The Poisson Geometry of Plancherel Formulas for Triangular Groups

TL;DR

This paper develops a canonical coordinate system for generic co-adjoint orbits of triangular groups, linking Poisson geometry, Plancherel formulas, and integrable systems like the Toda lattice.

Contribution

It generalizes known coordinate systems to generic orbits, connecting Plancherel measures with invariant theory and quantum integrability in Lie groups.

Findings

Established canonical coordinates for generic co-adjoint orbits.

Connected Plancherel formulas with invariant theory of Borel subgroups.

Discussed implications for quantum integrability of Toda lattices.

Abstract

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant-Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.