{
  "id": "math/0310140",
  "version": "v1",
  "published": "2003-10-09T17:44:08.000Z",
  "updated": "2003-10-09T17:44:08.000Z",
  "title": "Generalized Harish-Chandra Modules: A New Direction",
  "authors": [
    "Ivan Penkov",
    "Gregg Zuckerman"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\frak g$ be a reductive Lie algebra over $\\bold C$. We say that a $\\frak g$-module $M$ is a generalized Harish-Chandra module if, for some subalgebra $\\frak k \\subset\\frak g$, $M$ is locally $\\frak k$-finite and has finite $\\frak k$-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when $\\frak k$ is a Cartan subalgebra. We also review the recent determination of which reductive in $\\frak g$ subalgebras $\\frak k$ are essential to a classification. Finally, we present in detail the emerging picture for the case when $\\frak k$ is a principal 3-dimensional subalgebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-10-09T17:44:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "reductive lie algebra",
      "irreducible generalized harish-chandra modules",
      "cartan subalgebra",
      "multiplicities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10140P"
    }
  }
}