Modulation spaces on the Heisenberg group
In this article we show how certain irreducible unitary representation $ Π_λ$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $ M_λ^{p,q}(\R^{2n}).$ 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 $ \He^n.$ By lifting $ Π_λ$ we obtain another unitary representation $ Π$ 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 $ 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}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms.