Injective modules over some rings of Differential operators

Abstract

Let $R$ be a regular domain containing a field $K$ of characteristic zero and let $D$ be the ring of $K$-linear differential operators on $R$. Let $E$ be an injective left $D$-module. We ask the question, when is $E$ injective as a $R$-module? We show that this is indeed the case when $R = K[X_1,...,X_n]$ or $R = K[[X_1,...,X_n]]$ or $R = \mathbb{C}\{z_1,...,z_n\}$. We also give an application of our result to local cohomology.