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元形形式，这是一种自同构形式在分裂代数群的某些覆盖上的推广。具体而言，本文的研究重点是构建具有良好解析性质（如泛函方程和亚纯延拓）的复数变量Dirichlet级数，称为“多重Dirichlet级数”（Multiple Dirichlet series， MDS），并将这些级数与元群上的爱森斯坦级数的Fourier-Whittaker系数联系起来。由于元覆盖的构造与数域的互易律密切相关，Dirichlet级数及其极残数包含由幂残数符号构建的字符扭曲的自同构形式族。狄利克雷级数的解析性质随后转化为自同构形式的算术应用，包括自同构l函数的不消失结果。这些MDS的分析性质的构建和后续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标是更深入地理解这种表示理论与形而上学形式之间的联系。在此过程中，该结构有望进一步阐明组合表示理论与自同构形式的傅里叶-惠特克系数的特殊情况之间的关系。从历史上看，数论中的问题集中在多项式方程的整数解上，即所谓的“丢芬图方程”，可以简单地表述，但往往非常难以证明。这些问题一度被认为是纯粹思想实验的材料，因为整数是离散的，因此与世界及其连续现象没有关系。但在20世纪60年代末，罗伯特·朗兰兹提出了一系列影响深远的猜想，即今天的朗兰兹纲领，旨在研究数论、算术几何和谐波分析之间的联系；事实上，他的猜想是基于一种特殊情况下的高度对称函数的计算，即爱森斯坦级数，这是他的建议的主要研究对象。在过去的几十年里，朗兰兹猜想的范围得到了极大的扩展，现在已经从解决丢芬图方程的方法扩展到与量子场论和弦理论密切相关的几何版本，这些理论试图通过对包括重力和电磁力在内的基本力的统一处理来解释我们宇宙的起源和膨胀。也就是说，在关于整数多项式方程解的自然问题的激励下，人们对意义深远的重要物理现象得到了新的解释；在试图回答离散问题的过程中，人们找到了对连续世界的解释。该项目试图将数论、自同构形式、李群和组合学领域的研究人员和学生聚集在一起，通过研究更一般的爱森斯坦级数的大类及其与上述学科的关系，进一步研究类似的联系。

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.