Concentration of symplectic volumes on Poisson homogeneous spaces

For a compact Poisson–Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $\omega^s_\xi$, where $\xi \in \mathfrak{t}^{\ast}_{+}$ is in the positive Weyl chamber and $s \in \mathbb{R}$. The symplectic form $\omega^0_\xi$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $\xi$. The cohomology class of $\omega^s_\xi$ is independent of $s$ for a fixed value of $\xi$.

In this paper, we show that as $s \to -\infty$, the symplectic volume of $\omega^s_\xi$ concentrates in arbitrarily small neighborhoods of the smallest Schubert cell in $K/T \cong G/B$. This strengthens an earlier result of [10] and is a step towards a conjectured construction of global action-angle coordinates on $\operatorname{Lie}(K)^\ast$ [4, Conjecture 1.1].

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Received 13 September 2018

Accepted 20 November 2019

Published 29 October 2020