Bimahonian distributions

Motivated by permutation statistics, we define, for any complexreflection group W, a family of bivariate generating functionsW(t, q). They are defined either in terms of Hilbert seriesfor W-invariant polynomials when W acts diagonally on two setsof variables or, equivalently, as sums involving the fake degreesof irreducible representations for W. It is shown that W(t,q) satisfies a ‘bicyclic sieving phenomenon’ whichcombinatorially interprets its values when t and q are certainroots of unity.