Schwartz correspondence for real motion groups in low dimensions

Abstract

For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$ a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi-$K$-invariant Schwartz space $\mathcal{S}(K\backslash G/K)$ isomorphically onto the space $\mathcal{S}({\Sigma }_{\mathcal{D}})$, where ${\Sigma }_{\mathcal{D}}$ is an embedded copy of the Gelfand spectrum in ${\mathbb{R}}^{\ell }$, canonically associated to a generating system $\mathcal{D}$ of $G$-invariant differential operators on $G/K$, and $\mathcal{S}({\Sigma }_{\mathcal{D}})$ consists of restrictions to ${\Sigma }_{\mathcal{D}}$ of Schwartz functions on ${\mathbb{R}}^{\ell }$. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair $(M_n,S{O}_n)$ with $n=3,4$. The rather trivial case $n=2$ is included in previous work by the same authors.