Distinguished representations of reductive p-adic groups

******我的研究是朗兰兹计划的一部分，朗兰兹计划是数学的一个分支，涉及数论、自同构形式、表示理论和代数几何之间的深层联系。我研究了被称为约进群的某些矩阵群的表示的性质。这些矩阵群是用数论的***术语定义的——矩阵条目属于p进域——这些域是数字***域的非阿基米德补全（有理数的有限扩展）。近年来，人们对p进对称变分及其与朗兰兹规划的联系的谐波分析产生了浓厚的兴趣。我的工作涉及***研究和构建可约p进群的不同表征。这些不同的表示形式***相对于定义p进对称空间的对合表现出特定的对称性。区分表征理论的基本组成部分是所谓的相对超尖表征。我的主要研究重点是相对超尖表示的构造和它们的性质的研究，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.