{
  "id": "1106.0347",
  "version": "v2",
  "published": "2011-06-02T00:19:18.000Z",
  "updated": "2011-06-27T20:21:41.000Z",
  "title": "BGG reciprocity for current algebras",
  "authors": [
    "Matthew Bennett",
    "Vyjayanthi Chari",
    "Nathan Manning"
  ],
  "comment": "29 pages. Some minor corrections",
  "journal": "Adv. Math. 231 (2012), no. 1, 276-305",
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the category $\\cal I_{\\gr}$ of graded representations with finite--dimensional graded pieces for the current algebra $\\lie g\\otimes\\bc[t]$ where $\\lie g$ is a simple Lie algebra. This category has many similarities with the category $\\cal O$ of modules for $\\lie g$ and in this paper, we formulate and study an analogue of the famous BGG duality. We recall the definition of the projective and simple objects in $\\cal I_{\\gr}$ which are indexed by dominant integral weights. The role of the Verma modules is played by a family of modules called the global Weyl modules. We show that in the case when $\\lie g$ is of type $\\lie{sl}_2$, the projective module admits a flag in which the successive quotients are finite direct sums of global Weyl modules. The multiplicity with which a particular Weyl module occurs in the flag is determined by the multiplicity of a Jordan--Holder series for a closely associated family of modules, called the local Weyl modules. We conjecture that the result remains true for arbitrary simple Lie algebras. We also prove some combinatorial product--sum identities involving Kostka polynomials which arise as a consequence of our theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-27T20:21:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "current algebra",
      "bgg reciprocity",
      "global weyl modules",
      "arbitrary simple lie algebras",
      "weyl module occurs"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.0347B"
    }
  }
}