The combinatorics of $\mathbb{C}^*$-fixed points in generalized Calogero-Moser spaces and Hilbert schemes

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.