Bimahonian distributions

Motivated by permutation statistics, we define, for any complex reflection group W, a family of bivariate generating functions Wσ(t, q). They are defined either in terms of Hilbert series for W-invariant polynomials when W acts diagonally on two sets of variables or, equivalently, as sums involving the fake degrees of irreducible representations for W. It is shown that Wσ(t, q) satisfies a ‘bicyclic sieving phenomenon’ which combinatorially interprets its values when t and q are certain roots of unity.