Modulation spaces on the Heisenberg group

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.