Topics in Harmonic Analysis for Reductive P-adic Groups

DeBacker研究人员将致力于研究非阿基米德局域上约化群调和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，同质性问题会问：给出两个分布，这两个分布一致的最大函数集是什么？第一个主题涉及群中紧元素上所支持的不变分布的齐性问题。这是最后一个未经证实的(大体上)同质性陈述，它对“基本引理”(这是一个猜想)有影响。第二个主题是关于幂等集上支持的分布的稳定性问题。最后一个主题涉及Harish-Chandra-Howe局部特征标展式中出现的常量的意义。上面讨论的主题属于李群的调和分析范畴。这个扎根于物理学的数学领域是由哈里什-钱德拉在大约50年前开创的。自20世纪70年代以来，这一领域的许多工作都是针对朗兰兹计划中自然产生的问题(该计划试图在很大程度上联系到数论、表示理论和调和分析)。例如，Harish-Chandra研究了当一个人在共轭类上积分时出现的分布。从朗兰兹计划的角度来看，研究当一个人在某些稳定的共轭类上积分时产生的分布也是重要的，也就是说，在地场的代数闭包上成为共轭的类。这位研究人员希望他的研究计划将为这一领域的未来进步做出贡献。

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.