  In this paper we study the combinatorial consequences of the relationship
between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties
and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed
points in cyclic quiver varieties and calculate the corresponding characters of
tautological bundles. Furthermore, we give a combinatorial description of the
bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg
isomorphism and Nakajima reflection functors. We apply our results to obtain a
new proof as well as a generalization of the $q$-hook formula.
