Distinguished representations of reductive p-adic groups

我的研究是朗兰兹纲领的一部分，朗兰兹纲领是数学的一个分支，涉及到深层的联系 在数论自守形式表示论和代数几何之间。 我研究了 某些矩阵群的表示称为约化p进群。这些矩阵组定义在 数论术语-矩阵元素属于p-adic域-这些域是数的非阿基米德完备化 域（有理数的有限扩展）。近年来，谐波分析引起了人们的极大兴趣 关于p-adic对称簇和与朗兰兹程序的联系。我的工作包括 研究和构造约化p进群的特殊表示。这些杰出的代表 相对于定义p-adic对称空间的对合展现出特定的对称性质。 区别表示理论的基本组成部分是所谓的相对超尖点 表示。我的主要重点是相对超尖点表示的构造及其性质的研究， 尤其是那些与朗兰兹纲领有关的

My research forms part of the Langlands program, a branch of mathematics involving deep connections between number theory, automorphic forms, representation theory, and algebraic geometry. I study the properties of representations of certain matrix groups known as reductive p-adic groups. These matrix groups are defined in terms of number theory - the matrix entries belong to p-adic fields - these fields are nonarchimedean completions of number fields (finite extensions of the rational numbers). In recent years, there has been considerable interest in harmonic analysis on p-adic symmetric varieties and connections with the Langlands program. My work involves the study and construction of distinguished representations of reductive p-adic groups. These distinguished representations exhibit specific symmetry properties relative to the involution that defines the p-adic symmetric space. The basic building blocks in the theory of distinguished representations are the so-called relatively supercuspidal representations. My main focus is on construction of relatively supercuspidal representations and the study of their properties, particularly those that are relevant to the Langlands program.