On categories

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