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 \Phi_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 \Phi_p. First we show that \Phi_p has a canonical one sided inverse. Next we show that \Phi_0 =r\Phi_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_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.