Concentration of symplectic volumes on Poisson homogeneous spaces
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 …
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 …
For a compact Poisson-Lie group , the homogeneous space carries a family of symplectic forms , where is in the positive Weyl chamber and . The symplectic form is identified with the natural -invariant symplectic form on the coadjoint orbit corresponding to . The cohomology class of is independent of for a fixed value of . In this paper, we show that as , the symplectic volume of concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in . This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on [4, Conjecture 1.1].
arxiv.org
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