On the Commutativity of Weakly Commutative Riemannian Homogeneous Spaces

A Riemannian homogeneous space $X=G/H$ is said to be commutative if the algebra of $G$-invariant differential operators on $X$ is commutative and weakly commutative if the associated Poisson algebra is commutative. Clearly, the commutativity of $X$ implies its weak commutativity. The converse implication is proved in this paper.