{
  "id": "math/0005066",
  "version": "v1",
  "published": "2000-05-07T10:03:00.000Z",
  "updated": "2000-05-07T10:03:00.000Z",
  "title": "Banach space representations and Iwasawa theory",
  "authors": [
    "Peter Schneider",
    "Jeremy Teitelbaum"
  ],
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For example, Diarra showed that the abelian group $G=\\dZ$ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call ''admissibility''. We show that the category of all admissible $G$-representations is reasonable -- in fact, it is abelian and of a purely algebraic nature -- by showing that it is anti-equivalent to the category of all finitely generated modules over a certain kind of completed group ring $K[[G]]$. As an application of our methods we determine the topological irreducibility as well as the intertwining maps for representations of $GL_2(\\dZ)$ obtained by induction of a continuous character from the subgroup of lower triangular matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-05-07T10:03:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S80",
      "11R23",
      "22E35"
    ],
    "keywords": [
      "iwasawa theory",
      "adic haar measure",
      "traditional representation theory",
      "adic lie group",
      "topologically irreducible banach space representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......5066S"
    }
  }
}