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

Jeffrey Adams; Xuhua He; Sian Nie

paper

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

arXiv:2004.01337

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 $[G_{u}]$ 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.

25 pages