Extended McKay correspondence for quotient surface singularities

Let $G$ be a finite subgroup of $GL(2)$ acting on ${\mathbf{A}}^2\setminus \{0\}$ freely. The $G$-orbit Hilbert scheme $G-Hilb({\mathbf{A}}^2)$ is a minimal resolution of the quotient ${\mathbf{A}}^2\mathrm{/}G$. We determine the generator sheaf of the ideal defining the universal $G$-cluster over $G-Hilb({\mathbf{A}}^2)$, which somewhat strengthens the well-known McKay correspondence for a finite subgroup of $SL(2)$. We also study the quiver structure of $G-Hilb({\mathbf{A}}^2)$ at every $G$-cluster $O_{Z_y}=O_{{\mathbf{A}}^2}\mathrm{/}{I}_y$ in terms of a collection of sort of minimal $G$-submodules of $O_{Z_y}$ (called mono-special $O_{{\mathbf{A}}^2}$-submodules) and generating $G$-submodules of $I_y$.