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Lifting differential operators from orbit spaces. (English) Zbl 0836.14032
Let \(X\) an affine complex algebraic variety, and let \(\Delta (X)\) denote the (non-commutative) algebra of algebraic differential operators on \(X\). Then \(\Delta (X)\) has a filtration \(\{\Delta^n (X)\}\) by order of differentiation, and the associated graded \(\text{gr} \Delta (X)\) is commutative. Let \(X\) smooth and a \(G\)-variety \((G\) is a reductive complex algebraic group). Let \(\pi_X : X \to X/G\) the quotient morphism. Then one has a natural map \((\pi_X)_* : (\Delta^n (X))^G \to \Delta^n (X/G)\). The author studies conditions under which \((\pi_X)_*\) is surjective for all \(n\), and in which case \(\text{gr} \Delta (X/G)\) is finitely generated.
Reviewer: R.Salvi (Milano)

MSC:
14M17 Homogeneous spaces and generalizations
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
16W25 Derivations, actions of Lie algebras
14L30 Group actions on varieties or schemes (quotients)
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