On some exotic finite subgroups of E ₈ and Springer’s regular elements of the Weyl group

Abstract

A proof (by Serre and by Cohen, Griess and Lisser) verified, in the special case of E ₈, a conjecture of mine, that the finite projective group L ₂(61) embeds in ${E_8}\left( \mathbb{C} \right)$. Subsequently, Griess and Ryba have shown (using computers) that L ₂(49) and, in addition, (established by Serre without computers) L ₂(41) also embed in ${E_8}\left( \mathbb{C} \right)$. That is, if K = 30, 24, 20 and k ∈ K then L ₂(2k + 1) embeds in ${E_8}\left( \mathbb{C} \right)$. In this paper we show that the “Borel” subgroup B(k) of L ₂(2k + 1), k ∈ K, has a uniform construction. The theorem uses a result of T. Springer on the existence in ${E_8}\left( \mathbb{C} \right)$ of three regular elements of the Weyl group, having orders k ∈ K, and associated to the regular, subregular and subsubregular nilpotent elements. Springer’s result generalizes (in the E ₈ case) a 1959 general result of mine relating the principal nilpotent element with the Coxeter element.