Generic decompositions of Deligne--Lusztig representations

TL;DR

This paper extends the understanding of the decomposition patterns of Deligne--Lusztig representations for certain reductive groups over finite fields, providing optimal genericity bounds and insights into their Jordan--H"older factors.

Contribution

It generalizes Jantzen's generic decomposition pattern to a broader range of Deligne--Lusztig representations, achieving optimal bounds and analyzing their Jordan--H"older factors.

Findings

Extended decomposition pattern from (2h-1)-generic to h-generic representations.

Proved results on Jordan--H"older factors of Deligne--Lusztig representations.

Improved weight elimination results in related modular representation theory.

Abstract

Let $G_0$ be a reductive group over $\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--H\"older factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]