  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 the McKay correspondence for finite subgroups of SL(n,C) relates
to mirror symmetry. The main aim is to give numerical examples of how the 2
McKay correspondences
  (1) representations of G <--> cohomology of resolution
  (2) conjugacy classes of G <--> homology must work, and to restate my 1992
Conjecture as a tautology, like cohomology or K-theory of projective space.
Another aim is to give an introduction to Nakamura's results on the Hilbert
scheme of G-clusters, following his preprints and his many helpful
explanations. This is partly based on joint work with Y. Ito, and has benefited
from encouragement and invaluable suggestions of S. Mukai.
