Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

Abstract

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra \({\mathfrak {g}}\), we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of \({\mathscr {S}}({\mathfrak {g}})\) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in \({\mathfrak {g}}\simeq {\mathfrak {g}}^{*}\).