Generic decompositions of Deligne--Lusztig representations

Let G₀ be a reductive group over Fₚ 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