{
  "id": "1607.04145",
  "version": "v1",
  "published": "2016-07-14T14:15:02.000Z",
  "updated": "2016-07-14T14:15:02.000Z",
  "title": "Test vectors for local periods",
  "authors": [
    "U. K. Anandavardhanan",
    "Nadir Matringe"
  ],
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $\\pi$ of $GL(n,E)$ is said to be distinguished with respect to $GL(n,F)$ if it admits a non-trivial linear form that is invariant under the action of $GL(n,F)$. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the $F$-points of the mirabolic subgroup when $\\pi$ is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local $L$-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-14T14:15:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "22E50"
    ],
    "keywords": [
      "test vector",
      "local periods",
      "essential vector",
      "non-trivial linear form",
      "standard distinguishing linear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}