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.

摘要 DeBacker 研究者将继续研究非阿基米德域上李群调和分析的多个主题。 第一个目标是建立零深度超尖峰表示的 Murnaghan-Kirillov 理论。 在最基本的层面上，Murnaghan-Kirillov 理论要求我们群的超尖峰表示与某些共伴轨道积分的傅里叶变换之间存在联系。 由于它们与李型有限群的密切联系，为这些表示建立 Murnaghan-Kirillov 理论的问题简化为将正则半简单轨道积分与广义格林函数相关联的问题。 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理解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（如上所述）正则半简单轨道积分与广义格林函数相关联来解决。李群的调和分析可以追溯到以下物理学问题：描述弹拨吉他弦的运动。 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算术级数中素数的密度——可以通过研究圆（或其他群）上某些表现良好的函数来理解。 这些表现良好的函数称为特征，由此产生的理论称为调和分析。 到 20 世纪 30 年代，数学家对多种类型群（例如紧群或阿贝尔群）的调和分析有了深入的了解。 在 20 世纪 40 年代，相对论物理学的问题促使人们考虑对更一般的群（称为李群）进行调和分析。 由 Bargmann、Gelfand-Naimark 和 Harish-Chandra 的工作发起，对于吉他问题，目标是通过研究字符来理解群的函数。 这个目标很大程度上实现了，这主要归功于哈里什-钱德拉。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、非凡的猜想和想法网络——一种万物的数学理论。例如，哈里斯-泰勒、金-沙希迪和拉弗格的著名作品提供了其预期的深层结果的一小部分样本。 由于李群的调和分析在我们理解该领域的许多问题中发挥着核心作用，研究人员希望他的研究能够为未来的进展做出贡献。