On $p$-filtrations of Weyl modules

Abstract

This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p\not=2$. The main result proves, if $p\geq 2h-2$ (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 $Δ(λ)$ has a $Δ^p$-filtration, namely, a filtration with sections of the form $Δ^p(μ_0+pμ_1):=L(μ_0)\otimesΔ(μ_1)^{[1]}$, where $μ_0$ is restricted and $μ_1$ is arbitrary dominant. In case the highest weight $λ$ of the Weyl module $Δ(λ)$ 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 $Δ^p$-filtrations was first proposed as a worthwhile ("wünschenswert") problem in Jantzen's 1980 Crelle paper.