Journal of Lie Theory

\newcommand{\D}{\mathbb{D}} \newcommand{\tilG}{\widetilde{G}} \newcommand{\g}{\mathfrak{g}} We investigate the structure of the ring $\D_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\tilG$. We consider three natural subalgebras of $\D_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra $\D_{\tilG}(X)$ of $\tilG$-invariant differential operators on $X$ and two other subalgebras coming from the centers of the enveloping algebras of $\g$ and $\mathfrak{k}$, where $K$ is a maximal proper subgroup of $G$ containing $H$. We show that in most cases $\D_G(X)$ is generated by any two of these three subalgebras, and analyze when this may fail. Moreover, we find explicit relations among the generators for each possible triple $(\tilG,G,H)$, and describe \emph{transfer maps} connecting eigenvalues for $\D_{\tilG}(X)$ and for the center of the enveloping algebra of $\g_{\mathbb{C}}$.