Holomorphic extensions of eigenfunctions on $NA$ groups

Let $X = G/K$ be a rank one Riemannian symmetric space of noncompact type. In view of the Iwasawa decomposition $G = NAK$ of the underlying semisimple Lie group, we can also view $X$ as the solvable extension $S = NA$ of the Iwasawa group $N$ which is known to be a $H$-type group. In this work we study the holomorphic extendability of eigenfunctions of the Laplace-Beltrami operator $\Delta_S$ on $S$ to certain domains in the complexification of the nilpotent group $N$. We can also do the same for any $H$-type group $N$ not necessarily an Iwasawa group. The results are accomplished by making use of the connection with solutions of the extension problem for the Laplacian or the sublaplacian on the corresponding $N$.