Logarithmic forms and anti-invariant forms of reflection groups

Abstract

Let $W$ be a finite group generated by unitary reflections and $\mathcal{A}$ be the set of reflecting hyperplanes. We will give a characterization of the logarithmic differential forms with poles along $\mathcal{A}$ in terms of anti-invariant differential forms. If $W$ is a Coxeter group defined over $\mathbf{R}$, then the characterization provides a new method to find a basis for the module of logarithmic differential forms out of basic invariants.