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 $\Phi _{p}$ 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 $\Phi _{p}$. 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 and we show that this map depends only on the Coxeter group structure of $W$.