Double Coxeter arrangements, the Shi arrangements,
anti-invariant forms and logarithmic forms
Abstract: Let W be a finite crystallographic group and A(W) be the arrangement of its reflecting hyperplanes in the n-dimensional real vector space. The Shi arrangement is A(W) together with the hypeperlanes defined by \alpha = 1 (\alpha is moving on the set of positive roots of W). It has remarkable combinatorial properties. For example, the number of chambers is equal to (1+h)n (h is the Coxeter number). The "principal part" of the Shi arrangement is the double Coxeter arrangement. We prove that its derivation module is a free moduole of rank n with a basis consisting of derivations whose degrees are (h, h, ..., h). Explicit basis is described using the flat structure (the Frobenius system) of Coxeter group. The relation between the anti-invariant differential forms and the logarithmic differential forms is also discussed.