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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• Main Authors: Al Ameer, Amerah A., Kisil, Vladimir V.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211723
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration. Amerah A. Al Ameer and Vladimir V. Kisil. SIGMA 18 (2022), 065, 21 pages

We discuss a fine-tuning of the co- and contra-variant transforms through the construction of specific fiducial and reconstruction 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 ₂(ℝ) and the lattice (nilmanifold) representation on ₂(²). 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 that can be imposed on the reconstructing vector to obtain an intertwining operator from the induced contravariant transform.