  Suppose V is a finite dimensional, complex vector space, A is a finite set of
codimension one subspaces of V, and G is a finite subgroup of the general
linear group GL(V) that permutes the hyperplanes in A. In this paper we study
invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A)
denotes the complement in V of the hyperplanes in A and H^*( . ) denotes
rational singular cohomology, in the case when A is a reflection arrangement
and the pair (A,G) arises from a reflection coset. Our main result is the
construction of an explicit, natural (from the point of view of Coxeter groups)
basis of the space of invariants, H^*(M(A))^G. In addition to leading to a
proof of the description of the space of invariants conjectured by Felder and
Veselov for Coxeter groups that does not rely on computer calculations, this
construction provides an extension of this description of the space of
invariants to arbitrary finite, complex reflection groups. The main result also
leads to simplifications of some cohomology computations of Lehrer,
Callegaro-Marin, and Marin.
