Journal of Symplectic Geometry

Volume 18 (2020)

Number 5

Concentration of symplectic volumes on Poisson homogeneous spaces

Pages: 1197 – 1220

DOI: https://dx.doi.org/10.4310/JSG.2020.v18.n5.a1

Authors

Anton Alekseev (Section of Mathematics, University of Geneva, Switzerland)

Benjamin Hoffman (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Jeremy Lane (Section of Mathematics, University of Geneva, Switzerland)

Yanpeng Li (Section of Mathematics, University of Geneva, Switzerland)

Abstract

For a compact Poisson–Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $\omega^s_\xi$, where $\xi \in \mathfrak{t}^{\ast}_{+}$ is in the positive Weyl chamber and $s \in \mathbb{R}$. The symplectic form $\omega^0_\xi$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $\xi$. The cohomology class of $\omega^s_\xi$ is independent of $s$ for a fixed value of $\xi$.

In this paper, we show that as $s \to -\infty$, the symplectic volume of $\omega^s_\xi$ concentrates in arbitrarily small neighborhoods of the smallest Schubert cell in $K/T \cong G/B$. This strengthens an earlier result of [10] and is a step towards a conjectured construction of global action-angle coordinates on $\operatorname{Lie}(K)^\ast$ [4, Conjecture 1.1].

Received 13 September 2018

Accepted 20 November 2019