Semi-invariants of Finite Reflection Groups

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