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

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Anton Alekseev, Benjamin Hoffman, Jeremy Lane, Yanpeng Li

**30/3/20**Published in : arXiv:2003.13621

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 built using techniques of Poisson-Lie theory and the geometric crystals of Berenstein-Kazhdan. 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^\mathbb{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 arrive at a conjectured formula for Gromov width of regular coadjoint orbits. We prove similar results for multiplicity free K-spaces.