Concentration of symplectic volumes on Poisson homogeneous spaces

For a compact Poisson-Lie group $K$, the homogeneous space $K\mathrm{/}T$ carries a family of symplectic forms ${\omega }_{\xi }^s$, where $\xi \in {\mathfrak{t}}_{+}^{\ast }$ 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 $s\to -\mathrm{\infty }$, the symplectic volume of ${\omega }_{\xi }^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K\mathrm{/}T\cong G\mathrm{/}B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K{)}^{\ast }$ [4, Conjecture 1.1].