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

Abstract

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