q,t-Fu-Catalan numbers for finite reflection groups

In type A, the q,t-Fu-Catalan numbers can be defined as the bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with nonnegative integer coefficients. We prove the conjectures for the dihedral groups and for the cyclic groups. Finally, we present several ideas on how the q,t-Fu-Catalan numbers could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras and thereby generalize known connections.