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

对称性的研究具有基础性的科学意义。具体地说，许多宇宙和基本粒子的物理理论都是由称为李群的对称性集合来描述的。理解这些对称性可能作用的空间，为此类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令人惊讶的联系，这为理解这种空间提供了一种新的方法。更具体地说，研究人员和他的学生以及合作者将探索p-进群及其算术覆盖的表示的矩阵系数，称为亚普勒群。矩阵系数允许从表示中提取数值不变量。它们在构造自同构的L函数和证明其解析性质以及确定弦理论中的散射幅度方面都起着关键的作用。该项目将开发矩阵系数、量子群和统计力学之间的新关系，以及使用Hecke代数研究矩阵系数的新方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.