Primitive ideals of the ring of differential operators on an affine toric variety

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Let A be a d× n integer matrix whose column vectors generate the lattice ᵈ, and let D(Rₐ) be the ring of differential operators on the affine toric variety defined by A. We show that the classification of A-hypergeometric systems and that of ᵈ-graded simple D(Rₐ)-modules (up to shift) are the same. We then show that the set of ᵈ-homogeneous primitive ideals of D(Rₐ) is finite. Furthermore, we give conditions for the algebra D(Rₐ) being simple.