Weyl-Zak Transforms and Twisted Translations: A Representation Viewpoint on LCA Groups

Esmaeelzadeh, Fatemeh; Kamyabi Gol, Rajab Ali

doi:10.30504/jims.2026.572575.1317

Weyl-Zak Transforms and Twisted Translations: A Representation Viewpoint on LCA Groups

Articles in Press

Document Type : Research Article

Authors

Fatemeh Esmaeelzadeh ¹

Rajab Ali Kamyabi Gol ²

¹ Islamic Azad University, Bojnord Branch

² Ferdowsi University of Mashhad

10.30504/jims.2026.572575.1317

Abstract

This paper presents a unified approach to the Weyl-Zak transform on a locally compact abelian (LCA) group $G$, grounded in projective representation theory of the time-frequency plane $G \times \widehat{G}$. We define a projective representation and a twisted translation on this plane and prove their central intertwining relation with the Weyl transform. Using a uniform lattice $L \subset G$ and its annihilator $L^\perp \subset \widehat{G}$, we derive a generalized Weyl-Zak transform and show it is unitary from $L^2(G \times \widehat{G})$ to $L^2(S_L \times S_{L^\perp} \times G)$, where $S_L$ and $S_{L^\perp}$ are fundamental domains for $L$ and $L^\perp$. This provides a cohesive, representation-theoretic foundation beyond the Euclidean setting. Finally, we introduce a bracket product based on the Weyl-Zak transform and demonstrate its essential properties regarding twisted translations.

Keywords

projective representation&lrm

twisted translates&lrm

Weyl transforms&lrm

, &lrm

Weyl-Zak transforms

Subjects