Mckay Correspondence

1 Introduction Conjecture 1.1 (since 1992) G ⊂ SL(n, C) is a finite subgroup. Assume that the quotient X = C n /G has a crepant resolution f : Y → X (this just means that K Y = 0, so that Y is a " noncompact Calabi–Yau manifold "). Then there exist " natural " bijections {irreducible representations of G} → basis of H * (Y, Z) (1) {conjugacy classes of G} → basis of H * (Y, Z) (2) As a slogan " representation theory of G = homology theory of Y ". Moreover, these bijections satisfy " certain compatibilities " character table of G McKay quiver ↔ duality cup product As you can see, the statement is still too vague because I don't say what " natural " means, and what " compatibilities " to expect. At present it seems most useful to think of this statement as pointer towards the truth, rather than the truth itself (compare Main Conjecture 4.1). The conjecture is known for n = 2 (Kleinian quotient singularities, Du Val singularities). McKay's original treatment was mainly combinatorics [McK]. The other important proof is that of Gonzales-Sprinberg and Verdier [GSp-V], who introduced the GSp–V or tautological sheaves, also my main hope for the correspondence (1). For n = 3 a weak version of the correspondence (2) is proved in [IR]. We hope that a modification of this idea will work in general for (2); for details, see §3. Contents This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how McKay correspondence relates to mirror symmetry. The main aim is to give numerical examples of how the McKay correspondences