Geometric Construction of the McKay-Slodowy Correspondence

Abstract

Let $G\subseteq \operatorname{SL}_2(\mathbb{C})$ be a finite group and let $H\trianglelefteq G$ be a normal subgroup. The McKay correspondence for $H$ says there is a one-to-one correspondence between nontrivial irreducible representations of $H$ and irreducible components of the exceptional locus of the minimal resolution of $\mathbb{C}^2/H$. We prove that this correspondence for $H$ is $G$-equivariant, and it induces a one-to-one correspondence between induced representations and push-forwards of exceptional curves. We also identify the intersection pairing of curves with the inner product of representations under this correspondence.