Action-angle coordinates are a type of coordinate chart on symplectic manifolds originating from the theory of commutative completely integrable systems and the Liouville-Arnold theorem. For example, every toric manifold has global action-angle coordinates: the pre-image under the moment map of the interior of the Delzant polytope, $\mathring{\Delta}$, is symplectically isomorphic to the product $\mathring{\Delta}\times T$, equipped with the canonical symplectic structure.

Multiplicity-free spaces are the natural non-abelian generalization of toric manifolds. A multiplicity-free space is a symplectic manifold equipped with a Hamiltonian action of a compact connected Lie group that is ``completely integrable'' in the appropriate non-commutative sense. For example, every coadjoint orbit is a multiplicity-free space. Unlike toric manifolds, multiplicity-free spaces do not come readily equipped with nice global action-angle coordinates.

In this talk I will present recent work in collaboration with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li that gives a new construction of action-angle coordinates on a large family of multiplicity-free spaces.