Parabolic subgroup orbits on finite root systems

TL;DR

This paper generalizes Oshima's Lemma to all finite Coxeter group root systems, providing a new proof that does not rely on Lie algebra representation theory.

Contribution

It extends the classification of parabolic subgroup orbits from Weyl groups to all finite Coxeter groups with an independent proof.

Findings

Generalization of Oshima's Lemma to all finite Coxeter systems

A self-contained proof independent of Lie algebra representation theory

Broader understanding of parabolic subgroup orbits in finite Coxeter groups

Abstract

Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.