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

职业：多重狄利克雷级数、自同构形式和组合表示理论

• 批准号: 0844185
• 负责人: Benjamin Brubaker
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0844185&HistoricalAwards=false
• 关键词: CAREER; Multiple; Dirichlet; Series; Automorphic

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

在该提案中，首席研究员Brubaker打算在分裂，还原的代数群的有限覆盖范围内研究被称为元位形式的群体。更确切地说，他将研究由抛物线亚组引起的替代艾森斯坦系列的傅立叶旋转器系数。当覆盖范围很琐碎时，这将减少到兰格兰兹，沙希迪和其他人所研究的线性代数群的Eisenstein系列案例，这些群体在制定和证明兰格兰计划计划的部分方面发挥了作用。通过研究在所有有限覆盖物（包括微不足道）的家庭中的元容器形式，出乎意料的新结构。 Brubaker和他的合作者已经证明，由此产生的傅立叶旋风系数包含了几个复杂变量（所谓的``多个Dirichlet系列''）中的Dirichlet系列，它们的系数在Crystal图中进行了描述。这些晶体图编码有关量子基团表示的信息，这些信息是LIE代数的通用包围代数的变形。在这种情况下，相关的谎言代数与该小组的Langlands双重组相关联，其中建立了Eisenstein系列。该提案旨在更完整地发展该理论，并探索它在数字理论，量子群和组合代表理论之间提出的新颖联系。 Langlands的计划最初被认为是与有趣的算术特性（例如，将整数溶液的数量计算到方程式）与具有良好分析特性的函数（例如具有对称性和具有对称性和自然微分公式的解决方案）的功能相关的令人惊叹的猜想集合。但是在几何和数学物理学中已经观察到了类似的二元性，分别导致了兰兰兹计划的几何和量子版本。 简而言之，这些二元性已成为一个镜头，可以组织和理解大部分现代数学和数学物理学的镜头。但是，例如，与分析功能相关的显式基本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Brubaker与他的合作者和学生将使用上述特殊示例提供的数据来试图找到这种机制，并试图更好地了解Langlands计划在算术，几何和物理学中的各种化身之间的关系。项目的同样重要组成部分是通过创建分层的指导系统对学生进行各级培训。为了加强这些努力，将开发一组课程材料，以反映现代数字理论对分析技术的不断变化，重点关注计算方法和基于示例的学习，以增强概念