{
  "id": "2209.12566",
  "version": "v1",
  "published": "2022-09-26T10:33:50.000Z",
  "updated": "2022-09-26T10:33:50.000Z",
  "title": "Dirac cohomology for the BGG category $\\mathcal{O}$",
  "authors": [
    "Spyridon Afentoulidis-Almpanis"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We study Dirac cohomology $H_D^{\\mathfrak{g},\\mathfrak{h}}(M)$ for modules belonging to category $\\mathcal{O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for $H_D^{\\mathfrak{g},\\mathfrak{h}}(M)$ while we show that in the case of a Hermitian symmetric pair $(\\mathfrak{g},\\mathfrak{k})$ and an irreducible unitary module $M\\in\\mathcal{O}$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pand\\v{z}i\\'c and Somberg satisfy nice homological properties for $M\\in\\mathcal{O}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T10:33:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "17B56"
    ],
    "keywords": [
      "dirac cohomology",
      "bgg category",
      "finite dimensional complex semisimple lie",
      "dimensional complex semisimple lie algebra",
      "somberg satisfy nice homological properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}