Universal Abelian covers of rational surface singularities and multi-index filtrations

In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series.