  In a previous paper, a realization of the moduli space of framed torsion-free
sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that
result to construct ADHM data for the Hilbert scheme of points of the total
space of the line bundles $\mathcal O(-n)$ on $\mathbb P^1$, for $n \ge 1$,
i.e., the resolutions of the singularities of type $\frac1n(1,1)$. Basically by
implementing a version of the special McKay correspondence, this ADHM
description is in turn used to realize these Hilbert schemes as irreducible
connected components of quiver varieties. We obtain in this way new examples of
quiver varieties which are not of the Nakajima type.
