Hecke Algebras, Buldings and Harmonic Analysis on p-adic Groups

Hecke 代数、Buldings 和 p-adic 群的调和分析

• 批准号: 0223829
• 负责人: Ju-Lee Kim
• 金额: $ 1.93万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0223829&HistoricalAwards=false
• 关键词: Hecke; Algebras; Buldings; Harmonic; Analysis

9970454This project concerns problems in representation theory. Recent progress in the theory of "Types" suggests that the Harmonic analysis on p-adic groups can be completely understood via harmonic analysis on the associated Hecke algebras. Moreover these Hecke algebras are expected to be generalized affine Hecke algebras and so should be easier to analyze. Dr. Kim will try to establish the fact that many of the classical groups defined over p-adic fields give rise to algebras that are generalized affine Hecke algebras. Further she intends to study the relation between buildings of the classical groups and buildings of the general linear group. Finally she intends to consider how base changes relate to the representations of Hecke algebras.Representation theory is a major mathematical technique for exploiting the presence of symmetry. For example, the structure of the hydrogen atom, one of the fundamental computations in quantum mechanics, is controlled by representation theory. Another way of thinking about representation theory is as a generalization of the theory of eigenvalues and matrices that many people see in a college course on basic linear algebra. This project addresses two important questions in representation theory. One potentially has important applications to number theory. The other concerns a generalization of the theory of angular momentum in quantum mechanics. It will help describe how pair of interacting quantum mechanical systems of a rather general type breaks up into simpler systems.

这个项目涉及表象理论中的问题。在“类型”理论方面的最新进展表明，通过对相应的Hecke代数的调和分析，可以完全理解p-add群上的调和分析。此外，这些Hecke代数被认为是广义仿射Hecke代数，因此应该更容易分析。Kim博士将试图证明这样一个事实，即定义在p-ady域上的许多经典群产生的代数是广义仿射Hecke代数。此外，她还打算研究经典群的建筑物与一般线性群的建筑物之间的关系。最后，她打算考虑基数变化如何与Hecke代数的表示相关联。表示理论是利用对称性的存在的主要数学方法。例如，氢原子的结构，这是量子力学的基本计算之一，由表象理论控制。另一种思考表示论的方法是将许多人在基础线性代数的大学课程中看到的特征值和矩阵理论的推广。这个项目解决了表象理论中的两个重要问题。其中一个可能在数论中有重要的应用。另一个涉及量子力学中角动量理论的推广。它将有助于描述一对相当一般类型的相互作用的量子力学系统如何分解成更简单的系统。