Binomial $D$-modules

Abstract

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary $\mathbb{Z}^d$-graded binomial ideal $I$ in $\mathbb{C}[\partial_1,\ldots,\partial_n]$ along with Euler operators defined by the grading and a parameter $\beta \in \mathbb{C}^d$. We determine the parameters $\beta$ for which these $D$-modules (i) are holonomic (equivalently, regular holonomic, when $I$ is standard-graded), (ii) decompose as direct sums indexed by the primary components of $I$, and (iii) have holonomic rank greater than the rank for generic $\beta$. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in $\mathbb{C}^d$. In the special case of Horn hypergeometric $D$-modules, when $I$ is a lattice-basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated $A$-hypergeometric functions. This study relies fundamentally on the explicit lattice-point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article [DMM]