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

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map Φ_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 about Φ_p. First we show that Φ_p has a canonical one sided inverse. Next we show that Φ_0 =rΦ_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 Φ_0 with another natural map; we show that this map depends only on the Coxeter group structure of W. We also define the special conjugacy classes in W (in 1-1 correspondence with the special representations of W) and describe them explicitly for each simple type.