验，因为这些整数是离散的，因此应该与世界及其连续的现象没有关系。但在20世纪60年代末的S，罗伯特·朗兰兹发展了一系列影响深远的猜想，今天被称为朗兰兹计划，以研究数论、算术几何和调和分析之间的联系；事实上，他的猜想是基于涉及高度对称函数的一种特殊情况的计算，即艾森斯坦级数，这是本建议的主要研究对象。朗兰兹猜想的范围在过去几十年里得到了极大的扩展，现在已经从求解丢番图方程的方法扩展到与量子场论和弦理论有密切联系的几何版本，后者试图通过对包括重力和电磁在内的基本力的统一处理来解释我们宇宙的起源和扩张。也就是说，受到关于整数中多项式方程的解的自然问题的启发，人们对极其重要的物理现象获得了一种新的解释；在试图回答离散问题时，人们找到了对连续世界的解释。这个项目试图把以前完全不同的数论、自同构形式、李群和组合学的研究人员和学生聚集在一起，通过研究更一般的艾森斯坦级数的大类及其与上述学科的关系来进一步研究相似的联系。

In this project, Principal Investigator Brubaker and his collaborators seek to understand metaplectic forms, a generalization of automorphic forms to certain covers of split, reductive algebraic groups. Specifically, the proposed research focuses on constructing Dirichlet series in several complex variables, termed ``Multiple Dirichlet Series'' (MDS), with good analytic properties (e.g. functional equations and meromorphic continuation) and connecting these series to the Fourier-Whittaker coefficients of Eisenstein series on metaplectic groups. Since the construction of the metaplectic cover is intimately tied to reciprocity laws in number fields, the Dirichlet series and its polar residues contain families of automorphic forms twisted by characters built from power residue symbols. Analytic properties of the Dirichlet series then translate to arithmetic applications for automorphic forms, including non-vanishing results for automorphic L-functions. The construction and subsequent proof of analytic properties of these MDS uses new techniques in combinatorial representation theory, and another primary objective of this work is a deeper understanding of the connections between this representation theory and metaplectic forms. In the process, this structure is expected to illuminate further the relationship between combinatorial representation theory and the special case of Fourier-Whittaker coefficients of automorphic forms.Historically, problems in number theory have centered around integer solutions to polynomial equations, so called ``Diophantine equations,'' which could be simply stated, but often extraordinarily hard to prove. It once appeared that these questions were the stuff of pure thought experiments, since the integers are discrete and should therefore have no bearing on the world and its continuous phenomena. But in the late 1960's, Robert Langlands developed a series of far-reaching conjectures known today as the Langlands' Program to investigate connections among number theory, arithmetic geometry and harmonic analysis; in fact, his conjectures were based on calculations involving a special case of the highly symmetric functions known as Eisenstein series, which are the principal objects of study in this proposal. The reach of Langlands' conjectures has been greatly expanded in the last several decades and now extends from methods for solving Diophantine equations to geometric versions with intimate connections to quantum field theory and string theory, which attempt to explain the origins and expansion of our universe via a uniform treatment of fundamental forces including gravity and electromagnetism. That is, motivated by natural questions about solutions of polynomial equations in the integers, one obtains a new interpretation for profoundly important physical phenomena; in trying to answer discrete problems, one finds explanations of the continuous world. This project attempts to bring together a previously disparate community of researchers and students in number theory, automorphic forms, Lie groups, and combinatorics to further investigate analogous connections by studying large classes of more general Eisenstein series and their relations to the aforementioned disciplines.