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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, ) if both dim V = n and the number of loops are greater than 1. We show that when n ≤ 3 there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category O and compute the multiplicities of simples in standards for 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. Contents 1. Introduction 431 2. First results on O ν (A λ (n, )) 436 3. Symplectic leaves and slices 446 4. Category O ξ (S λ (2, )) for the slice SL p 450 5. Harish-Chandra bimodules, ideals and localization theorems 457 6. Structure of the category O ν (A λ (2, )) 460 7. Singular parameters 469 Acknowledgments 471 References 471