On categories 𝒪 of quiver varieties overlying the bouquet graphs

Tsvelikhovskiy, Boris

doi:10.1090/ert/644

On categories $\mathcal {O}$ of quiver varieties overlying the bouquet graphs
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by Boris Tsvelikhovskiy;

Represent. Theory 27 (2023), 431-472

DOI: https://doi.org/10.1090/ert/644

Published electronically: July 10, 2023

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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 $\overline {\mathcal {A}}_{\lambda }(n, \ell )$ if both $\operatorname {dim}V=n$ 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.

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