{
  "id": "1704.04018",
  "version": "v1",
  "published": "2017-04-13T07:29:52.000Z",
  "updated": "2017-04-13T07:29:52.000Z",
  "title": "The Fourier transform on the group $GL_2(R)$ and the action of the overalgebra $gl_4$",
  "authors": [
    "Yury A. Neretin"
  ],
  "comment": "16pp",
  "categories": [
    "math.RT",
    "math.FA",
    "math.GR"
  ],
  "abstract": "We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$. We transfer the corresponding action of the Lie algebra $gl_4$ to the Plancherel decomposition of $GL_2(R)$, the Lie algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of $gl_4\\oplus gl_4$ in the Plancherel decomposition of $GL_2(C)$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-13T07:29:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "22E46",
      "43A30",
      "43A80",
      "43A85"
    ],
    "keywords": [
      "fourier transform",
      "overalgebra",
      "plancherel decomposition",
      "open dense chart",
      "lie algebra acts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}