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Concentration of symplectic volumes on Poisson homogeneous spaces

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 نشر من قبل Benjamin Hoffman
 تاريخ النشر 2018
  مجال البحث
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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 $sto -infty$, 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].



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