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Concentration of symplectic volumes on Poisson homogeneous spaces
Journal of Symplectic Geometry  (IF0.707),  Pub Date : 2020-01-01, DOI: 10.4310/jsg.2020.v18.n5.a1
Anton Alekseev, Benjamin Hoffman, Jeremy Lane, Yanpeng Li

For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $\omega_\xi^s$, where $\xi \in \mathfrak{t}^*_+$ is in the positive Weyl chamber and $s \in \mathbb{R}$. The symplectic form $\omega_\xi^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $\xi$. The cohomology class of $\omega_\xi^s$ is independent of $s$ for a fixed value of $\xi$. In this paper, we show that as $s\to -\infty$, the symplectic volume of $\omega_\xi^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T \cong G/B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K)^*$ [4, Conjecture 1.1].