LIFTING DIFFERENTIAL-OPERATORS FROM ORBIT SPACES

Let X be an affine complex algebraic variety, and let D(X) denote the (non-commutative) algebra of algebraic differential operators on X. Then D(X) has a filtration {D-n(X)} by order of differentiation, and the associated graded gr D(X) is commutative. Now assume that X is smooth and a G-variety, where G is a reductive complex algebraic group. Let pi(X) : X --> X//G be the quotient morphism. Then we have a natural map (pi(x))* : (D-n(X))(G) --> D-n(X//G). We find conditions under which (pi(X))* is surjective for all n, in which case gr D(X//G) is finitely generated. We conjecture that the latter is always true. We also consider generalizations to algebras of differential operators on sections of G-vector bundles.