The irreducible complex representations of nite groups of Lie type are divided into series by means of the Harish-Chandra philosophy, whose starting point is the concept of cuspidal characters (see e.g..Sr]). The classiication of the irreducible characters of such a group G can be done in two steps: First nd all cuspidal irreducible characters of Levi subgroups L of G. Then apply Harish-Chandra...

We introduce a basis of the symmetric functions that evaluates to (irreducible) characters group, just as Schur evaluate irreducible G L n modules. Our main result gives three different characterizations for this basis. One shows structure coefficients (outer) product these are stable Kronecker coefficients. The results in paper focus on developing fundamental properties