  Windowing a Fourier transform is a useful tool, which gives us the similarity
between the signal and time frequency signal, and it allows to get sense
when/where ceratin frequencies occur in the input signal, this method is
introduced by Dennis Gabor. In this paper, we generalize the classical
Gabor-Fourier transform(GFT) to the Riemannian symmetric space called the
Helgason Gabor Fourier transform (HGFT). We continue with proving several
important properties of HGFT, like the reconstruction formula, the Plancherel
formula, and Parseval formula. Finally we establish some local uncertainty
principle such as Benedicks-type uncertainty principle
