Logarithmic Forms and Anti-invariant Forms of Reeection Groups

Let W be a nite group generated by unitary re ections and A be the set of re ecting hyperplanes. We will give a characterization of the logarithmic di erential forms with poles along A in terms of antiinvariant di erential forms. If W is a Coxeter group de ned over R, then the characterization provides a new method to nd a basis for the module of logarithmic di erential forms out of basic invariants. Basic de nitions. Let V be an `-dimensional unitary space. Let W GL(V ) be a nite group generated by unitary re ections and A be the set of re ecting hyperplanes. We say that W is a unitary re ection group and A is the corresponding unitary re ection arrangement. Let S be the algebra of polynomial functions on V: The algebra S is naturally graded by S = L 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 2 DerS is homogeneous of degree q if (S1) Sq : Choose for each hyperplane H 2 A a linear form H 2 V such that H = ker( H): De ne Q 2 S by Q = Y