Special representations and the two - dimensional McKay correspondence

{ finite subgroups Γ ⊂ SL (2, C) }/ conjugacy {CDW-diagrams of type ADE } l l {Klein singularities XΓ = C/Γ }/ ∼ ←→ {minimal resolutions X̃Γ }/ ∼ Here, the symbol ∼ in the last row denotes complex–analytic equivalence. The arrow in the second column is given in the upward direction by associating to a minimal resolution X̃Γ of XΓ the dual graph of its exceptional set E ⊂ X̃Γ . In 1979, McKay [16] constructed directly via representation theory the resulting bijection in the first row of this diagram (see Section 1). In particular, according to this so called McKay correspondence, each (nontrivial) irreducible complex representation of Γ corresponds uniquely to an irreducible component of the exceptional set E . Of course, geometers wanted to understand this phenomenon geometrically , and the first who succeeded in this attempt were Gonzales-Sprinberg and Verdier [8] in 1983. They associated to each nontrivial irreducible representation of Γ a vector bundle F on X̃Γ whose first Chern class c1(F) hits precisely one component of E transversally. Their proof was not completely satisfying since they had to check the details case by case. But in 1985, Artin and Verdier [1] gave a conceptual proof using only standard facts on rational singularities, and in combination with the so called multiplication formula contained in the paper [6] of Hélène Esnault and Knörrer from the same year it became clear how to understand the full strength of the correspondence, i. e. how to