  Let $G$ be a reductive group over an algebraically closed field and let $W$
be its Weyl group. In a series of papers, Lusztig introduced a map from the set
$[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of
$G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$
of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to
[G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$
arising from combinatorics and the natural partial order on $[G_u]$ arising
from geometry.
