Invariant differential operators on spherical homogeneous spaces with overgroups

TL;DR

This paper studies the structure of invariant differential operators on spherical homogeneous spaces with overgroups, identifying generators, relations, and transfer maps connecting eigenvalues across different algebraic structures.

Contribution

It provides explicit generators, relations, and transfer maps for the ring of invariant differential operators on spherical spaces with overgroups, expanding understanding of their algebraic structure.

Findings

Most cases, ${ m D}_G(X)$ is generated by two subalgebras.

Explicit relations among generators are derived for various triples $( ilde{G}, G, H)$.

Transfer maps linking eigenvalues of ${ m D}_{ ilde{G}}(X)$ and the center of ${ m U}(rak g)$ are described.

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 ${\mathbb g}_{\mathbb C}$.