Generic weights for finite reductive groups

TL;DR

This paper introduces the concept of generic weights in finite reductive groups, aiming to advance the understanding of Alperin's weight conjecture by generalizing existing notions and establishing their role in representation theory.

Contribution

It generalizes $e$-cuspidality and defines generic weights in non-defining characteristic, linking them to Alperin's weights and the inductive Alperin weight condition.

Findings

Generic weights mirror Alperin's weights in simple groups of Lie type.

The approach provides a framework for progress towards proving Alperin's weight conjecture.

Establishes a connection between $e$-Harish-Chandra theory and weight conjectures.

Abstract

This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of $e$-cuspidality in the $e$-Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.