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Journal of Lie Theory 29 (2019), No. 3, 663--754
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 Bures-sur-Yvette, France

Toshiyuki Kobayashi
Grad. School of Math. Sciences, and: Kavli Inst. for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Tokyo 153-8914, Japan


\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

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