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

圆周率将研究类型理论及其在p-进群调和分析中的应用。首先，派将与余久康一起在一定的驯服情况下构造K型，并证明它们是Bushnell-Kutzko意义上的K型。这些类型有与之相关的Hecke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型在调和分析中的各种应用。类型理论的最新进展表明，通过相关Hecke代数上的调和分析，可以更好地理解p-add群上的调和分析。这些项目是推进类型理论的必要步骤。在这种情况下，人们可以预期它们会显著地提高对p-add群表示理论的理解，它们在更一般的情况下也应该被证明是有用的。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。研究人员研究某些无限维对称的无限群。这个项目解决了表示论中的重要问题，它可能在数论中有应用。

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.