{
  "id": "1604.07333",
  "version": "v1",
  "published": "2016-04-25T17:12:49.000Z",
  "updated": "2016-04-25T17:12:49.000Z",
  "title": "A filtration on rings of representations of non-Archimedean $GL_n$",
  "authors": [
    "Maxim Gurevich"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $F$ be a $p$-adic field. Let $\\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\\{GL_n(F)\\}_{n=0}^\\infty$ taken together, with multiplication defined in the sense of parabolic induction. We introduce a width invariant for elements of $\\mathcal{R}$ and show that it gives an increasing filtration on the ring. Irreducible representations of width $1$ are precisely those known as ladder representations. We thus obtain a necessary condition on irreducible factors of a product of two ladder representations. For such a product we further establish a multiplicity-one phenomenon, which was previously observed in special cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-25T17:12:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G25",
      "22E50"
    ],
    "keywords": [
      "filtration",
      "non-archimedean",
      "complex smooth finite-length representations",
      "ladder representations",
      "necessary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407333G"
    }
  }
}