Reproducing formulas associated to translation generated systems on Nilpotent Lie Groups

Let $G$ be a connected, simply connected, nilpotent Lie group whose irreducible unitary representations are square-integrable modulo the center. We obtain characterization results for reproducing formulas associated with the left translation generated systems in $L^2(G)$. Unlike the previous study of discrete frames on the nilpotent Lie groups, the current research occurs within the set up of continuous frames, which means the resulting reproducing formulas are given in terms of integral representations instead of discrete sums. As a consequence of our results for the Heisenberg group, a reproducing formula associated with the orthonormal Gabor systems of $L^2(\mathbb R^d)$ is obtained.