On transitive action on quiver varieties

Abstract

Associated with each finite subgroup $\Gamma$ of $\rm{SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O_\tau(\Gamma)$ quantizing $\mathbb{C}^2/\!\!/\Gamma$. Let $G_\Gamma$ be the group of $\Gamma$-equivariant automorphisms of $O_\tau$. One of the authors earlier defined and studied a natural action of $G_\Gamma$ on certain quiver varieties associated with $\Gamma$. He established a $G_\Gamma$-equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of $O_\tau$-ideals. The main theorem in this paper states that when $\Gamma$ is a cyclic group, the action of $G_\Gamma$ on each quiver variety is transitive. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the first Weyl algebra on the Calogero-Moser spaces. Our result has two important implications. First, it confirms the Bockland-Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to $O_\tau(\Gamma)$.