Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup Γ ⊂ SL(2,C) and n ≥ 1, we construct the (reduced scheme under-lying the) Hilbert scheme of n points on the Kleinian singularity C2/Γ as a Nakajimaquiver variety for the framed McKay quiver of Γ, taken at a specific non-generic stabil-ity parameter. We deduce that this Hilbert scheme is irreducible (a result previouslydue to Zheng), normal and admits a unique symplectic resolution. More generally, weintroduce a class of algebras obtained from the preprojective algebra of the framedMcKay quiver by removing an arrow and then 'cornering', and we show that fine mod-uli spaces of cyclic modules over these new algebras are isomorphic to quiver varietiesfor the framed McKay quiver and certain non-generic choices of the stability parameter.