Matrix Coefficients of Covering Groups, Quantum Groups, and Lie Superalgebras

对称的研究在科学上具有重要的基础。特别是，许多关于宇宙和基本粒子的物理理论都是用称为李群的对称性集合来描述的。了解这些对称可能作用的空间，为这些物理理论提供了重要的见解。这个项目将研究数论、量子群和数学物理之间的惊人联系，为理解这些空间提供一种新的方式。更具体地说，研究者和他的学生和合作者将探索p进群的表示及其算术覆盖的矩阵系数，被称为元群。矩阵系数允许从表示中提取数值不变量。它们在自同构l -函数的构造和解析性质的证明以及弦理论中散射振幅的确定中起着关键作用。本项目将发展矩阵系数、量子群和统计力学之间的新关系，以及利用Hecke代数研究矩阵系数的新方法。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。

The study of symmetry is of fundamental scientific importance. In particular, many physical theories of the universe and of elementary particles are described by collections of symmetries called Lie groups. Understanding the spaces on which these symmetries may act provides important insights into such physical theories. This project will study surprising connections between number theory, quantum groups, and mathematical physics that provide a new way of understanding such spaces.More specifically, the investigator and his students and collaborators will explore matrix coefficients of representations of p-adic groups and their arithmetic covers, known as metaplectic groups. Matrix coefficients allow one to extract numerical invariants from representations. They play a key role in both the construction of automorphic L-functions and the proofs of their analytic properties, and also in the determination of scattering amplitudes in string theory. The project will develop new relations between matrix coefficients, quantum groups, and statistical mechanics, and new methods to study matrix coefficients using Hecke algebras.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.