On categories $\mathcal{O}$ of quiver varieties overlying the bouquet graphs

TL;DR

This paper investigates the representation theory of quantized Nakajima quiver varieties over bouquet graphs, revealing conditions for finite-dimensional representations, fixed points, and homological properties, with explicit computations for specific cases.

Contribution

It provides new results on the structure and representations of quantizations of Nakajima quiver varieties associated with bouquet graphs, including conditions for finite-dimensionality and explicit calculations in low-dimensional cases.

Findings

No finite-dimensional representations for dim V > 1 and loops

Existence of Hamiltonian torus action with finitely many fixed points for n ≤ 3

Explicit dimensions of Hom-spaces and multiplicities in specific cases

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 dim $V=n$ is greater than $1$ and so is the number of loops $\ell$. We find that there is a Hamiltonian torus action with finitely many fixed points in case $n\leq 3$, 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 localisation theorem and find the values of parameters, for which the quantizations have infinite homological dimension.