Topics in Harmonic Analysis for Reductive P-adic Groups

研究者将致力于非阿基米德局部域上约化群的调和分析研究中出现的一些课题。 这些主题的统一主题是它们与同质性陈述的关系。 在最基本的层次上，齐性问题是：给定两个分布，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所支持的不变分布的齐性问题。 这是最后一个未经证明的（一般）同质性陈述，它对“基本引理”（这是一个猜想）有影响。 第二个主题涉及的问题的稳定性分布支持的幂幺集。 最后一个主题涉及Harish-Chandra-Howe局部特征标展开式中常数的意义。上面讨论的主题属于李群上的调和分析的范畴。 这一领域的数学，这是坚实的植根于物理学，开创了哈里什钱德拉开始约50年前。 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中自然出现的问题（在很强的意义上，朗兰兹纲领试图将数论、表示论和调和分析联系起来）。 例如，哈里什-钱德拉研究的分布时出现的一个整合了共轭类。 从朗兰兹纲领的角度来看，研究在某些稳定的共轭类上积分时产生的分布也很重要，这些共轭类是在基场的代数闭包上共轭的类。 研究人员希望他的研究计划将有助于这一领域的未来进展。

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.