Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems

TL;DR

This paper constructs new integrable systems on the dual of compact Lie algebras, generalizing Gelfand-Zeitlin systems, with properties like Hamiltonian torus actions, convexity, and fiber connectedness, extending geometric quantization methods.

Contribution

It introduces a novel construction of integrable systems on Lie algebra duals using stratified gradient Hamiltonian flows, extending toric degeneration techniques to singular varieties.

Findings

Constructed integrable systems with Hamiltonian torus actions

Proved convexity and fiber connectedness of the collective moment map

Extended toric degeneration methods to singular quasi-projective varieties

Abstract

We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian $K$-manifold defines a Hamiltonian torus action on an open dense subset, B) if the $K$-manifold is multiplicity-free, then the resulting torus action is \textit{completely} integrable, and C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand-Zeitlin systems and Gelfand-Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map.