## F. Kassel and T. Kobayashi, *Invariant differential operators on spherical
homogeneous spaces with overgroups*, Journal of Lie Theory **29**
(2019), 663-754, arXiv: 1810.02803.

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 \tilde{*G*}.
We consider three natural subalgebras of **D**_{G(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 **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 (\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