Metaplectic automorphic forms and matrix coefficients

这项研究项目将探索与几何、代数组合学和统计力学有关的数论和表示论的主题。在过去的几十年里，数论中一些最深刻的问题和猜想，尤其是那些与朗兰兹计划有关的问题和猜想，在几何和物理上有着强大的相似性，这一点已经变得很明显。然而，这种关系和相关猜测背后的机制在很大程度上仍然是个谜。这个项目的目的是通过拓宽所考虑的对象的类别并使用适用于这一更大类别的筛选的技术集来探索可能的联系来源。特别是，许多提出的项目都围绕着研究p-进代数群及其算术覆盖的矩阵系数。这些矩阵系数在构造自同构的L函数中起着关键作用。它们在亚素覆盖的背景下的显式计算导致了与Schubert簇的几何、Macdonald多项式的各种特化以及通过正则基和晶格模型的量子群的惊人联系。这些将在拟议的工作中进一步发展，并将寻求将代数群上的矩阵系数分类为Hecke代数模块的交织算子的框架。算术函数的新分布结果将是这些研究的另一个副产品。

This research project will explore topics in number theory and representation theory with connections to geometry, algebraic combinatorics, and statistical mechanics. Over the past several decades it has become clear that some of the deepest questions and conjectures in number theory, most notably those connected with the Langlands program, have powerful analogs in geometry and physics. However, the mechanism behind this relationship and associated conjectures remains largely mysterious. This project aims to explore possible sources of the connections by broadening the class of objects under consideration and using the winnowed set of techniques that apply to this larger class.In particular, many of the projects proposed center around the investigation of matrix coefficients for p-adic algebraic groups and their arithmetic covers. These matrix coefficients play a key role in the construction of automorphic L-functions. Their explicit computation in the context of metaplectic covers leads to surprising connections with geometry of Schubert varieties, to various specializations of Macdonald polynomials, and to quantum groups via both canonical bases and lattice models. These will be further developed in the proposed work and a framework for classifying matrix coefficients on algebraic groups as intertwining operators for Hecke algebra modules will be pursued. New distribution results for arithmetic functions will be another byproduct of these investigations.