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，涉及数论、自守形式、表示论和代数几何之间的深刻联系。 我研究某些矩阵群的表示的性质，这些矩阵群被称为约化p进群。这些矩阵群是用数论的 * 术语定义的-矩阵元素属于p-adic域-这些域是数 * 域的非阿基米德完备化（有理数的有限扩展）。近年来，对p-adic对称簇的调和分析 * 以及与朗兰兹程序的联系有了相当大的兴趣。我的工作涉及 * 的研究和建设的区别表示约化p-adic集团。相对于定义p-adic对称空间的对合，这些特殊的表示 * 表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，特别是那些与朗兰兹纲领相关的性质。

******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.