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

Abstract

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.