Mathematical Sciences: RUI: Gelfand Pairs in Nilpotent Analysis

本文主要研究由紧李群作用于幂零李群而产生的Gelfand对。具体的目标包括这些对的分类，Gelfand对的几何准则的发展，以及相关的球函数和微分算子的详细分析。该项目是在现代分析的一般领域，并关注非交换调和分析的经典问题在一个新的设置Gelfand对涉及幂零群。该研究大量借鉴了著名的紧李群和幂零李群理论，并与黎曼对称空间的研究提供了一个很好的对比。这项研究有机会加深我们对微分算子的可解性和李群轨道几何与表示理论之间的充分关系的理解。

This research focuses on the study of Gelfand pairs arising from actions of compact Lie groups on nilpotent Lie groups. Specific goals include the classification of such pairs, development of geometric criteria for Gelfand pairs, and a detailed analysis of the associated spherical functions and differential operators. The project is in the general area of modern analysis and concerns classical questions from noncommutative harmonic analysis in a new setting of Gelfand pairs involving nilpotent groups. The research draws heavily on the well-known theories of compact and nilpotent Lie groups and provides a nice contrast to the study of Riemannian symmetric spaces. This research has the opportunity of deepening our understanding of the solvability of differential operators and the full relationship between the geometry of the orbits of Lie groups and representation theory.