On Lusztig's map for spherical unipotent conjugacy classes

TL;DR

This paper offers a new perspective on Lusztig's map for spherical unipotent conjugacy classes, analyzing its image and properties in relation to Weyl group conjugacy classes, with implications for understanding their structure.

Contribution

It provides an alternative description of Lusztig's map restriction and characterizes the maximal elements in conjugacy classes of finite Weyl groups.

Findings

Unique maximal length element exists if and only if the conjugacy class has a maximum.

Analysis of the image of Lusztig's map for irreducible root systems.

New insights into the structure of spherical unipotent conjugacy classes.

Abstract

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.