  Recent work by a number of people has shown that complex reflection groups
give rise to many representation-theoretic structures (e.g., generic degrees
and families of characters), as though they were Weyl groups of algebraic
groups. Conjecturally, these structures are actually describing the
representation theory of as-yet undescribed objects called ''spetses'', of
which reductive algebraic groups ought to be a special case.
  In this paper, we carry out the Lusztig--Shoji algorithm for calculating
Green functions for the dihedral groups. With a suitable set-up, the output of
this algorithm turns out to satisfy all the integrality and positivity
conditions that hold in the Weyl group case, so we may think of it as
describing the geometry of the ''unipotent variety'' associated to a spets.
  From this, we determine the possible ''Springer correspondences'', and we
show that, as is true for algebraic groups, each special piece is rationally
smooth, as is the full unipotent variety.
