Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

TL;DR

This paper explores the fine-tuning of co- and contra-variant transforms within the Heisenberg group, illustrating their application through various induced representations and transforms like Zak and theta transforms.

Contribution

It introduces a method for fine-tuning covariant and contravariant transforms using specific vectors, with applications to multiple representations of the Heisenberg group.

Findings

Zak transform as an induced covariant transform

Connection between induced transforms and analytic function spaces

Conditions for reconstructing vectors to produce intertwining operators

Abstract

We discuss a fine tuning of the co- and contra-variant transforms through construction of specific fiducial and reconstructing vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operators between pairs of representations. In particular, we obtain the Zak transform as an induced covariant transform intertwining the Schr\"odinger representation on $\mathsf{L}_2(\mathbb{R})$ and the lattice (nilmanifold) representation on $\mathsf{L}_2\big(\mathbb{T}^2\big)$. Induced covariant transforms in other pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe peelings which map the group-theoretical induced representations to convenient representation spaces of analytic functions. Finally, we provide a condition which can be imposed on the reconstructing vector in order to obtain an intertwining operator from the induced contravariant transform.