CAREER: Multiple Dirichlet Series, Automorphic Forms, and Combinatorial Representation Theory

在这项提议中，首席研究员Brubaker打算研究分裂的约化代数群的有限覆盖上的自同构形式，称为亚可解型。更确切地说，他将研究由抛物子群诱导出的亚可解Eisenstein级数的傅立叶-惠特克系数。当覆盖的次数很小时，这就归结为朗兰兹、沙希迪和其他人所研究的线性代数群上的艾森斯坦级数的情况，他们在公式化和证明朗兰兹程序的部分方面起到了重要作用。通过研究涵盖所有有限覆盖(包括平凡覆盖)的族中的亚分解形式，出现了令人惊讶的新结构。Brubaker和他的合作者已经证明了产生的傅立叶-惠特克系数包含几个复变量的Dirichlet级数(所谓的‘多个Dirichlet级数’)，这些Dirichlet级数的系数用晶格图来描述。这些晶图编码有关量子群表示的信息，量子群是李代数的万能包络代数的变形。在这种情况下，相关的李代数与建立爱森斯坦级数的群的朗兰兹对偶群有关。该提案试图更全面地发展这一理论，并探索它所提出的数论、量子群和组合表示理论之间的新联系。朗兰兹的程序最初被设想为一个令人惊叹的猜想集合，它将具有有趣的算术性质的函数(例如，计算方程的整数解的数量)与具有良好分析性质的函数(例如，具有对称性并且是自然微分方程式的解)联系起来。但在几何学和数学物理中也观察到了类似的二元性，导致了朗兰兹计划的几何和量子版本。简而言之，这些二元性已经成为一个透镜，通过它可以组织和理解很大一部分现代数学和数学物理。然而，例如，将算术函数与解析函数相关联的显式潜在机制在很大程度上仍然是一个谜。在这些项目中，首席调查员布鲁贝克和他的合作者和学生将使用上述特殊例子提供的数据，试图找到这样一种机制，并试图更好地理解朗兰兹计划在算术、几何和物理方面的不同化身之间的关系。这些项目的一个同样重要的组成部分是通过建立分级指导制度对各级学生进行培训。为了支持这些努力，将开发一套课程材料，以反映现代数论对分析技术的重视程度的变化，重点是计算方法和基于实例的学习，以加强概念。

In this proposal, Principal Investigator Brubaker intends to study automorphic forms on finite covers of split, reductive algebraic groups known as metaplectic forms. More precisely, he will investigate the Fourier-Whittaker coefficients of metaplectic Eisenstein series induced from parabolic subgroups. When the degree of the cover is trivial, this reduces to the case of Eisenstein series on linear algebraic groups as studied by Langlands, Shahidi, and others, which have been instrumental in formulating and proving portions of the Langlands program. By studying metaplectic forms in families ranging over all finite covers (including the trivial one), surprising new structure emerges. Brubaker and his collaborators have demonstrated that the resulting Fourier-Whittaker coefficients contain Dirichlet series in several complex variables (so-called ``multiple Dirichlet series'') whose coefficients are described in terms of crystal graphs. These crystal graphs encode information about representations of quantum groups, which are deformations of the universal enveloping algebra of a Lie algebra. In this situation, the relevant Lie algebra is associated to the Langlands dual group of the group on which one builds the Eisenstein series. The proposal seeks to develop this theory more completely and explore the novel connections it suggests between number theory, quantum groups and combinatorial representation theory. Langlands' program was initially conceived as a stunning collection of conjectures relating functions with interesting arithmetic properties (e.g., counting the number of integer solutions to an equation) to functions with good analytic properties (e.g., having symmetries and being the solution of a natural differential equation). But similar kinds of duality have been observed in geometry and mathematical physics, leading to geometric and quantum versions of the Langlands programs, respectively. In short, these dualities have become a lens through which a large portion of modern mathematics and mathematical physics can be organized and understood. However, the explicit underlying mechanisms which relate, for example, arithmetic functions to analytic functions remain largely a mystery. In these projects, Principal Investigator Brubaker with his collaborators and students will use the data provided by the above special examples to attempt to find such a mechanism and attempt to better understand the relationships between various incarnations of the Langlands program in arithmetic, geometry, and physics. An equally important component of the projects is the training of students at all levels by creating a tiered system of mentoring. To bolster these efforts, a set of course materials will be developed to reflect the changing emphasis in modern number theory on analytic techniques, focusing on computational approaches and example-based learning to reinforce concepts