Generalized Calogero-Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W -orbits of the intersections of mirrors of a finite reflection group W . When W is either a real reflection group or the complex reflection group G(m, p,N) we determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra at certain values of the parameters hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W . This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify.