Some Properties of Character Sheaves

In 1986, George Lusztig stated some interesting properties of character sheaves with non-zero restriction to the unipotent variety and proved them for the group SO2n+1(Fq) under the assumption that the characteristic p of Fq is good. In February 1992, the author has given lectures on that proof of Lusztig in the Seminar “Faisceaux Caractères” at the École Normale Supérieure and it was no doubt for her that the same method should work for the other types of groups (both classical and exceptional), as it has been already noticed by Lusztig. It is one purpose of this paper to show that the question can be reduced to statements uniquely involving representations of some Coxeter groups. In particular we shall show that the method employed by Lusztig for the group SO2n+1(Fq) provides a general process (based on both “parabolic induction” of character sheaves and generalized Springer correspondence) for associating a well-defined unipotent class to any character sheaf which has non-zero restriction to the unipotent variety.