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

9970454这个项目涉及表征理论中的问题。“类型”理论的最新进展表明，p进群的调和分析可以通过对相关Hecke代数的调和分析来完全理解。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。金博士将试图证明这样一个事实，即在p进域上定义的许多经典群会产生广义仿射赫克代数。她打算进一步研究古典建筑群与一般线性建筑群之间的关系。最后，她打算考虑基的变化如何与赫克代数的表示有关。表征理论是利用对称性存在的主要数学技术。例如，氢原子的结构，量子力学的基本计算之一，是由表征理论控制的。另一种思考表征理论的方法是将特征值理论和矩阵理论一般化很多人在大学基本线性代数课程中都看到过。这个项目解决了表征理论中的两个重要问题。其中一个对数论有潜在的重要应用。另一个是关于量子力学中角动量理论的推广。它将有助于描述一对相互作用的一般类型的量子力学系统如何分解成更简单的系统。

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.