Mathematical Sciences: RUI: Gelfand Pairs in Nilpotent Analysis

本研究主要研究Gelfand对的产生 紧李群在幂零李群上的作用。 具体目标包括对这些对进行分类， Gelfand对的几何标准的发展，以及详细的 相关球函数和微分的分析 运营商 该项目属于现代分析的一般领域， 涉及非对易调和分析的经典问题 在一个新的设置Gelfand对涉及幂零群。 的 研究在很大程度上借鉴了众所周知的理论紧凑， 幂零李群，并提供了一个很好的对比，研究 黎曼对称空间 这项研究有机会 加深了我们对微分方程可解性的理解 运算符和几何之间的全部关系， 李群的轨道和表示论。

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.