On categories 𝒪 of quiver varieties overlying the bouquet graphs

Abstract

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations A ¯ λ ( n , ℓ ) \overline {\mathcal {A}}_{\lambda }(n, \ell ) if both dim ⁡ V = n \operatorname {dim}V=n and the number of loops ℓ \ell are greater than 1 1 . We show that when n ≤ 3 n\leq 3 there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category O \mathcal {O} and compute the multiplicities of simples in standards for n = 2 n=2 in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension.