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Unformatted text preview: MCKAY’S CORRESPONDENCE FOR COCOMPACT DISCRETE SUBGROUPS OF SU (1 , 1) IGOR V. DOLGACHEV To John McKay Abstract. The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup Π of SU(2) and the geometry of the minimal resolution of the affine surface V = C 2 / Π. In this paper we discuss a possible generalization of the McKay correspondence to the case when Π is replaced with a discrete cocompact subgroup of the universal cover of SU(1 , 1) such that its image Γ in PSU(1 , 1) is a fuchsian group of signature (0 , e 1 , . . . , e n ). We establish a correspondence between a certain class of finite-dimensional unitary representations of Π and vector bundles on an algebraic surface with trivial canonical class canonically associated to Γ. 1. Introduction It has been known since the work of P. Du Val in the thirties that Coxeter- Dynkin diagrams of type ADE are in bijective correspondence with the conjugacy classes of finite subgroups Π of SU(2) in such a way that the intersection graph of a minimal resolution of C 2 / Π is the diagram corre- sponding to the group Π. In the early eighties John McKay added more to this mysterious connection by introducing a certain graph attached to any finite group. When the group is equal to a binary polyhedral group Π the graph coincides with the affine extension of the Dynkin diagram at- tached to Π ,. The vertices of the graph correspond to irreducible representations of Π where the extended vertex corresponds to the trivial representation. The first geometric explanations of the McKay correspon- dence were given independently by G. Gonzalez-Sprinberg and J.-L. Verdier  and H. Kn¨ orrer . Other, more algebraic, interpretations were given later by B. Kostant , T. Springer , R. Steinberg ,, M. Artin and J.-L. Verdier , H. Esnault and H. Kn¨ orrer . Modern development reveals a more general context of the correspon- dence. The current slogan is that the McKay correspondence establishes an isomorphism of the Grothendieck group K G ( X ) of G-equivariant coherent sheaves on an algebraic variety X on which a finite group G acts and the Grothendieck group K ( Y ) of coherent sheaves on a crepant resolution of the quotient X/G (when it exists), or more generally, an equivalence of the The author was supported in part by NSF grant 0245203. 1 2 IGOR V. DOLGACHEV corresponding derived categories. For example, when X = C n , n = 2 , 3 a crepant resolution exists if and only if G is a subgroup of SU( n ) and an equivalence of the categories was established by M. Kapranov and Vesselot ( n = 2) and T. Bridgeland, A. King and M. Reid ( n = 3). We refer for all this and much more to an excellent survey of M. Reid ....
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- Fall '04
- Geometry, Vector bundle, John McKay, Igor V. Dolgachev, mckay correspondence