Modulation spaces on the Heisenberg group

TL;DR

This paper introduces and studies modulation spaces on the Heisenberg group using irreducible unitary representations, establishing their properties and invariance under specific transformations.

Contribution

It defines new modulation spaces on the Heisenberg group via irreducible unitary representations and explores their fundamental properties and invariance features.

Findings

Spaces are invariant under Heisenberg translations and modulations.

Established completeness and Fourier invariance of the modulation spaces.

Connected twisted modulation spaces to representations of nilpotent Lie groups.

Abstract

In this article we show how certain irreducible unitary representation $ \Pi_\lambda $ of the twisted Heisenberg group $ \He_\lambda^n(\C)$ leads to the twisted modulation spaces $ M_\lambda^{p,q}(\R^{2n}).$ These $ \Pi_\lambda $ also turn out to be irreducible unitary representations of another nilpotent Lie group $ G_n $ which contains two copies of the Heisenberg group $ \He^n.$ By lifting $ \Pi_\lambda $ we obtain another unitary representation $ \Pi $ of $ G_n $ acting on $ L^2(\He^n).$ We define our modulation spaces $ M^{p,q}(\He^n) $ in terms of the matrix coefficients associated to $ \Pi.$ 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_\lambda^{p,q}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms.