Topics in Harmonic Analysis for Reductive P-adic Groups

还原 P 进群的调和分析主题

• 批准号: 0345121
• 负责人: Stephen DeBacker
• 金额: $ 7.56万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0345121&HistoricalAwards=false
• 关键词: Topics; Harmonic; Analysis; Reductive; adic

AbstractDeBackerThe investigator will work on a number of topics which arise in the study of harmonic analysis for reductive groups over nonarchimedian local fields. The unifying theme for these topics is their relation to homogeneity statements. At their most basic level, homogeneity questions ask: given two distributions, what is the largest set of functions on which the two distributions will agree? The first topic concerns the homogeneity question for invariant distributions supported on the compact elements in the group. This is the last unproven (in general) homogeneity statement, and it has implications for the "fundamental lemma" (which is a conjecture). The second topic concerns questions of stability for distributions supported on the unipotent set. The final topic concerns the meaning of the constants occurring in the Harish-Chandra-Howe local character expansion.The topics discussed above fall under the rubric of harmonic analysis on Lie groups. This area of mathematics, which is solidly rooted in physics, was pioneered by Harish-Chandra beginning some fifty years ago. Since the 1970s, much of the work in this area has been directed toward questions which arise naturally in the Langlands program (which seeks to relate, in a strong sense, number theory, representation theory, and harmonic analysis). For example, Harish-Chandra studied the distributions which arise when one integrates over a conjugacy class. From the perspective of the Langlands program, it is also important to study the distributions which arise when one integrates over certain stable conjugacy classes, that is, classes which become conjugate over an algebraic closure of the ground field. The investigator hopes his program of research will contribute to future progress in this area.

研究者将研究非阿基米德局部域上约化群的谐波分析中出现的一些问题。这些主题的统一主题是它们与同质性陈述的关系。在最基本的层面上，同质性问题问的是：给定两个分布，这两个分布一致的最大函数集是什么？第一个主题是关于群中紧元素支持的不变分布的同质性问题。这是最后一个未经证明的（一般的）同质性陈述，它对“基本引理”（这是一个猜想）有影响。第二个主题涉及在幂偶集上支持的分布的稳定性问题。最后一个主题是关于在Harish-Chandra-Howe局部字符扩展中出现的常量的含义。上面讨论的题目属于李群的调和分析范畴。这个牢牢扎根于物理学的数学领域，是大约50年前由哈里什-钱德拉开创的。自20世纪70年代以来，这一领域的大部分工作都是针对朗兰兹纲领（在强烈的意义上，它试图将数论、表示理论和谐波分析联系起来）中自然产生的问题。例如，Harish-Chandra研究了在共轭类上积分时产生的分布。从朗兰兹纲领的角度来看，研究在某些稳定共轭类上积分时产生的分布也很重要，即在地场的代数闭包上变成共轭的类。研究者希望他的研究计划将有助于这一领域未来的发展。