The gamma point represents waves with $k=0$, infinite wavelength. In the tight-binding approximation, this means a constant value of the phase factor for the atomic orbitals.
Germanium is not an easy example. It has an indirect band gap. But many salts have direct band gaps at the gamma point, for example MgO. In a tight-binding picture, at $\Gamma$ the phase factors are the same throughout the crystal. The lower band corresponds to electron wave functions with mostly oxygen $2p$ character, the upper band to wave functions with mainly magnesium $3sp$ character.
When waves have the same periodicity as the lattice in some direction, this will give rise to standing waves. There will be a gap at the zone boundary between waves with the same $k$ but locations of nodes and antinodes interchanged. So then there is a direction involved: along cubic directions for gaps at the X point, along the body diagonals for gaps at the L point. This may be relevant for surface properties at different cuts (image states on silver single crystals come to mind), but it does not affect bulk phenomena. And there may be complications due to surface reconstruction. Is this about a photoemission experiment?