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

Abstract

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 $\ell \nmid q$ be a good prime for $G$. We determine the number of irreducible Brauer characters in the quasi-isolated $\ell$-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 $G^{\ast F}$.

Rewritten using new results of Geck-Malle, fixed mistakes and added more information in the tables