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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)

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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