Generalized Green functions and unipotent classes for finite reductive   groups, I
  The algorithm of computing generalized Green functions of a finite reductive group contains some unkonwn scalars occuring from the F_q structure of irreducible local systems on unipotent classes on G. In this paper, we determine such scalars in the case where G = SL_n with Frobenius map F of split or non-split type. In the case where F is of non-split type, we use the theory of graded Hecke algebras and their representations on the equivariant homology group due to Lusztig. 
Toshiaki Shoji