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 6= 2. The main result proves, if p ≥ 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 ∆(λ) has a ∆-filtration, namely, a filtration with sections of the form ∆(μ0 + pμ1) := L(μ0) ⊗ ∆(μ1), where μ0 is restricted and μ1 is arbitrary dominant. In case the highest weight λ of the Weyl module ∆(λ) is p-regular, the p-filtration is compatible with the G1-radical series of the module. The problem of showing that Weyl modules have ∆-filtrations was first proposed as a worthwhile (“wünschenswert”) problem in Jantzen’s 1980 Crelle paper.