The most general version of the question I want to ask is:
Let $R$ be a regular (commutative, Noetherian) ring containing a field $k$ of characteristic $0$, let $D = D(R,k)$ be the ring of $k$-linear differential operators on $R$, let $M$ be a (left) $D$-module, and let $E^{\bullet}(M)$ be the minimal injective resolution of $M$ as an $R$-module. Is $E^{\bullet}(M)$ a complex of $D$-modules?
More specifically, I would be happy with special cases:
Is the answer to the previous question positive if $R = k[x_1, \ldots, x_n]$ or $k[[x_1, \ldots, x_n]]$? What if we further require $M$ to be a holonomic (or finite length) $D$-module?
It is enough to show that if $M \subset E(M)$ is a chosen injective hull of $M$ as an $R$-module, then $E(M)$ can be given a structure of $D$-module in such a way that the inclusion is $D$-linear. Since $R$ is Gorenstein, each indecomposable summand $E(R/\mathfrak{p})$ of $E(M)$ is isomorphic as an $R_{\mathfrak{p}}$-module to the local cohomology $H^h_{\mathfrak{p} R_{\mathfrak{p}}}(R_{\mathfrak{p}})$, a localization of the $D$-module $H^h_{\mathfrak{p}}(R)$ (here $h$ is the height of $\mathfrak{p}$), so there is no trouble in abstractly giving $E(M)$ a $D$-module structure-- but this says nothing about the inclusion of the submodule $M$.
For those familiar with Lyubeznik's theory of "$F$-modules" in positive characteristic, the analogous statement is true for an arbitrary $F$-module over any regular positive-characteristic (Noetherian) ring; this is what motivated the question.
Here is what I know so far.
If $M = R$ is the ring itself, then $E^{\bullet}(R)$ is isomorphic to the Cousin complex of $R$, by a theorem of Sharp (true for any Gorenstein ring). It is easy from the explicit description of the Cousin complex, which is built from localization maps, to see that this complex is a complex of $D$-modules.
If the only associated primes of $M$ as an $R$-module are minimal primes, then $E(M) = \oplus_{\mathfrak{p}} M_{\mathfrak{p}}$ is an injective hull of $M$ (here $\mathfrak{p}$ runs over the set of associated primes) and the natural map $M \rightarrow \oplus_{\mathfrak{p}} M_{\mathfrak{p}}$ is an injection. In this case the map is a map of $D$-modules. However, since the injective hull of $E(M)/M$ may have embedded primes, we can't continue the inductive argument.
Finally, if the modules $P^j_{R/k}$ of principal parts are flat over $R$ (true in the polynomial and power series cases), then we can use the correspondence (from EGA IV 16.8) between differential operators $M \rightarrow M$ of order at most $j$ and $R$-linear maps $P^j_{R/k} \otimes_R M \rightarrow M$ to prove that any $k$-linear differential operator on $M$ lifts to a $k$-linear differential operator of the same order on $E(M)$. However, I see no reason to expect this lifting to be unique, or that it can be done for all differential operators in $D$ at once in a manner that preserves the necessary relations.
Edit (5/5/2018): Here is a slightly stronger form of the question. Let $J = E_D(M)$ be the injective hull of $M$ as a left $D$-module. Since $D$ is $R$-projective, $J$ is $R$-injective. Let $E$ be the maximal essential extension of $M$ inside $J$. As an $R$-module, $E$ is isomorphic to $E_R(M)$, since the ambient $R$-module $J$ is injective. Is $E$ a $D$-submodule of $J$? In the polynomial or power series case, this is the same as asking: do the partial derivatives $\partial_1, \ldots, \partial_n$, which act on $J$, stabilize $E$? I would be interested to know the answer even for $n=1$.
