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 space

X=k^r × (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, ... , x_r, x_r+1^{± 1}, ... , x_n^{± 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; S^H=k; every graded component S_a, a ∈ A is finite dimensional. When S_a is finite dimensional we can give a bounded polyhedron (polytope) whose number of lattice points equals the dimension of S_a. 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".