We define spaces of entire functions in several variables that generalize the classical Paley–Wiener spaces. We introduce various notions of exponential type in $\mathbb{C}^d$ and require that their restrictions to a given submanifold are $L^p$-integrable with respect to a natural measure. A natural class of submanifolds arises in this setting; we call this class the Siegel CR submanifolds. For these spaces, we prove several basic properties. In particular, we prove a Paley–Wiener theorem in terms of the non-commutative Fourier transform on a nilpotent Lie group. the Plancherel–Pólya inequality, a Bernstein inequality, and a sufficient condition for a sequence to be sampling. This is a report on joint work with Mattia Calzi.