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Differential operators on quotients of simple groups. (English) Zbl 0835.14019
Let $$X$$ be 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 algebra $$\text{gr} \Delta (X)$$ is commutative. Let $$X$$ be smooth and a $$G$$-variety $$(G$$ is a reductive complex algebraic group). Let $$\pi_X : X \to X/G$$ be the quotient morphism. Then one has a natural map $$(\pi_X)_* : (\Delta^n (X))^G \to \Delta^n (X/G)$$. The main aim of the paper is the following:
Theorem. Let $$G$$ be simple and connected, and let $$V$$ be an irreducible nontrivial $$G$$-module such that $$\dim V/G > 0$$. Then the following are equivalent
(1) $$V$$ is 2-modular;
(2) $$V$$ is 2-large;
(3) $$(\pi_V)_*$$ is graded surjective;
(4) $$(\pi_V)_*$$ is surjective;
(5) $$V/G$$ has no codimension one strata;
(6) $$V$$ is not coregular.
Reviewer: R.Salvi (Milano)

##### MSC:
 14M17 Homogeneous spaces and generalizations 16S32 Rings of differential operators (associative algebraic aspects) 13N10 Commutative rings of differential operators and their modules 14L30 Group actions on varieties or schemes (quotients)
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