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 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.
