  In this article, we study the decomposition into irreducible components of
the fixed point locus under the action of $\Gamma$ a finite subgroup of
$\mathrm{SL}_2(\mathbb{C})$ of the smooth Nakajima quiver variety of the Jordan
quiver. The quiver variety associated with the Jordan quiver is either
isomorphic to the punctual Hilbert scheme in $\mathbb{C}^2$ or to the
Calogero-Moser space. We describe the irreducible components using quiver
varieties of McKay's quiver associated with the finite subgroup $\Gamma$ and we
have given a general combinatorial model of the indexing set of these
irreducible components in terms of certain elements of the root lattice of the
affine Lie algebra associated with $\Gamma$. Finally, we prove that for every
projective, symplectic resolution of a wreath product singularity, there exists
an irreducible component of the fixed point locus of the punctual Hilbert
scheme in the plane that is isomorphic to the resolution.
