On $p$-filtrations of Weyl modules

TL;DR

This paper proves that Weyl modules for certain algebraic groups over fields of positive characteristic have a specific type of filtration, under conditions on the characteristic and Lusztig's conjecture, advancing understanding of their structure.

Contribution

It establishes the existence of $ ext{Δ}^p$-filtrations for Weyl modules when the characteristic is sufficiently large and Lusztig's character formula holds, linking module filtrations to representation theory conjectures.

Findings

Weyl modules admit $ ext{Δ}^p$-filtrations under certain conditions.

The $ ext{Δ}^p$-filtration aligns with the $G_1$-radical series for $p$-regular weights.

The result confirms a longstanding problem proposed by Jantzen in 1980.

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 $\Delta(\lambda)$ has a $\Delta^p$-filtration, namely, a filtration with sections of the form $\Delta^p(\mu_0+p\mu_1):=L(\mu_0)\otimes\Delta(\mu_1)^{[1]}$, where $\mu_0$ is restricted and $\mu_1$ is arbitrary dominant. In case the highest weight $\lambda$ of the Weyl module $\Delta(\lambda)$ 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 $\Delta^p$-filtrations was first proposed as a worthwhile ("w\"unschenswert") problem in Jantzen's 1980 Crelle paper.