The Bruhat order on abelian ideals of Borel subalgebras

TL;DR

This paper studies the structure of Borel subgroup orbits on abelian subalgebras within the nilradical of a Borel subalgebra in algebraic groups, providing a parametrization and closure relations.

Contribution

It introduces a parametrization of B-orbits on B-stable abelian subalgebras of the nilradical and describes their closure relations, advancing understanding of the Bruhat order in this context.

Findings

Parametrization of B-orbits on abelian subalgebras

Description of closure relations among orbits

Insights into the Bruhat order structure

Abstract

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations.