Coxeter exponents and orthogonal complements of highest roots

Abstract

In this note, we study the subroot system of an irreducible (reduced, crystallographic) root system obtained by taking the orthogonal complement of the highest root. We describe root theoretic data (the Coxeter number and the largest coefficient of the highest root when expressed in terms of the simple roots) of this subroot system in terms of that of As an application, we obtain a new description of the Coxeter exponents of a complex simple Lie algebra.