Logarithmic Forms and Anti - Invariant Forms of Reflection Groups

Let W be a finite group generated by unitary reflections and A be the set of reflecting hyperplanes. We will give a characterization of the logarithmic differential forms with poles along A in terms of antiinvariant differential forms. If W is a Coxeter group defined over R, then the characterization provides a new method to find a basis for the module of logarithmic differential forms out of basic invariants. Basic definitions. Let V be an l-dimensional unitary space. Let W ⊂ GL(V ) be a finite group generated by unitary reflections and A be the set of reflecting hyperplanes. We say that W is a unitary reflection group and A is the corresponding unitary reflection arrangement. Let S be the algebra of polynomial functions on V. The algebra S is naturally graded by S = ⊕ q≥0 Sq where Sq is the space of homogeneous polynomials of degree q. Thus S1 = V ∗ is the dual space of V. Let DerS be the S-module of C-derivations of S. We say that θ ∈ DerS is homogeneous of degree q if θ(S1) ⊆ Sq. Choose for each hyperplane H ∈ A a linear form αH ∈ V ∗ such that H = ker(αH). Define Q ∈ S by Q = ∏