Finite dimensional representations of invariant differential operators

Let be an algebraically closed field of characteristic , , and let be an algebraic torus acting diagonally on the ring of algebraic differential operators . We give necessary and sufficient conditions for to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if is a representation of a reductive group and if zero is not a weight of a maximal torus of on , then has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension .