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Suppose V is a finite dimensional, complex vector space, A is a finite set ofcodimension one subspaces of V, and G is a finite subgroup of the generallinear group GL(V) that permutes the hyperplanes in A. In this paper we studyinvariants 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^*( . ) denotesrational singular cohomology, in the case when A is a reflection arrangementand the pair (A,G) arises from a reflection coset. Our main result is theconstruction 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 aproof of the description of the space of invariants conjectured by Felder andVeselov for Coxeter groups that does not rely on computer calculations, thisconstruction provides an extension of this description of the space ofinvariants to arbitrary finite, complex reflection groups. The main result alsoleads to simplifications of some cohomology computations of Lehrer,Callegaro-Marin, and Marin.

Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement