Catalan numbers for complex reflection groups

We construct $(q,t)$-Catalan polynomials and $q$-Fuss-Catalan polynomials for any irreducible complex reflection group $W$. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik algebras of $W$, and Opdam's analysis of permutations of the irreducible representations of $W$ arising from the Knizhnik-Zamolodchikov connection.