Rings of differential operators on invariant rings of tori

Abstract:

Let $k$ be an algebraically closed field of characteristic zero and $G$ a torus acting diagonally on ${k^s}$. For a subset $\beta$ of ${\mathbf {s}} = \{ 1, 2, \ldots , s\}$, set ${U_\beta } = \{ u \in {k^s}|{u_j} \ne 0\;{\text {if}}\;j \in \beta \}$. Then $G$ acts on $\mathcal {O}({U_\beta })$, the ring of regular functions on ${U_\beta }$, and we study the ring $D(\mathcal {O}{({U_\beta })^G})$ of all differential operators on the invariant ring. More generally suppose that $\Delta$ is a set of subsets of s, such that each invariant ring $\mathcal {O}{({U_\beta })^G}$, $\beta \in \Delta$, has the same quotient field. We prove that ${ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})$ is Noetherian and finitely generated as a $k$-algebra. Now $G$ acts on each $D(\mathcal {O}({U_\beta }))$ and there is a natural map \[ \theta :\bigcap \limits _{\beta \in \Delta } {D{{(\mathcal {O}({U_\beta }))}^G} \to \bigcap \limits _{\beta \in \Delta } {D(\mathcal {O}{{({U_\beta })}^G}) = D({Y_\Delta } / G)} } \] obtained by restriction of the differential operators. We find necessary and sufficient conditions for $\theta$ to be surjective and describe the kernel of $\theta$. The algebras ${ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})$ carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of ${ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})$.