  We investigate the cohomology of the Milnor fibre of a reflection arrangement
as a module for the group $\Gamma$ generated by the reflections, together with
the cyclic monodromy.
  Although we succeed completely only for unitary reflection groups of rank
two, we establish some general results which relate the isotypic componenents
of the monodromy on the cohomology, to the Hodge structure and to the
cohomology degree.
  Using eigenspace theory for reflection groups, we prove some sum formulae for
additive functions such as the equivariant weight polynomial and certain
polynomials related to the Euler characteristic, such as the Hodge-Deligne
polynomials. We also use monodromy eigenspaces to determine the spectrum in
some cases, which in turn throws light on the Hodge structure of the
cohomology. These methods enable us to compute the complete story, including
the representation of $\Gamma$ on the Hodge components in each cohomology
degree, for some groups of low rank.
