$\mathcal{D}$-modules with finite support are semi-simple

Abstract

Let $(R, \frak{m}, k_{R})$ be a regular local k-algebra satisfying the weak Jacobian criterion, and such that k_R/k is an algebraic field extension. Let $\mathcal{D}_{R}$ be the ring of k-linear differential operators of R. We give an explicit decomposition of the $\mathcal{D}_{R}$-module $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}$ as a direct sum of simple modules, all isomorphic to $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}$, where certain “Pochhammer” differential operators are used to describe generators of the simple components.