{
  "id": "1708.03014",
  "version": "v1",
  "published": "2017-08-09T21:10:48.000Z",
  "updated": "2017-08-09T21:10:48.000Z",
  "title": "The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic $\\textrm{GL}_n$",
  "authors": [
    "Karol Koziol"
  ],
  "comment": "33 pages. Comments welcome!",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $p\\geq 5$ be a prime number and $F$ a $p$-adic field. Let $I_1$ denote the pro-$p$-Iwahori subgroup of $\\textrm{GL}_n(F)$, and $\\mathcal{H}$ the pro-$p$-Iwahori--Hecke algebra of $\\textrm{GL}_n(F)$ with respect to $I_1$ (over a coefficient field of characteristic $p$). We compute the structure of $\\textrm{H}^1(I_1,\\pi)$ as an $\\mathcal{H}$-module, where $\\pi$ is a mod-$p$ principal series representation of $\\textrm{GL}_n(F)$. We also give some partial results about the structure of $\\textrm{H}^1(I_1,\\pi)$ for a general split reductive group with irreducible root system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-09T21:10:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "iwahori cohomology",
      "principal series representation",
      "general split reductive group",
      "iwahori subgroup",
      "coefficient field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}