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.References
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Bibliographic Information
- Boris Tsvelikhovskiy
- Affiliation: Department of Mathematics, University of Pittsburgh, 15260 Pittsburgh, Pennsylvania
- Email: bdt18@pitt.edu
- Received by editor(s): February 1, 2022
- Received by editor(s) in revised form: September 25, 2022, December 19, 2022, and February 14, 2023
- Published electronically: July 10, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory 27 (2023), 431-472
- MSC (2020): Primary 16S99, 53D55
- DOI: https://doi.org/10.1090/ert/644
- MathSciNet review: 4613930