Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan

Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this paper, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This is a completion of the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems.