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 $${\\mathfrak {g}}$$ g , we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of $${\\mathscr {S}}({\\mathfrak {g}})$$ S ( 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 $${\\mathfrak {g}}\\simeq {\\mathfrak {g}}^{*}$$ g ≃ g ∗ .