{
  "id": "1907.09069",
  "version": "v1",
  "published": "2019-07-22T01:11:41.000Z",
  "updated": "2019-07-22T01:11:41.000Z",
  "title": "Geometric and algebraic parameterizations for Dirac cohomology of simple modules in $\\mathcal{O}^\\mathfrak{p}$ and their applications",
  "authors": [
    "Ho-Man Cheung"
  ],
  "comment": "41 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we show that the Dirac cohomology $H_{D}(L(\\lambda))$ of a simple highest weight module $L(\\lambda)$ in $\\mathcal{O}^\\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\\mathcal{W}_I(\\lambda)$ of the orbit of the Weyl group $W$ acting on $\\lambda+\\rho$. As an application, we show that any simple module in $\\mathcal{O}^\\mathfrak{p}$ is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(\\lambda))$ when $\\lambda$ is regular. Two of these parameterizations are geometric in terms of a partial ordering on the dual of the Cartan subalgebra and a generalization of strong linkage, respectively. Using these geometric parameterizations, we derive two algebraic parameterizations in terms of the multiplicities of the composition factors of a Verma module and the embeddings between Verma modules, respectively. As an application, for Verma modules with regular infinitesimal character, we obtain an extended version of the Verma-BGG Theorem. We also investigate Dirac cohomology of Kostant modules. Using Dirac cohomology, we give a new proof of the simplicity criterion for Verma modules and describe a new simplicity criterion for parabolic Verma modules with regular infinitesimal character.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-22T01:11:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirac cohomology",
      "simple module",
      "algebraic parameterizations",
      "application",
      "regular infinitesimal character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}