{
  "id": "1912.07071",
  "version": "v1",
  "published": "2019-12-15T17:06:57.000Z",
  "updated": "2019-12-15T17:06:57.000Z",
  "title": "Fourier transforms on the basic affine space of a quasi-split group",
  "authors": [
    "Nadya Gurevich",
    "David Kazhdan"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "For a quasi-split group $G$ over a local field $F$, with Borel subgroup $B=TU$ and Weyl group $W$, there is a natural geometric action of $G\\times T$ on $L^2(X),$ where $X=G/U$ is the basic affine space of $G$. For split groups, Gelfand and Graev have extended this action to an action of $G\\times (T\\rtimes W)$ by generalized Fourier transforms $\\Phi_w$. We define an analog of these operators for quasi-split groups. We also extend the construction of the Schwartz space $\\mathcal S (X)$ by Braverman and Kazhdan to the case of quasi-split groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-15T17:06:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50"
    ],
    "keywords": [
      "basic affine space",
      "quasi-split group",
      "natural geometric action",
      "generalized fourier transforms",
      "split groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}