Complete families of commuting functions for coisotropic Hamiltonian actions

Abstract

Let G be an algebraic group defined over a field F of characteristic zero with g=LieG. The dual space g⁎ equipped with the Lie–Poisson bracket is a Poisson variety and each irreducible G-invariant subvariety X⊂g⁎ carries the induced Poisson structure. We prove that there is a set {f1,...,fl}⊂F[X] of algebraically independent polynomial functions, which pairwise commute with respect to the Poisson bracket, such that l=(dim⁡X+tr.degF(X)G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varieties of corank zero and 2.