A bijection between dominant Shi regions and core partitions

It is well-known that Catalan numbers Cn=1n+1(2nn) count the number of dominant regions in the Shi arrangement of type AA, and that they also count partitions which are both nn-cores as well as (n+1)(n+1)-cores. These concepts have natural extensions, which we call here the mm-Catalan numbers and mm-Shi arrangement. In this paper, we construct a bijection between dominant regions of the mm-Shi arrangement and partitions which are both nn-cores as well as (mn+1)(mn+1)-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group.