Green functions of finite groups of Lie type provide an important input for computing character values of these groups.
Permutation characters from parabolic subgroups can be computed when all Green functions of centralizers of semisimple elements are known.
The other way round, we have computed values of such permutation characters by computations in the groups - and from this we could determine the Green functions in the last unknown cases (for exceptional groups of type E8(q)).