  This paper studies the poset of eigenspaces of elements of an imprimitive
unitary reflection group, for a fixed eigenvalue, ordered by the reverse of
inclusion. The study of this poset is suggested by the eigenspace theory of
Springer and Lehrer. The posets are shown to be isomorphic to certain subposets
of Dowling lattices (the `d-divisible, k-evenly coloured Dowling lattices').
This enables us to prove that these posets are Cohen-Macaulay, and to determine
the dimension of their top homology.
