{
  "id": "math/0206119",
  "version": "v2",
  "published": "2002-06-11T18:42:33.000Z",
  "updated": "2002-06-21T16:59:56.000Z",
  "title": "Normalized intertwining operators and nilpotent elements in the Langlands dual group",
  "authors": [
    "Alexander Braverman",
    "David Kazhdan"
  ],
  "comment": "22 pages, Latex",
  "categories": [
    "math.RT",
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $F$ be a local non-archimedian field and let $G$ be a group of points of a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set $X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component $M$ we construct an explicit unitary isomorphism $L^2(X_P)\\to L^2(X_Q)$ (which depends on a choice of an additive character of $F$). The formula for the above isomorphism involves the action of the principal nilpotent element in the Langlands dual group of $M$ on the unipotent radicals of the corresponding dual parabolics. We use the above isomorphisms to define a new space $\\calS(G,M)$ of functions on $X_P$ (which depends only on $P$ and not on $M$). We explain how this space may be applied in order to reformulate in a slightly more elegant way the construction of $L$-functions associated with the standard representation of a classical group due to Gelbart, Piatetski-Shapiro and Rallis.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-06-21T16:59:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "langlands dual group",
      "normalized intertwining operators",
      "local non-archimedian field",
      "principal nilpotent element",
      "explicit unitary isomorphism"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......6119B"
    }
  }
}