From conjugacy classes in the Weyl group to unipotent classes, II

Abstract

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map Phi[subscript p] from the set [underline W] of conjugacy classes in the Weyl group W to the set of unipotent classes in G. Here we prove three results for Phi[subscript p]. First we show that Phi[subscript p] has a canonical one-sided inverse. Next we show that Phi[subscript 0]=r Phi[subscript p for a unique map r. Finally, we construct a natural surjective map from [underline W] to the set of special representations of W which is the composition of Phi[subscript 0] with another natural map and we show that this map depends only on the Coxeter group structure of W.