On Transitive Action on Quiver Varieties

Abstract

Abstract Associated with each finite subgroup $\Gamma $ of ${\textrm{SL}}_2({\mathbb{C}})$ 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 $. In [ 16], one of the authors 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. In this paper we prove that the action of $G_\Gamma $ on the quiver variety is transitive when $\Gamma $ is a cyclic group. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the 1st Weyl algebra on the Calogero–Moser spaces. Our result has two important implications. First, it confirms the Bocklandt–Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to $O_\tau (\Gamma ),$ thus answering the question of Hodges. At the end of the introduction we explain why the result of this paper does not extend when $\Gamma $ is not cyclic.