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

Abstract

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 [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.