Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_{3,2}

TL;DR

This paper characterizes the Gelfand transforms of SO(3)-invariant Schwartz functions on a specific free nilpotent group, extending known results from the Heisenberg group to a broader class of nilpotent groups.

Contribution

It proves that Schwartz functions on the spectrum correspond to Gelfand transforms of invariant Schwartz functions on the group, specifically for the pair involving the free two-step nilpotent group N_{3,2}.

Findings

Spectrum of the Gelfand pair embeds in Euclidean space.

Schwartz functions on the spectrum are Gelfand transforms of invariant Schwartz functions.

Extension of results from the Heisenberg group to N_{3,2}.

Abstract

The spectrum of a Gelfand pair $(K\ltimes N, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group.