
Journal of Lie Theory 29 (2019), No. 3, 663754 Copyright Heldermann Verlag 2019 Invariant Differential Operators on Spherical Homogeneous Spaces with Overgroups Fanny Kassel CNRS and Institut des Hautes Etudes Scientifiques, Lab. Alexander Grothendieck, 91440 BuressurYvette, France kassel@ihes.fr Toshiyuki Kobayashi Grad. School of Math. Sciences, and: Kavli Inst. for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Tokyo 1538914, Japan toshi@ms.utokyo.ac.jp [Abstractpdf] \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 $\g_{\mathbb{C}}$. Keywords: Branching law, spherical variety, real spherical variety, symmetric space, invariant differential operator, enveloping algebra. MSC: 22E46; 17B10, 16S30, 16S32, 17B35 [ Fulltextpdf (541 KB)] for subscribers only. 