Analysis, Dynamics and Applications Seminar

The Poisson Geometry of Plancherel Theorems for Triangular Groups

Plancherel theorems/formulae are at the heart of much of harmonic analysis. In the setting of commutative groups, such results underlie classical Fourier analysis. For non-commutative but compact groups, generalizations of the classical theorems enter fundamentally into applications of spherical-type harmonics to differential equations with symmetries, including many of the basic models for quantum mechanics. The situation for non-compact groups is much more challenging.

In this talk we will focus on the particular (non-abelian, non-compact)  case of triangular groups, an example of the more general setting of maximal solvable (Borel) subgroups of semisimple Lie groups. Thanks to the pioneering work of Moore, Wolf and their collaborators, one knows how to formulate Plancherel-type results for these groups and their generalizations. We present recent results, based on the Kirillov-Kostant orbit philosophy, that provide an explicit geometric and Hamiltonian perspective on these Plancherel theorems and their potential applications.