Concentration of symplectic volumes on Poisson homogeneous spaces

TL;DR

This paper demonstrates that as a parameter tends to negative infinity, the symplectic volume on certain homogeneous spaces concentrates near the smallest Schubert cell, advancing understanding of symplectic geometry and action-angle coordinates.

Contribution

It proves volume concentration on the smallest Schubert cell in Poisson homogeneous spaces as a parameter approaches negative infinity, extending previous results.

Findings

Symplectic volume concentrates near the smallest Schubert cell as s→ -∞.

Strengthens earlier results on symplectic forms on Poisson-Lie homogeneous spaces.

Progresses towards constructing global action-angle coordinates on Lie(K)*.

Abstract

For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $\omega_\xi^s$, where $\xi \in \mathfrak{t}^*_+$ 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].