 
Summary: POISSON STRUCTURES ON AFFINE SPACES
AND FLAG VARIETIES. II. GENERAL CASE
K. R. Goodearl and M. Yakimov
Dedicated to the memory of our colleague XuDong Liu (19622005)
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 di#erent 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 PoissonDirac 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 Lorbits on G/P
constructed by Richardson, R˜ohrle, and Steinberg.
Introduction
0.1. First we fix some notation. Let G be a connected reductive algebraic group over C.
Fix a pair of dual Borel subgroups B ± in G, and set H = B +
