Stratified gradient Hamiltonian vector fields and collective integrable systems

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 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 piecewise, has a flow whose limit exists and defines a continuous degeneration map.