The freeness of Shi–Catalan arrangements

Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H ∈ A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H . We say that a deformation is W -equivariant if the number of parallel hyperplanes of each hyperplane H ∈ A depends only on the W -orbit of H . We prove that the conings of the W -equivariant deformations are free arrangements under a Shi–Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga’s theorem conjectured by Edelman–Reiner.