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

## TL;DR

This paper investigates the structure of quasi-isolated blocks in finite groups of exceptional Lie type, establishing generalized $e$-Harish-Chandra theory to determine the number of irreducible Brauer characters.

## Contribution

It proves that generalized $e$-Harish-Chandra theory applies to Lusztig series of quasi-isolated elements, enabling enumeration of irreducible Brauer characters in these blocks.

## Key findings

- Number of irreducible Brauer characters in quasi-isolated $oldsymbol{ extit{ ext{l}}}$-blocks determined.
- Generalized $e$-Harish-Chandra theory validated for Lusztig series of quasi-isolated elements.
- Framework established for analyzing blocks of exceptional Lie type groups.

## 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 $G^{*F}$.