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

摘要本文将研究类型理论及其在p进群谐波分析中的应用。首先，与余九康一起，PI将在特定的情况下构造k类型，并证明它们是Bushnell-Kutzko意义上的类型。这些类型都有与其相关的赫克代数。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。这些项目是推进类型理论的必要步骤。在温和的情况下，人们可以期望它们能显著提高对p进群的表征理论的理解，而且它们也应该被证明在更一般的情况下是有用的。表征理论是利用对称性存在的一种主要数学技术。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，这是很多人在大学基本线性代数课程中看到的。研究了无限维对称的无限群。该项目解决了表示理论中的重要问题，这可能会应用于数论。

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.