Invariant differential operators on spherical homogeneous spaces with overgroups
Abstract
We investigate the structure of the ring ${\mathbb D}_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\widetilde{G}$. We consider three natural subalgebras of ${\mathbb D}_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra ${\mathbb D}_{\widetilde{G}}(X)$ of $\widetilde{G}$-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 ${\mathbb D}_G(X)$ 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 $(\widetilde{G},G,H)$, and describe "transfer maps" connecting eigenvalues for ${\mathbb D}_{\widetilde{G}}(X)$ and for the center $Z({\mathfrak g}_{\mathbb C})$ of the enveloping algebra of ${\mathbb g}_{\mathbb C}$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.02803
- arXiv:
- arXiv:1810.02803
- Bibcode:
- 2018arXiv181002803K
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Group Theory;
- 22E46;
- 16S30;
- 16S32;
- 17B10;
- 17B35
- E-Print:
- 90 pages. Corrected a few typos. Final form