Representations of p-adic Covering Groups and Integrable Systems

这个项目将在数学和物理之间建立新的联系。在数学方面，它探索的对象起源于数论，拥有一个复杂的集合的对称性。在物理方面，它涉及到统计力学中的晶格模型。这些晶格模型试图通过确定每个原子相互作用中的能量来研究物质的行为（例如，从固体到液体到气体的相变）。虽然这种还原主义的物理方法看起来令人生畏，但它经常令人惊讶地工作，粗略地说，在这种方法中成功的模型被称为“可解决的”。在这个项目中，可解的晶格模型被用来表示数论中的特殊函数，并证明它们以前未知的性质，数学和物理之间的联系为这两个学科提供了信息。该项目将为本科生和研究生提供广泛适用于广泛STEM职业的培训机会。更确切地说，该项目的目标是为局部域及其覆盖上的群的矩阵系数的研究提供代数结构。PI和他的合作者已经证明了p进群和量子群上的元惠特克函数之间的基本联系，使用上面描述的可解晶格模型作为两个理论之间的桥梁。研究将证明这种联系在多个方向上扩展和推广，并且在元塑性Whittaker模型中，Iwahori固定向量的可解晶格模型现在已经触手可及。一旦完成，该项目将通过用于演示Yang-Baxter方程的r矩阵，在元表示和量子仿射超代数之间提供强有力的联系。要研究的其他主题包括寻找p进表示理论和量子群之间的直接联系；利用Hecke代数特征对矩阵系数进行分类，以更好地理解局部理论中的非分支计算，这在l函数的积分表示中是必不可少的；描述了对超代数的Kashiwara晶体理论和几何晶体上的阿基米德矩阵系数的改进。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will establish new connections between mathematics and physics. On the mathematics side, it explores objects originating from number theory that possess a sophisticated collection of symmetries. On the physics side, it involves lattice models from statistical mechanics. These lattice models are attempts to study the behavior of matter (e.g., its phase transitions from solid to liquid to gas) by determining the energy in each individual atomic interaction. While such a reductionist physical approach may seem daunting, it works surprisingly often and, roughly speaking, the models which succeed in this approach are termed "solvable." In this project solvable lattice models are used to represent special functions from number theory and to demonstrate previously unknown properties of them and the connections between mathematics and physics serve to inform both subjects. The project will provide training opportunities for undergraduate and graduate students that are broadly applicable to a wide range of STEM careers. More precisely, the goal of the project is to provide algebraic structure to the study of matrix coefficients on groups over local fields and their covers. The PI and his collaborators have demonstrated a fundamental connection between metaplectic Whittaker functions on p-adic groups and quantum groups, using the solvable lattice models described above as a bridge between the two theories. The research will demonstrate that this connection extends and generalizes in multiple directions, and a solvable lattice model for Iwahori fixed vectors in a metaplectic Whittaker model is now within reach. Once completed, the project will provide a powerful connection between metaplectic representations and quantum affine superalgebras via the R-matrices used to demonstrate Yang-Baxter equations. Additional topics to be investigated include finding direct links between p-adic representation theory and quantum groups; using Hecke algebra characters to categorize matrix coefficients to better understand unramified calculations in the local theory, which are essential in integral representations of L-functions; and describing enhancements to the theory of Kashiwara crystals for superalgebras and to archimedean matrix coefficients on geometric crystals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.