On the T-leaves of some Poisson structures related to products of flag varieties

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• 作者：Jiang-Hua Lu^a ; ^{jhlu@maths.hku.hk} ; Victor Mouquin^b ; ^{mouquinv@math.toronto.edu}
• 关键词：Poisson geometry ; Flag varieties ; T-leaves ; Generalized Bruhat cells
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：14 January 2017
• 年：2017
• 卷：306
• 期：Complete
• 页码：1209-1261
• 全文大小：905 K
• 卷排序：306

文摘

For a connected abelian Lie group TT acting on a Poisson manifold (Y,π)(Y,π) by Poisson isomorphisms, the TT-leaves of π in Y are the orbits of the symplectic leaves of π   under TT, and the leaf stabilizer of a TT-leaf is the subspace of the Lie algebra of TT that is everywhere tangent to all the symplectic leaves in the TT-leaf. In this paper, we first develop a general theory on TT-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces for a complex semi-simple Lie group G. We describe their T-leaf decompositions, where T is a maximal torus of G, in terms of (open) generalized Richardson varieties and generalized double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T-leaf.