Generic decompositions of Deligne--Lusztig representations

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–Hölder factors of general Deligne–Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]