K-Types and Harmonic Analysis on p-Adic Reductive Groups

PI将研究类型理论及其在p-adic群上的调和分析中的应用。首先，PI与余久康一起构造了某些驯服情形下的K-型，并证明它们是Bushnell-Kutzko意义下的型。这些类型都有Hecke代数与之相关。在第二个项目中，PI将研究这些Hecke代数的结构。最后，PI将考虑这些K型在谐波分析中的各种应用。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。这些项目是推进类型理论的必要步骤。在驯服的情况下，人们可以期望它们导致对p-adic群的表示理论的显著改进的理解，表示论是利用对称性的一种主要的数学方法。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。另一种思考表示论的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。该项目研究无限维对称的某些无限群。该项目解决了表示论中的重要问题，可能应用于数论。

AbstractKim The PI will study the theory of types, and their applications in Harmonicanalysis on p-adic groups. First, jointly with Jiu-Kang Yu, the PI willconstruct K-types in certain tame case and show that they are types in thesense of Bushnell-Kutzko. These types have Hecke algebras associated tothem. In the second project, the PI will study the structure ofthese Hecke algebras. Lastly, the PI will consider various applications ofthese K-types in Harmonic analysis. Recent progress in the theory oftypes suggests that the Harmonic analysis on p-adic groups can becompletely understood via harmonic analysis on the associated Hecke algebras.These projects are necessary steps towards advancing the theory of types.In the tame case, one can expect them to lead to a significantly improvedunderstanding of representation theory of p-adic groups, andthey should also prove useful in the more general situation.Representation theory is a major mathematical technique for exploiting thepresence of symmetry. For example, the structure of the hydrogen atom, oneof the fundamental computations in quantum mechanics, is controlled byrepresentation theory. Another way of thinking about representation theoryis as a generalization of the theory of eigenvalues and matrices that manypeople see in a college course on basic linear algebra. The investigatorstudies certain infinite groups of infinite dimensional symmetries.This project addresses important questions in representation theory,which potentially has applications to number theory.