We investigate the structure of the ring DG(X) of G-invariant differential operators on a reductive spherical homogeneous space X=G/H with an overgroup \tilde{G}. We consider three natural subalgebras of DG(X) which are polynomial algebras with explicit generators, namely the subalgebra D\tilde{G}(X) of \tilde{G}-invariant differential operators on X and two other subalgebras coming from the centers of the enveloping algebras of \mathfrak{g} and \mathfrak{k}, where K is a maximal proper subgroup of G containing H. We show that in most cases DG(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 (\tilde{G},G,H), and describe transfer maps connecting eigenvalues for D\tilde{G}(X) and for the center Z(\mathfrak{g}C) of the enveloping algebra of \mathfrak{g}C.
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© Toshiyuki Kobayashi