Finite fans, actions of tori and D-modules

Let G be a finite dimensional torus acting diagonally on the smooth affine variety X = k^r x (k^x)^s, with k an algebraically closed field k of characteristic 0. We denote the ring of regular functions on X by O(X) and the ring of differential operators by D(X). Let D(X)^G be the subring of D(X) of invariants under the action of G.The goal of this poster is to show how finite fans of cones can be used to study D(X)^G-modules. We associate a finite fan of cones to the action of G on X, in such a way that the study of the fan will allow us to get conclusions about the finite dimensional D(X)^G-modules. We describe next the basic ingredients of our construction.