Structure of the Center of the Algebra of Invariant Differential Operators on Certain Riemannian Homogeneous Spaces

We study Duflo's conjecture on the isomorphism between the center of the algebra of invariant differential operators on a homogeneous space and the center of the associated Poisson algebra. For a rather wide class of Riemannian homogeneous spaces, which includes the class of (weakly) commutative spaces, we prove the "weakened version" of this conjecture. Namely, we prove that some localizations of the corresponding centers are isomorphic. For Riemannian homogeneous spaces of the form X = (H ⋌ N)/H, where N is a Heisenberg group, we prove Duflo's conjecture in its original form, i.e., without any localization.