  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, R\"ohrle, and Steinberg.
