Using a representation theoretic parameterization for the orbits in the enhanced cyclic nilpotent cone as derived by the authors in a previous article, we compute the fundamental group of these orbits. This computation has several applications to the representation theory of the category of admissible -modules on the space of representations of the framed cyclic quiver. First and foremost, we compute precisely when this category is semisimple. We also show that the category of admissible -modules has enough projectives. Finally, the support of an admissible -module is contained in a certain Lagrangian in the cotangent bundle of the space of representations. Thus, taking these characteristic cycles defines a map from the K-group of the category of admissible -modules to the -span of the irreducible components of this Lagrangian. We show that this map is always injective, and is a bijection if and only if the monodromicity parameter is integral.
The authors would like to thank K. Brown and T. Schedler for helpful remarks on the subject. We would also like to thank the referees for an extremely detailed and constructive review of an earlier version of the article.
The first author was partially supported by Engineering and Physical Sciences Research Council (EPSRC) grant EP/N005058/1.
Gwyn Bellamy. Magdalena Boos. "Semisimplicity of the category of admissible -modules." Kyoto J. Math. 61 (1) 115 - 170, April 2021. https://doi.org/10.1215/21562261-2020-0006