Classification of the root systems $R(m)$

Abstract: Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple of $m$. In a paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems $R(m)$; in particular, they introduced a constant $d_m$ which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to $R(m)$ and equals $1$ if the roots of height $m$ form a base of $R(m)$, and proved this property when $R$ is of type $A$ or $C$, and also in type $B$ if $m$ is odd. In this paper, we complete their analysis by determining a base of $R(m)$ and computing the constant $d_m$ in all cases.