The construction of the homogeneous coordinate ring of a toric variety, that I.M. Musson, D.A.Cox and others discovered in the early 1990s, takes a fan Δ and creates a torus action on an open subset of an affine space whose quotient is the toric variety of Δ. We reverse this process. Let k be an algebraically closed field of characteristic 0. Let H be an algebraic torus times a finite abelian group acting diagonally on the affine spaceX=kr × (k×)s. We create various fans whose toric varieties are the quotients under the action of an open subset of X. Let S be the Laurent polynomial ring k[x1, ... , xr, xr+1± 1, ... , xn± 1],with n=r+s. Then S is multi-graded by the finitely generated group A=Hom(H,k×). We prove the following statements to be equivalent: the fan is not contained in a half-space; SH=k; every graded component Sa, a ∈ A is finite dimensional. When Sa is finite dimensional we can give a bounded polyhedron (polytope) whose number of lattice points equals the dimension of Sa. Let D(X) be the ring of differential operators on S and let D(X)H be the subring of invariants under the action of H. We give results on the existence of finite dimensional representations of D(X)H and we use the graded components of S to give families of finite dimensional D(X)H-modules with "enough members".