Modulation spaces on the Heisenberg group

2025/01/22 by Md Hasan Ali Biswas, Sundaram Thangavelu, Biswas, Md Hasan Ali +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2501.12845

openalex publication_date 2025/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we show how certain irreducible unitary representation Π_λ of the twisted Heisenberg group \He_λⁿ(\C) leads to the twisted modulation spaces M_λ^p,q(\R²ⁿ). These Π_λ also turn out to be irreducible unitary representations of another nilpotent Lie group G_n which contains two copies of the Heisenberg group \Heⁿ. By lifting Π_λ we obtain another unitary representation Π of G_n acting on L²(\Heⁿ). We define our modulation spaces M^p,q(\Heⁿ) in terms of the matrix coefficients associated to Π. These spaces are shown to be invariant under Heisenberg translations and Heisenberg modulations which are different from euclidean modulations. We also establish some of the basic properties of M_λ^p,q(\R²ⁿ) and M^p,q(\Heⁿ) such as completeness and invariance under suitable Fourier transforms.

Related

• Heisenberg-Modulation Spaces at the Crossroads of Coorbit Theory and Decomposition Space Theory
• Algebras of entire functions and representations of the twisted Heisenberg group
• Modulation spaces of symbols for representations of nilpotent Lie groups
• Fourier transform, Schrödinger representation, and Heisenberg modules
• Shift-invariant Spaces with Countably Many Mutually Orthogonal Generators on the Heisenberg group