Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization.

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K = U n , SO n. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie (K) ⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G = K C. We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K , we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K -spaces. An essential tool in our study is the dual Poisson-Lie group K ⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K ⁎ , and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K ⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie (K) ⁎ → K ⁎. [ABSTRACT FROM AUTHOR]