{
  "id": "1706.09044",
  "version": "v1",
  "published": "2017-06-21T15:52:32.000Z",
  "updated": "2017-06-21T15:52:32.000Z",
  "title": "Fourier Transform of Schwartz Algebras on Groups in the Harish-Chandra class",
  "authors": [
    "Olufemi O. Oyadare"
  ],
  "comment": "9 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "It is well-known that the Harish-Chandra transform, $f\\mapsto\\mathcal{H}f,$ is a topological isomorphism of the spherical (Schwartz) convolution algebra $\\mathcal{C}^{p}(G//K)$ (where $K$ is a maximal compact subgroup of any arbitrarily chosen group $G$ in the Harish-Chandra class and $0<p\\leq2$) onto the (Schwartz) multiplication algebra $\\bar{\\mathcal{Z}}({\\mathfrak{F}}^{\\epsilon})$ (of $\\mathfrak{w}-$invariant members of $\\mathcal{Z}({\\mathfrak{F}}^{\\epsilon}),$ with $\\epsilon=(2/p)-1$). The same cannot however be said of the full Schwartz convolution algebra $\\mathcal{C}^{p}(G),$ except for few specific examples of groups (notably $G=SL(2,\\mathbb{R})$) and for some notable values of $p$ (with restrictions on $G$ and/or on $\\mathcal{C}^{p}(G)$). Nevertheless the full Harish-Chandra Plancherel formula on $G$ is known for all of $\\mathcal{C}^{2}(G)=:\\mathcal{C}(G).$ In order to then understand the structure of Harish-Chandra transform more clearly and to compute the image of $\\mathcal{C}^{p}(G)$ under it (without any restriction) we derive an absolutely convergent series expansion (in terms of known functions) for the Harish-Chandra transform by an application of the full Plancherel formula on $G.$ This leads to a computation of the image of $\\mathcal{C}(G)$ under the Harish-Chandra transform which may be seen as a concrete realization of Arthur's result and be easily extended to all of $\\mathcal{C}^{p}(G)$ in much the same way as it is known in the work of Trombi and Varadarajan.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-21T15:52:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "22E30",
      "22E46"
    ],
    "keywords": [
      "harish-chandra class",
      "harish-chandra transform",
      "schwartz algebras",
      "fourier transform",
      "full schwartz convolution algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}