Generalized Green functions and unipotent classes for finite reductive groups, I

Abstract

The algorithm of computing generalized Green functions of a reductive group $G$ contains some unknown scalars occurring from the $\mathbf{F}_{q}$-structure of irreducible local systems on unipotent classes of $G$. In this paper, we determine such scalars in the case where $G = SL_{n}$ with Frobenius map $F$ of split type or non-split type. In the case where $F$ is of non-split type, we use the theory of graded Hecke algebras due to Lusztig.