{
  "id": "2404.03918",
  "version": "v1",
  "published": "2024-04-05T07:04:41.000Z",
  "updated": "2024-04-05T07:04:41.000Z",
  "title": "Dirac cohomology, branching laws and Wallach modules",
  "authors": [
    "Chao-Ping Dong",
    "Yongzhi Luan",
    "Haojun Xu"
  ],
  "comment": "17 pages, 4 figures, 5 tables",
  "categories": [
    "math.RT"
  ],
  "abstract": "The idea of using Dirac cohomology to study branching laws was initiated by Huang, Pandzi\\'c and Zhu in 2013 [HPZ]. One of their results says that the Dirac cohomology of $\\pi$ completely determines $\\pi|_{K}$, where $\\pi$ is any irreducible unitarizable highest weight $(\\mathfrak{g}, K)$ module. This paper aims to develop this idea for the exceptional Lie groups $E_{6(-14)}$ and $E_{7(-25)}$: we recover the $K$-spectrum of the Wallach modules from their Dirac cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-05T07:04:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46"
    ],
    "keywords": [
      "dirac cohomology",
      "wallach modules",
      "exceptional lie groups",
      "irreducible unitarizable highest weight",
      "study branching laws"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}