Metaplectic automorphic forms and matrix coefficients

本研究计画将探讨与几何学、代数组合学及统计力学相关的数论与表示论的主题。在过去的几十年里，人们已经清楚地认识到，数论中一些最深刻的问题和理论，特别是那些与朗兰兹纲领有关的问题和理论，在几何和物理学中有着强大的相似之处。然而，这种关系背后的机制和相关的假设仍然很神秘。 该项目旨在通过扩大所考虑的对象类别并使用适用于这一更大类别的精选技术来探索可能的联系来源。特别是，提出的许多项目都围绕着对p-adic代数群及其算术覆盖的矩阵系数的调查。这些矩阵系数在自守L-函数的构造中起着关键作用。他们的显式计算的上下文中的metaplectic覆盖导致令人惊讶的连接与几何的舒伯特品种，各种专业化的麦克唐纳多项式，量子群通过规范的基础和晶格模型。这些将进一步发展，在拟议的工作和一个框架，分类矩阵系数代数群交织运营商Hecke代数模块将被追求。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.