### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 065, 21 pages      arXiv:2105.13811      https://doi.org/10.3842/SIGMA.2022.065

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

Amerah A. Al Ameer a and Vladimir V. Kisil b
a) School of Science, Mathematics Department, University of Hafr Al Batin, Hafr Al Batin 31991 P.O Box 1803, Saudi Arabia
b) School of Mathematics, University of Leeds, Leeds LS29JT, UK

Received December 26, 2021, in final form August 26, 2022; Published online September 01, 2022

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 Schrödinger 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.

Key words: Heisenberg group; covariant transform; coherent states; Zak transform; Fock-Segal-Bargmann space.

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