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

在这个提议中，首席研究员Brubaker打算研究分裂，还原代数群的有限覆盖上的自守形式，称为元形式。更准确地说，他将调查的Fourier-Whittaker系数metaplectic爱森斯坦系列诱导抛物子群。当覆盖的度是平凡的，这减少到线性代数群上的爱森斯坦级数的情况，正如朗兰兹、沙希迪和其他人所研究的那样，他们在朗兰兹纲领的制定和证明中起了重要作用。通过研究所有有限覆盖（包括平凡覆盖）上的族中的元复形，出现了令人惊讶的新结构。Brubaker和他的合作者已经证明，所得到的傅里叶-惠特克系数包含多个复变量的狄利克雷级数（所谓的“多重狄利克雷级数”），其系数用晶体图来描述。这些晶体图编码了关于量子群表示的信息，量子群是李代数的通用包络代数的变形。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相关联。该提案旨在更完整地发展这一理论，并探索它在数论，量子群和组合表示论之间提出的新联系。朗兰兹的程序最初被认为是一个惊人的集合，它将函数与有趣的算术属性（例如，对方程的整数解的数目进行计数）到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的解）。但是在几何和数学物理中也观察到了类似的对偶性，分别导致了几何和量子版本的朗兰兹纲领。 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大部分现代数学和数学物理学。然而，明确的潜在机制，例如，算术函数与解析函数的关系在很大程度上仍然是一个谜。在这些项目中，首席研究员Brubaker与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纲领在算术，几何和物理学中的各种化身之间的关系。这些项目的一个同样重要的组成部分是通过建立一个分层次的指导制度，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不断变化的重视，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