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In a previous paper, a realization of the moduli space of framed torsion-freesheaves on Hirzebruch surfaces in terms of monads was given. We build upon thatresult to construct ADHM data for the Hilbert scheme of points of the totalspace 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 byimplementing a version of the special McKay correspondence, this ADHMdescription is in turn used to realize these Hilbert schemes as irreducibleconnected components of quiver varieties. We obtain in this way new examples ofquiver varieties which are not of the Nakajima type.

Hilbert schemes of points of $\mathcal O_{\mathbb P^1}(-n)$ as quiver varieties