Harmonic analysis on some generalized Gelfand pairs attached to Heisenberg groups
Volume 241 / 2018
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.
