Modulation spaces on the Heisenberg group

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