McKay Correspondence over Non Algebraically Closed Fields

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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.