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)


14E15 Global theory and resolution of singularities (algebro-geometric aspects)
20G15 Linear algebraic groups over arbitrary fields
14M17 Homogeneous spaces and generalizations
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