Derived Reid's recipe for abelian subgroups of SL3(C)

Abstract

For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid's recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one isolated singularity.