Bimahonian distributions

Motivated by permutation statistics, we define for any complex reflection group W a family of bivariate generating functions. 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 also shown that they satisfy a ``bicyclic sieving phenomenon'', which combinatorially interprets their values when the two variables are set equal to certain roots of unity.