Metaplectic automorphic forms and matrix coefficients

这个研究项目将探讨数论和表示理论与几何学、代数组合学和统计力学的联系。在过去的几十年里，数论中一些最深刻的问题和猜想，尤其是与朗兰兹纲领有关的问题和猜想，在几何学和物理学中有着强大的相似之处，这一点已经变得越来越清楚。然而，这种关系和相关猜想背后的机制在很大程度上仍然是神秘的。该项目旨在通过扩大所考虑的对象类别和使用适用于更大类别的精选技术来探索连接的可能来源。特别是，许多提出的项目围绕着对p进代数群的矩阵系数及其算术覆盖的研究。这些矩阵系数在构造自同构l函数中起着关键作用。它们在元复盖的背景下的显式计算导致了与舒伯特变量的几何，麦克唐纳多项式的各种专门化以及通过正则基和晶格模型的量子群的惊人联系。这些将在提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。新的算术函数的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.