On $e$-cuspidal pairs of finite groups of exceptional Lie Type

Let $G$ be a simple, simply connected algebraic group of exceptional type defined over ${\mathbb{F}}_q$ with Frobenius endomorphism $F:G\to G$. Let $\ell \nmid q$beagoodprimefor$G$. We determine the number of irreducible Brauer characters in the quasi-isolated $\mathrm{\ell }$-blocks of $G^F$. This is done by proving that generalized $e$-Harish-Chandra theory holds for the Lusztig series associated to quasi-isolated elements of $G^{\ast F}$.