{
  "id": "2204.10480",
  "version": "v1",
  "published": "2022-04-22T03:39:20.000Z",
  "updated": "2022-04-22T03:39:20.000Z",
  "title": "Restricting Representations from a Complex Group to a Real Form",
  "authors": [
    "Lucas Mason-Brown"
  ],
  "comment": "Comments are welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex connected reductive algebraic group and let $G_{\\mathbb{R}}$ be a real form of $G$. We construct a sequence of functors $L_i\\mathcal{R}$ from admissible (resp. finite-length) representations of $G$ to admissible (resp. finite-length) representations of $G_{\\mathbb{R}}$. We establish many basic properties of these functors, including their behavior with respect to infinitesimal character, associated variety, and restriction to a maximal compact subgroup. We deduce that each $L_i\\mathcal{R}$ takes unipotent representations of $G$ to unipotent representations of $G_{\\mathbb{R}}$. Taking the alternating sum of $L_i\\mathcal{R}$, we get a well-defined homomorphism on the level of characters. We compute this homomorphism in the case when $G_{\\mathbb{R}}$ is split.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-22T03:39:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46"
    ],
    "keywords": [
      "real form",
      "complex group",
      "restricting representations",
      "unipotent representations",
      "complex connected reductive algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}