On the Schwartz correspondence for Gelfand pairs of polynomial growth

TL;DR

This paper investigates the Schwartz correspondence property for Gelfand pairs of polynomial growth, extending known results from nilpotent cases to more general pairs, and establishing foundational settings for strong Gelfand pairs.

Contribution

It broadens the analysis of property (S) beyond nilpotent Gelfand pairs, providing a framework for understanding the Schwartz transform in more general polynomial growth contexts.

Findings

Property (S) holds for many nilpotent Gelfand pairs.

The paper establishes a basic setting for general polynomial growth Gelfand pairs.

Focuses on strong Gelfand pairs to extend existing results.

Abstract

Let $(G,K)$ be a Gelfand pair, with $G$ a Lie group of polynomial growth, and let $\Sigma\subset{\mathbb R}^\ell$ be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system $D_1,\dots,D_\ell$ of $G$-invariant differential operators on $G/K$ and associating to a bounded spherical function $\varphi$ the $\ell$-tuple of its eigenvalues under the action of the $D_j$'s. We say that property (S) holds for $(G,K)$ if the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto ${\mathcal S}(\Sigma)$, the space of restrictions to $\Sigma$ of the Schwartz functions on ${\mathbb R}^\ell$. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where $G=K\ltimes N$, with $N$ nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs.