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

9970454这个项目关注的问题在表征理论。 “类型”理论的最新进展表明，p-adic群上的调和分析可以通过相应的Hecke代数上的调和分析得到完全理解。 此外，这些Hecke代数预计是广义仿射Hecke代数，因此应该更容易分析。 金博士将试图建立这样一个事实，即许多经典的群体定义在p-adic领域产生的代数是广义仿射Hecke代数。 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，她打算考虑如何基地的变化涉及到陈述的Hecke代数。代表性理论是一个主要的数学技术，利用存在的对称性。 例如，氢原子的结构，量子力学的基本计算之一，是由表示论控制的。 关于表示论的另一种思考方式是把它作为特征值和矩阵理论的推广，很多人在大学的基础线性代数课程中看到过这些理论。 这个项目解决了表征理论中的两个重要问题。其中一个可能对数论有重要的应用。 另一个是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裂成更简单的系统。

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.