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

TL;DR

This paper investigates the order relations between conjugacy classes in the Weyl group and unipotent classes in a reductive group, revealing that Lusztig's map is order-reversing on elliptic classes.

Contribution

It proves that Lusztig's map from elliptic conjugacy classes in the Weyl group to unipotent classes is order-reversing, connecting combinatorial and geometric partial orders.

Findings

Lusztig's map is injective on elliptic classes.

The map is order-reversing with respect to natural partial orders.

Establishes a new relationship between combinatorial and geometric structures.

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.