Differential operators on quotients of simple groups.

*(English)*Zbl 0835.14019Let \(X\) 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.

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) |