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

Understand

In type A, the q,t-Fuss-Catalan numbers can be defined as a 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 non-negative integer coefficients.
• We prove the conjectures for the dihedral groups and for the cyclic groups.
• Finally, we present several ideas how the q,t-Fuss-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.