Semiinvariants of Finite Reflection Groups

Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and \chi be a multiplicative character of G. Let \Omega^\chi be the R-module of \chi-invariant differential forms. We define a multiplication in \Omega^\chi and show that under this multiplication \Omega^\chi has an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between \chi-invariant forms and logarithmic forms.