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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 $[G_u]$ arisingfrom geometry.

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes