INVARIANT DIFFERENTIAL OPERATORS ON SPHERICAL HOMOGENEOUS SPACES WITH OVERGROUPS

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 (G) over tilde. We consider three natural subalgebras of D-G (X) which are polynomial algebras with explicit generators, namely the subalgebra D-(G) over tilde(X) of (G) over tilde -invariant differential operators on X and two other subalgebras coming from the centers of the enveloping algebras of g and t, 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 ((G) over tilde, G, H), and describe transfer maps connecting eigenvalues for D-(G) over tilde(X) and for the center Z(g(C)) of the enveloping algebra of g(C).