ON THE PRINCIPAL BLOCKS OF FINITE GENERAL LINEAR GROUPS IN NON-DEFINING CHARACTERISTIC

Let $k $ be a field of characteristic $\ell>0 $. In this note, we consider the $\ell$-modular representation of a finite general linear group $\mathrm{G}\mathrm{L}_{n}(q) $ with abelian Sylow $p$-subgroup of rank 2 where $q $ is a prime power which is not divided by $\ell $. We fix a positive integer $e$ such that $1<e<\ell $. Let $e(q) $ be the minimal $a>0 $ such that $\ell|q^{a}-1 $. Let $r(q) $ be th