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 $\overline{\mathcal{A}}_{\lambda}(n, \ell)$ if both $\mbox{dim}~V=n$ and the number of loops $\ell$ are greater than $1$. We show that when $n\leq 3$ there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ 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.--Author's abstract
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