Semi-simplicity of the category of admissible D-modules

TL;DR

This paper investigates the structure of admissible D-modules on cyclic quiver representation spaces, determining conditions for semi-simplicity, projective objects, and the relationship between characteristic cycles and irreducible components.

Contribution

It provides a precise criterion for the semi-simplicity of the category of admissible D-modules and establishes a connection between characteristic cycles and the geometry of the cotangent bundle.

Findings

The category is semi-simple if and only if the monodromicity parameter is integral.

The category of admissible D-modules has enough projectives.

The characteristic cycle map is injective and bijective under certain conditions.

Abstract

Using a representation theoretic parameterization for the orbits in the enhanced cyclic nilpotent cone, 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 $D$-modules on the space of representations of the framed cyclic quiver. First, and foremost, we compute precisely when this category is semi-simple. We also show that the category of admissible $D$-modules has enough projectives. Finally, the support of an admissible $D$-module is contained in a certain Lagrangian in the cotangent bundle of the space of representations. Thus, taking characteristic cycles defines a map from the $K$-group of the category of admissible $D$-modules to the $\mathbb{Z}$-span of the irreducible components of this Lagrangian. We show that this map is always injective, and a bijection if and only if the monodromicity parameter is integral.