{
  "id": "1707.02592",
  "version": "v1",
  "published": "2017-07-09T15:15:03.000Z",
  "updated": "2017-07-09T15:15:03.000Z",
  "title": "The Submodule Structure of Spherical Principal Series Module of Reductive Groups with Frobenius Maps in Cross Characteristic",
  "authors": [
    "Xiaoyu Chen"
  ],
  "comment": "7 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let ${\\bf G}$ be a connected reductive group over $\\bar{\\mathbb{F}}_q$, the algebraically closure of $\\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\\bf B}$ be an $F$-stable Borel subgroup. Let $\\Bbbk$ be a field of characteristic $r\\neq p$. In this paper, we completely determine the composition factors of the induced module $\\op{Ind}_{\\bf B}^{\\bf G}\\op{tr}=\\Bbbk{\\bf G}\\otimes_{\\Bbbk{\\bf B}}\\op{tr}$ (here $\\Bbbk{\\bf H}$ is the group algebra of the group ${\\bf H}$, and $\\op{tr}$ is the trivial ${\\bf B}$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of ${\\bf G}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-09T15:15:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05"
    ],
    "keywords": [
      "spherical principal series module",
      "frobenius map",
      "reductive group",
      "submodule structure",
      "cross characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}