Topics in Harmonic Analysis on Reductive p-adic Groups

约简 p 进群调和分析专题

• 批准号: 0500667
• 负责人: Stephen DeBacker
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500667&HistoricalAwards=false
• 关键词: Topics; Harmonic; Analysis; Reductive; adic

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.

研究者将继续研究非阿基米德场上李群谐波分析的一些课题。第一个目标是建立深度零超尖表示的Murnaghan-Kirillov理论。在最基本的层面上，Murnaghan-Kirillov理论要求在我们群的超尖态表示和某些协伴随轨道积分的傅里叶变换之间建立联系。由于它们与Lie型有限群的密切联系，为这些表示建立Murnaghan-Kirillov理论的问题简化为将正则半单轨道积分与广义Green函数联系起来的问题。第二个目标是调查有关稳定性的问题。例如，通过Bruhat-Tits理论，以统一的方式明确地理解在幂零集上支持的稳定分布的空间将是有用的。由于各种同质性结果，这个问题可以通过将正则半单轨道积分与广义格林函数相关联（如上所述）来解决。李群的谐波分析可以追溯到物理学中的以下问题：描述拨弦吉他弦的运动。最终，人们意识到，这个问题——以及更纯粹的问题，比如计算高斯和或研究等差数列中的素数密度——可以通过研究圆（或其他群）上的某些表现良好的函数来理解。这些表现良好的函数被称为特征，由此产生的理论被称为谐波分析。到20世纪30年代，数学家对许多类型群（例如紧群或阿贝尔群）的调和分析有了坚定的理解。在20世纪40年代，相对论物理学的问题使人们开始考虑对一种更一般的群——李群——进行谐波分析。由Bargmann， Gelfand- Naimark和Harish-Chandra的工作所发起，对于吉他问题，目标是通过研究字符来理解组的功能。多亏了哈里什-钱德拉，这个目标基本上实现了。至少部分基于他对这项工作的理解，在20世纪60年代末，朗兰兹被引导制定了他的计划；这个程序是一个巨大的、非凡的猜想和想法的网络——一种关于一切的数学理论。例如，Harris- Taylor， Kim- Shahidi和Lafforgue的著名作品提供了它所预期的深度结果的小样本。由于李群的谐波分析在我们理解这一领域的许多问题中起着核心作用，研究者希望他的研究将有助于未来的进展。