Concentration of symplectic volumes on Poisson homogeneous spaces

Abstract

For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $ω_ξ^s$, where $ξ\in \mathfrak{t}^*_+$ is in the positive Weyl chamber and $s \in \mathbb{R}$. The symplectic form $ω_ξ^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $ξ$. The cohomology class of $ω_ξ^s$ is independent of $s$ for a fixed value of $ξ$. In this paper, we show that as $s\to -\infty$, the symplectic volume of $ω_ξ^s$ 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].