  We investigate the fine structure of the simplectic foliations of Poisson
homogeneous spaces. Two general results are proved for weak splittings of
surjective Poisson submersions from Heisenberg and Drinfeld doubles. The
implications of these results are that the torus orbits of symplectic leaves of
the quotients can be explicitly realized as Poisson-Dirac submanifolds of the
torus orbits of the doubles. The results have a wide range of applications to
many families of real and complex Poisson structures on flag varieties. Their
torus orbits of leaves recover important families of varieties such as the open
Richardson varieties.
