On $p$-filtrations of Weyl modules

This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p≠2$. The main result proves, if $p\ge 2h-2$ (where $h$ is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) regular restricted highest weights, then any Weyl module $\mathrm{\Delta }(\lambda )$ has a ${\mathrm{\Delta }}^p$-filtration, namely, a filtration with sections of the form ${\mathrm{\Delta }}^p({\mu }_0+p{\mu }_1):=L({\mu }_0)\otimes \mathrm{\Delta }({\mu }_1{)}^{[1]}$, where ${\mu }_0$ is restricted and ${\mu }_1$ is arbitrary dominant. In case the highest weight $\lambda$ of the Weyl module $\mathrm{\Delta }(\lambda )$ is $p$-regular, the $p$-filtration is compatible with the $G_1$-radical series of the module. The problem of showing that Weyl modules have ${\mathrm{\Delta }}^p$-filtrations was first proposed as a worthwhile (“w"unschenswert”) problem in Jantzen’s 1980 Crelle paper.