{
  "id": "1910.08286",
  "version": "v1",
  "published": "2019-10-18T07:17:40.000Z",
  "updated": "2019-10-18T07:17:40.000Z",
  "title": "Contravariant forms on Whittaker modules",
  "authors": [
    "Adam Brown",
    "Anna Romanov"
  ],
  "comment": "14 pages; preliminary version, comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker $\\mathfrak{g}$-modules $Y(\\chi, \\eta)$ introduced by Kostant. We prove that the set of all contravariant forms on $Y(\\chi, \\eta)$ forms a vector space whose dimension is given by the cardinality of the Weyl group of $\\mathfrak{g}$. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules $M(\\chi, \\eta)$ introduced by McDowell.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-18T07:17:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "17B35"
    ],
    "keywords": [
      "complex semisimple lie algebra",
      "degenerate whittaker modules",
      "vector space",
      "weyl group",
      "parabolically inducing contravariant forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}