Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$ is the preprojective algebra of $Q$.

In Geiss, Leclerc, and Schröer's paper, they gave a geometric realization of irreducible $\lambda$ highest weight module by considering constructible function on the variety of nilpotent $\Lambda$-module which are isormorphic to submodule of $q_\lambda$. Here $q_\lambda$ is a direct sum of some $q_i$, where $q_i$ is the injective hull of $s_i$.

I want to calculate some small examples which are mainly about ADE type quiver. Are there any reference finding the injective hull $q_i$ of simple module $s_i$? Thank you so much.

EDIT: I have found a reference. In Alistair Savage, Peter Tingley's paper:Quiver grassmannians, quiver varieties and the preprojective algebra definition 2.7. Assume $Q$ is a quiver of finite type. For any vertex $i$, let $$q^i=\text{Hom}_{\mathbb{C}}(e_i\mathcal{P},\mathbb{C}).$$The left $\mathcal{P}$-module structure on $q^i$ is by setting $a\cdot f(x)=f(xa)$.

EDIT 2: I have found a way to draw $q^i$ in type A. The next thing I want to know is about indecomposable module. For ADE type quiver, what is the injective hull of an indecomposable module $M$ over the preprojective algebra $\mathcal{P}$? Is its injective hull equal to its socle?