Poisson structures on affine spaces and flag varieties. II. General case
Poisson structures on affine spaces and flag varieties. II. General case
复制标题
DOI：
10.1090/s0002-9947-09-04654-6
复制
发表时间：
2005-09
期刊：
影响因子：
--
通讯作者：
K. Goodearl;M. Yakimov
中科院分区：
文献类型：
--
作者：
K. Goodearl;M. Yakimov
研究了复约化代数群G的旗簇G/P上的标准Poisson结构。证明了G/P中的辛叶在G的一个固定的极大环面下的轨道是G/P的光滑不可约局部闭子簇，同构于G的满旗簇G/B中的对偶Schubert胞腔的交，并明确地计算了它们的Zapriki闭包.给出了前一结果的两种不同的证明。第一个是在泊松齐性空间的框架和第二个使用的满射泊松淹没弱分裂的想法，基于泊松-狄拉克子流形的概念。对具有交换幂幺根的抛物子群P（其中G/P是紧致型Hermite对称空间），证明了P的标准Levi因子L在G/P上的所有轨道都是完备Poisson子簇，它们是L的直积，具有标准Poisson结构.证明了G/P上的Poisson结构在Richardson，Rohrle和Steinberg构造的G/P上的L-轨道的所有特殊基点处为零.
The standard Poisson structures on the flag varieties G/P of a complex reductive algebraic group G are investigated. It is shown that the orbits of symplectic leaves in G/P under a fixed maximal torus of G are smooth irreducible locally closed subvarieties of G/P, isomorphic to intersections of dual Schubert cells in the full flag variety G/B of G, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson-Dirac submanifolds. For a parabolic subgroup P with abelian unipotent radical (in which case G/P is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor L of P on G/P are complete Poisson subvarieties which are quotients of L, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on G/P vanishes at all special base points for the L-orbits on G/P constructed by Richardson, Rohrle, and Steinberg.