The (cyclic) enhanced nilpotent cone via quiver representations

Gwyn Bellamy, Magdalena Boos

Published 2016-09-15Version 1

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 is different to the previous ones that have appeared in the literature. Our parameterization generalizes naturally to the enhanced cyclic nilpotent cone. The new, representation theoretic, parameterization of the orbits we obtain allows us to compute the fundamental group of these orbits. We describe the consequence of these computations for the category of admissible $D$-modules on the space of representations of the framed cyclic quiver.