McKay Correspondence over Non Algebraically Closed Fields

Abstract

The classical McKay correspondence for finite subgroups G of $\mathrm{SL}(2, \mathbb{C})$ gives a bijection between isomorphism classes of nontrivial irreducible representations of G and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity $\mathbb{A}_{\mathbb{C}}^{2}/G$. Over non algebraically closed fields K there may exist representations irreducible over K which split over $\overline{K}$. The same is true for irreducible components of the exceptional divisor. In this paper we show that these two phenomena are related and that there is a bijection between nontrivial irreducible representations and irreducible components of the exceptional divisor over non algebraically closed fields K of characteristic 0 as well.