Classical dynamical r-matrices and homogeneous Poisson structures on $G/H$ and $K/T$

Abstract

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.