# Concentration of symplectic volumes on Poisson homogeneous spaces

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Anton Alekseev, Benjamin Hoffman, Jeremy Lane, Yanpeng Li

Anton Alekseev, Benjamin Hoffman, Jeremy Lane, Yanpeng Li

**21/8/18**Published in : arXiv:1808.06975

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 ass\to -\infty, the symplectic volume of ωsξ 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].