Operators preserving the representation of a semidirect product group

TL;DR

This paper characterizes operators that commute with the group actions of a semidirect product involving a locally compact abelian group and a lattice, focusing on shift-dilation preserving operators.

Contribution

It provides a new characterization of operators that preserve the representation of a semidirect product group acting on L^2 of a locally compact abelian group.

Findings

Operators commuting with the group action are characterized.

Shift-dilation preserving operators are explicitly described.

The work advances understanding of symmetries in harmonic analysis.

Abstract

For a locally compact abelian group $\textbf{R}$ with a uniform lattice $\Lambda$ and a group $G$ that acts on $\textbf{R}$ by continuous automorphisms, we study operators commuting with the representation of $G \rtimes \Lambda$ on $L^2(\textbf{R})$. As a consequence, we give a characterization of shift-dilation preserving operators.