Semisimplicity of the category of admissible D-modules

Abstract

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 $\mathcal{D}$-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 $\mathcal{D}$-modules has enough projectives. Finally, the support of an admissible $\mathcal{D}$-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 $\mathcal{D}$-modules to the $\mathbb{Z}$-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.