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Rings of differential operators on invariant rings of tori. (English) Zbl 0628.13019
Let k be an algebraically closed field with $$char(k)=0$$ and G a torus acting diagonally on $$k^ S$$. For a subset $$\beta$$ of $$\bar S=\{1,2,...,s\}$$, set $$U_{\beta}=\{u\in k^ S| \quad u_ j\neq 0\;if\;j\in \beta \}.$$ Then G acts on the ring $$O_{\beta}$$ of regular functions on $$U_{\beta}$$, and the author studies the ring $${\mathcal D}(O^ G_{\beta})$$ of all differential operators on the invariant ring. More generally, suppose that $$\Delta$$ is a set of subsets of $$\bar S,$$ such that each invariant ring $$O^ G_{\beta}$$ ($$\beta\in \Delta)$$ has the same quotient field. Then $$\cap_{\beta \in \Delta}D(O^ G_{\beta})\quad is$$ Noetherian and finitely generated as an R-algebra. Now G acts on each $$D(O_{\beta})$$ and there is a natural map $$\theta$$ :$$\cap_{\beta \in \Delta}D(O_{\beta})^ G\to \cap_{\beta \in \Delta}D(O^ G_{\beta})=D(Y_{\Delta}/G)\quad obtained$$ by restriction of the differential operators. The author finds necessary and sufficient conditions for $$\theta$$ to be surjective and describes the kernel of $$\theta$$. The algebras $$\cap_{\beta \in \Delta}D(O_{\beta})^ G\quad and\cap_{\beta \in \Delta}D(O^ G_{\Gamma})\quad carry$$ a natural filtration given by the order of the differential operators. The author shows that the associated graded rings are finitely generated commutative algebras and determines the centers of $$\cap_{\beta \in \Delta}D(O_{\beta})^ G\quad and\cap_{\beta \in \Delta}D(O^ G_{\beta})$$.
Reviewer: S.V.Mihovski

##### MSC:
 13N05 Modules of differentials 14L24 Geometric invariant theory 16W20 Automorphisms and endomorphisms 13B10 Morphisms of commutative rings
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