Mathematical Sciences: RUI: Gelfand Pairs in Nilpotent Analysis

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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.