Regularity of Unipotent Elements in Total Positivity

TL;DR

This paper proves that all unipotent elements in the totally nonnegative part of a split reductive group are regular in some Levi subgroup, confirming a conjecture by Lusztig.

Contribution

It establishes a key property of unipotent elements in total positivity, confirming Lusztig's conjecture for split reductive groups.

Findings

Unipotent elements are regular in some Levi subgroup.

Confirms Lusztig's conjecture on total positivity.

Advances understanding of structure in reductive groups.

Abstract

Let $G$ be a connected reductive group split over R. We show that every unipotent element in the totally nonnegative monoid of G is regular in some Levi subgroups, confirming a conjecture of Lusztig.