{
  "id": "1709.08986",
  "version": "v1",
  "published": "2017-09-26T12:55:45.000Z",
  "updated": "2017-09-26T12:55:45.000Z",
  "title": "Semi-simplicity of the category of admissible D-modules",
  "authors": [
    "Gwyn Bellamy",
    "Magdalena Boos"
  ],
  "comment": "34 pages. Previously this was the second half of our article on \"The (cyclic) enhanced nilpotent cone via quiver representations\" arXiv:1609.04525 . The main result has been strengthened, and new material added",
  "categories": [
    "math.RT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-26T12:55:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "admissible d-modules",
      "semi-simplicity",
      "characteristic cycles defines",
      "enhanced cyclic nilpotent cone",
      "representation theoretic parameterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}