  Let G be a finite dimensional simple complex group equipped with the standard
Poisson Lie group structure. We show that all G-homogeneous (holomorphic)
Poisson structures on $G/H$, where $H \subset G$ is a Cartan subgroup, come
from solutions to the Classical Dynamical Yang-Baxter equations which are
classified by Etingof and Varchenko. A similar result holds for the maximal
compact subgroup K, and we get a family of K-homogeneous Poisson structures on
$K/T$, where $T = K \cap H$ is a maximal torus of K. This family exhausts all
K-homogeneous Poisson structures on $K/T$ up to isomorphisms. We study some
Poisson geometrical properties of members of this family such as their
symplectic leaves, their modular classes, and the moment maps for the T-action.
