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Transactions of the American Mathematical Society

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Poisson structures on affine spaces and flag varieties. II

Authors: K. R. Goodearl and M. Yakimov
Journal: Trans. Amer. Math. Soc. 361 (2009), 5753-5780
MSC (2000): Primary 14M15; Secondary 53D17, 14L30, 17B20, 17B63, 53C35
Published electronically: June 19, 2009
MathSciNet review: 2529913
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Abstract: 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, Röhrle, and Steinberg.

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Additional Information

K. R. Goodearl
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

M. Yakimov
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

Received by editor(s): June 13, 2007
Published electronically: June 19, 2009
Additional Notes: The research of the first author was partially supported by National Science Foundation grant DMS-0401558.
The research of the second author was partially supported by National Science Foundation grant DMS-0406057 and an Alfred P. Sloan research fellowship.
Dedicated: Dedicated to the memory of our colleague Xu-Dong Liu (1962-2005)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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