{
  "id": "1909.09873",
  "version": "v1",
  "published": "2019-09-21T19:01:10.000Z",
  "updated": "2019-09-21T19:01:10.000Z",
  "title": "Some harmonic analysis on commutative nilmanifolds",
  "authors": [
    "Andrea L. Gallo",
    "Linda. V. Saal"
  ],
  "categories": [
    "math.FA",
    "math.RT"
  ],
  "abstract": "In this work, we consider a family of Gelfand pairs $(K \\ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost all these 2-step nilpotent Lie group have square integrable representations. In this cases, following Moore-Wolf's theory, we find an explicit expression for the inversion formula of $N$, and as a consequence, we decompose the regular action of $K \\ltimes N$ on $L^{2}(N)$. This result completes the analysis carried out by Wolf, where the inversion formula is obtained in the case that $N$ has not square integrable representation. When $N$ is the Heisenberg group, we obtain the decomposition of $L^{2}(N)$ under the action of $K \\ltimes N$ for all $K$ such that $(K,N)$ is a Gelfand pair. Finally, we also give a parametrization for the generic spherical functions associated to the pair $(K,N)$, and we give an explicit expression for these functions in some cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-21T19:01:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A80",
      "22E25"
    ],
    "keywords": [
      "harmonic analysis",
      "commutative nilmanifolds",
      "square integrable representation",
      "step nilpotent lie group",
      "explicit expression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}