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.
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Acknowledgements
The suggestion to investigate McKay correspondence over non algebraically closed fields is due to Victor Batyrev.
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Blume, M. (2014). McKay Correspondence over Non Algebraically Closed Fields. In: Frühbis-Krüger, A., Kloosterman, R., Schütt, M. (eds) Algebraic and Complex Geometry. Springer Proceedings in Mathematics & Statistics, vol 71. Springer, Cham. https://doi.org/10.1007/978-3-319-05404-9_3
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