Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert of $n$ points on Kleinian singularity $\mathbb{C}^2/\Gamma$ as Nakajima quiver variety for framed McKay $\Gamma$, taken at specific non-generic stability parameter. We deduce that this is irreducible (a result previously due to Zheng), normal, admits unique symplectic resolution. More generally, introduce class algebras obtained from preprojective algebra by process called cornering, show fine moduli spaces cyclic modules over these new are isomorphic varieties certain choices