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].

中文翻译：

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泊松齐次空间上辛体积的集中

对于紧泊松-李群 $K$，齐次空间 $K/T$ 携带一族辛形式 $\omega_\xi^s$，其中 $\xi \in \mathfrak{t}^*_+$是在正外尔室和 $s \in \mathbb{R}$。辛形式$\omega_\xi^0$ 与$\xi$ 对应的$K$ coadjoint 轨道上的自然$K$-不变辛形式相同。$\omega_\xi^s$ 的上同调类对于 $\xi$ 的固定值独立于 $s$。在本文中，我们证明 $s\to -\infty$，$\omega_\xi^s$ 的辛体积集中在 $K/T \cong G/B$ 中最小舒伯特单元的任意小邻域中. 这加强了早期的结果 [9,10] 并且是朝着在 $Lie(K)^*$ [4, Conjecture 1.1] 上推测构建全局动作角坐标迈出的一步。