The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of SL₂ (C)

In this article, we study the decomposition into irreducible components of the fixed point locus under the action of a finite subgroup of SL₂ (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 C² or to the Calogero-Moser space. We describe the irreducible components using quiver varieties of McKay's quiver associated with the finite subgroup 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. 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.