Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?
Personally, I am interested in the following questions: if $M$ is an indecomposable injective $A$-module, is there a prime ideal $q\subset \mathbb{C}[x_1,\dots,x_n]=R$ and a left injective module $N$ over $A\otimes_RR_q$ such that $N=M$ as $A$-modules? Is the support of the $R$-module $M$ an irreducible closed subset of $\mathrm{Spec}(R)=A^n_{\mathbb{C}}$?
From Thm. 2.4 of "Injective modules over Noetherian rings" by Matlis, we know that every indecomposable injective module over $A$ is the injective hull of $A/J$, where $J$ is a left irreducible ideal. But I don't know a classification of left irreducible ideals, neither.
