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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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