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Semi-invariant Forms


Anne V. Shepler

Let $G$ be a finite group of complex $n \times n$ unitary matrices generated by reflections acting on $C^n$ . Let $R$ be the ring of invariant polynomials, and $\chi$ be a multiplicative character of $G$. Consider the $R$-module of $\chi$ -invariant differential forms and the $R$-module of $\chi$-invariants in the exterior algebra of derivations. We define a natural multiplication on these modules using ideas from arrangements of hyperplanes. We show that this multiplication gives each module the structure of an exterior algebra. We also define a multiarrangement associated to $\chi$, and formulate the relationship between $\chi$-invariants and logarithmic forms. We introduce a new method of computing basic derivations and the generating $\chi$-invariants and give explicit constructions for the exceptional irreducible reflection groups.

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