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Harmonic analysis on some generalized Gelfand pairs attached to Heisenberg groups

Volume 241 / 2018

Fernando Levstein, Linda Saal Studia Mathematica 241 (2018), 135-158 MSC: Primary 43A80; Secondary 35A08. DOI: 10.4064/sm8592-5-2017 Published online: 20 November 2017

Abstract

Let $H_n$ be the $2n+1$-dimensional Heisenberg group. We consider the generalized Gelfand pairs $(\mathbb {R}^*\ltimes H_1,\mathbb {R}^*)$ and $((\mathbb {R}_{ \gt 0}\times \mathrm {SO}(n))\ltimes H_n, \mathbb {R}_{ \gt 0}\times \mathrm {SO}(n))$ for $n\ge 2$. We describe the spherical distributions corresponding to these pairs and we obtain inversion formulæ in terms of them for the spaces of Schwartz functions on $\mathbb {R}^{2n}$ and $H_n$. We use the Tengstrand transform to compute the spherical distributions for $n=1$ explicitly.

Authors

  • Fernando LevsteinFacultad de Matemática Astronomía y Física
    Universidad Nacional de Córdoba
    Córdoba, Argentina
    e-mail
  • Linda SaalFacultad de Matemática Astronomía y Física
    Universidad Nacional de Córdoba
    Córdoba, Argentina

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