{
  "id": "1606.09640",
  "version": "v1",
  "published": "2016-06-30T19:54:05.000Z",
  "updated": "2016-06-30T19:54:05.000Z",
  "title": "Highest weight modules to first order",
  "authors": [
    "Gurbir Dhillon",
    "Apoorva Khare"
  ],
  "comment": "57 pages. Comments welcome!",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We give positive formulas for the weights of every simple highest weight module $L(\\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild condition on the highest weight, we express the weights of $L(\\lambda)$ as an alternating sum similar to the Weyl-Kac character formula. For general highest weight modules, we answer questions of Bump and Lepowsky on weights, a question of Brion on convex hulls of the weights, and a question of Brion on the corresponding $D$-modules. Many of these results are new in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group. These results are obtained as elaborations of two ideas. First, the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, and (iii) when a localization theorem is available, its behavior on certain codimension 1 Schubert cells. Second, in favorable circumstances the above data are equivalent to (iv) its weights.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-30T19:54:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B67",
      "22E47",
      "17B37",
      "20G42",
      "52B20"
    ],
    "keywords": [
      "first order",
      "convex hull",
      "simple highest weight module",
      "general highest weight modules",
      "weyl-kac character formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}