Co-quasi-invariant spaces for generalized symmetric groups

We study, in a global uniform manner, the quotient of the ring of polynomials in ℓ sets of n variables, by the ideal generated by diagonal quasi-invariant polynomials for generalized permutation groups W = G ( r , n ) . We show that, for each such group W, there is an explicit universal symmetric function that gives the N ℓ -graded Hilbert series for these spaces. This function is universal in that its dependence on ℓ only involves the number of variables it is calculated with.