Sutherland models for complex reflection groups

There are known to be integrable Sutherland models associated to every real root system, or, which is almost equivalent, to every real reflection group. Real reflection groups are special cases of complex   reflection groups. In this paper we associate certain integrable Sutherland models to the classical family of complex reflection groups. Internal degrees of freedom are introduced, defining dynamical spin chains, and the freezing limit taken to obtain static chains of Haldane–Shastry type. By considering the relation of these models to the usual BCN$B{C}_N$ case, we are led to systems with both real and complex reflection groups as symmetries. We demonstrate their integrability by means of new Dunkl operators, associated to wreath products of dihedral groups.