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

Abstract

Let $A$ be a $d\times n$ integer matrix whose column vectors generate the lattice ${\mathbb{Z}}^d$, and let $D(R_A)$ 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 ${\mathbb{Z}}^d$-graded simple $D(R_A)$-modules (up to shift) are the same. We then show that the set of ${\mathbb{Z}}^d$-homogeneous primitive ideals of $D(R_A)$ is finite. Furthermore, we give conditions for the algebra $D(R_A)$ being simple.

29 pages