Reid, Miles McKay’s correspondence. (La correspondance de McKay.) (English) Zbl 0996.14006 Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque. 276, 53-72, Exp. No. 867 (2002). For \(G\subset \text{SL}(2,{\mathbb C})\) a finite group, the quotient variety \(X={\mathbb C}^2/G\) is called a Klein quotient singularity. The resolution of singularities \(Y\rightarrow X\) has exceptional locus consisting of \(-2\)-curves \(E_i\) (i.e. isomorphic to \({\mathbb P}_{{\mathbb C}}^1\), with self-intersection \(E_i^2=-2\)), and whose intersections \(E_iE_j\) are given by one of the Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\) or \(E_8\). The classical McKay correspondence begins in the late 1970s with the observation that the same graph arises in connection with the representation theory of \(G\), i.e. there is a one-to-one correspondence between the components of the exceptional locus of \(Y\rightarrow X\) and the nontrivial irreducible representations of \(G\subset \text{SL}(2,{\mathbb C})\). The paper explains this coincidence in several ways, and discusses higher dimensional generalizations.For the entire collection see [Zbl 0981.00011]. Reviewer: V.P.Kostov (Nice) Cited in 3 ReviewsCited in 35 Documents MSC: 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 20G15 Linear algebraic groups over arbitrary fields 14M17 Homogeneous spaces and generalizations Keywords:group action; \(K\)-theory; derived category; quotient variety; resolution of singularity; motivic integration; McKay correspondence; Hilbert schemes of \(G\)-orbits; crepant resolution; discrepancy divisor; Klein quotient singularity PDFBibTeX XMLCite \textit{M. Reid}, in: Séminaire Bourbaki. Volume 1999/2000. Exposés 865--879. Paris: Société Mathématique de France. 53--72, Exp. No. 867 (2002; Zbl 0996.14006) Full Text: arXiv Numdam EuDML