The (cyclic) enhanced nilpotent cone via quiver representations

TL;DR

This paper introduces a new way to classify orbits in the enhanced nilpotent cone using quiver representations, extending to cyclic cases and providing explicit translations between different parametrizations.

Contribution

It offers a novel parameterization of orbits in the enhanced nilpotent cone via quiver representations, generalizing to cyclic cases and differing from previous methods.

Findings

New parameterization of orbits in the enhanced nilpotent cone

Extension of parameterization to enhanced cyclic nilpotent cone

Explicit translations between different parametrizations

Abstract

The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.