On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let 8 be an irreducible crystallographic root system in a Euclidean space V , with 8+ the set of positive roots. For α ∈8, k ∈Z, let H(α, k) be the hyperplane {v ∈ V : 〈α, v〉 = k}. We define a set of hyperplanes H = {H(δ, 1) : δ ∈ 8+} ∪ {H(δ, 0) : δ ∈ 8+}. This hyperplane arrangement is significant in the study of the affine Weyl groups. In this paper it is shown that the Poincaré polynomial of H is (1+ ht)n , where n is the rank of 8 and h is the Coxeter number of the finite Coxeter group corresponding to 8.