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)

MSC:

 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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