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 ${\omega }_{\xi }^s$, 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 $s\to -\infty$, 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 $Lie(K{)}^{\ast }$ [4, Conjecture 1.1].

arXiv comment15 pages, 3 figures

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