The Procesi bundle over the $\Gamma$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$

For $\Gamma$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibers of the Procesi bundle over the $\Gamma$-fixed points of the Hilbert scheme of $n$ points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibers of the Procesi bundle, as an $(\mathfrak{S}_n \times \Gamma)$-module, to the study of the fibers of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When $\Gamma$ is of type $A$, our main result shows, as a corollary, that the fiber of the Procesi bundle over the monomial ideal associated with a partition $\lambda$ is induced, as an $(\mathfrak{S}_n \times \Gamma)$-module, from the fiber of the Procesi bundle over the monomial ideal associated with the core of $\lambda$. We give different proofs of this corollary in two edge cases, using only representation theory and symmetric functions.