Topics in Harmonic Analysis on Reductive p-adic Groups

摘要DeBacker这位研究者将继续研究非阿基米德区域上李群的调和分析的一些主题。第一个目标是建立深度为零的超尖球面表示的Murnaghan-Kirillov理论。在最基本的层面上，Murnaghan-Kirillov理论要求我们的群的超尖球面表示与某些共轭轨道积分的傅里叶变换之间的联系。由于它们与李型有限群的密切联系，为这些表示建立Murnaghan-Kirillov理论的问题归结为将正则半单轨道积分与广义格林函数联系起来的问题。第二个目标是调查有关稳定的问题。例如，通过Bruhat-Tits理论以统一的方式明确地理解支持在幂零集上的稳定分布空间将是有用的。由于不同的齐性结果，这个问题可以通过将正则半单轨道积分与广义格林函数联系起来(如上所述)来解决。关于李群的调和分析将其根源追溯到以下物理问题：描述拨动的吉他弦的运动。最终，人们意识到这个问题-以及更纯粹的问题，如计算高斯和或研究算术级数中素数的密度-可以通过研究圆(或其他群)上的某些行为良好的函数来理解。这些行为良好的函数被称为特征，由此产生的理论被称为调和分析。到20世纪30年代，数学家们对许多类型的群(例如，紧群或交换群)的调和分析有了坚实的理解。在20世纪40年代，来自相对论物理学的问题导致人们考虑对一类更一般的群进行调和分析，这类群被称为李群。由Bargmann，Gelfand-Naimark和Harish-Chandra的工作发起，目标是，对于吉他问题，通过研究人物来理解群体的功能。这主要归功于哈里什-钱德拉，这一目标在很大程度上实现了。至少部分基于他对这项工作的理解，朗兰兹在20世纪60年代末被引导制定了他的计划；这个计划是一个巨大的、非凡的猜测和想法的网络-一种关于一切的数学理论。例如，哈里斯-泰勒、金-沙希迪和拉夫格的著名作品提供了它所预期的深刻结果的一个小样本。由于李群上的调和分析在我们理解这一领域的许多问题中起着核心作用，研究人员希望他的研究将有助于未来的进步。

Abstract DeBackerThe investigator will continue his research into a number of topics in harmonic analysis for Lie groups over nonarchimedian fields. The first goal is to establish Murnaghan-Kirillov theory for depth zero supercuspidal representations. At its most basic level, Murnaghan-Kirillov theory asks for a connection between the supercuspidal representations of our group and the Fourier transforms of certain coadjoint orbital integrals. Because of their intimate connection to finite groups of Lie type, the problem of establishing Murnaghan-Kirillov theory for these representations reduces to the problem of associating regular semisimple orbital integrals to generalized Green functions. The second objective is to investigate questions about stability. For example, it would be useful to explicitly understand, in a uniform way via Bruhat-Tits theory, the space of stable distributions supported on the nilpotent set. Thanks to various homogeneity results, this problem can be addressed by associating (as above) regular semisimple orbital integrals to generalized Green functions.Harmonic analysis on Lie groups traces its roots to the following problem from physics: Describe the motion of a plucked guitar string. Eventually, people realized that this problem --- and rather more pure problems like calculating Gauss sums or studying the density of primes in arithmetic progressions --- could be understood by studying certain well-behaved functions on the circle (or other groups). These well-behaved functions are called characters, and the resulting theory is called harmonic analysis. By the 1930s mathematicians had a firm understanding of harmonic analysis on many types of groups (for example, compact or abelian groups). During the 1940s problems from relativistic physics led people to think about harmonic analysis on a more general class of groups, called Lie groups. Initiated by the work of Bargmann, Gelfand--Naimark, and Harish-Chandra, the goal was, as for the guitar problem, to understand functions on the group by studying characters. Thanks mostly to Harish-Chandra this goal was largely realized. Based at least partially on his understanding of this work, in the late 1960s Langlands was led to formulate his program; this program is a vast, remarkable web of conjectures and ideas --- a kind of mathematical theory of everything. For example, the celebrated works of Harris--Taylor, Kim--Shahidi, and Lafforgue provide a small sampling of the deep results it anticipates. As harmonic analysis on Lie groups plays a central role in our understanding of many of the problems in this area, the investigator hopes his research will contribute to future progress.