Admissible and sectorial convergence of generalized Poisson integrals on Harmonic $NA$ groups

We prove a converse of Fatou type result for certain eigenfunctions of the Lalplace-Beltrami operator on Harmonic NA groups relating sectorial convergence and admissible convergence of Poisson type integrals of complex (signed) measures. This result extends several results of this kind proved eariler in the context of the classical upper half space ${\mathbb{R}}_{+}^{n+1}$. Similar results are also obtained in the degenerate case of the real hyperbolic spaces.