Semiinvariants of Finite Reflection Groups

Abstract

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 χbe a multiplicative character of G. Let Ω^χbe the R-module of χ-invariant differential forms. We define a multiplication in Ω^χand show that under this multiplication Ω^χhas 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.