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Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization. (English) Zbl 1510.37092

Summary: 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 \(K = \operatorname{U}_n, \operatorname{SO}_n\). 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 \(\operatorname{Lie} ( K )^\ast\) 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 are able to complete the proof of a long-standing conjecture due to Y. Karshon and S. Tolman [Algebr. Geom. Topol. 5, 911–922 (2005; Zbl 1092.53062)] 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 \(K^\ast\) equipped with the Berenstein-Kazhdan potential \(\Phi \). Partial tropicalizations arise as tropical limits of \(K^\ast \), and the potential \(\Phi\) 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 \(K^\ast\) and Ginzburg-Weinstein Poisson isomorphisms \(\operatorname{Lie} ( K )^\ast \to K^\ast \).

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
37J06 General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants
17B08 Coadjoint orbits; nilpotent varieties
53D05 Symplectic manifolds (general theory)
53D17 Poisson manifolds; Poisson groupoids and algebroids
14T90 Applications of tropical geometry

Citations:

Zbl 1092.53062
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References:

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