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

Abstract

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 ${\mathbb{C}}^{\ast }$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between ${\mathbb{C}}^{\ast }$-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.

A few minor typos corrected. To appear in the Journal of Algebra

References in corpus (2)

• Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
• Generalized graded Hecke algebra for complex reflection group of type G(r,1,n)

Cited by in corpus (1)

• Fixed points in smooth Calogero-Moser spaces