$p$-filtrations of dual Weyl modules

TL;DR

This paper proves that dual Weyl modules for semisimple algebraic groups over fields of characteristic p have p-filtrations when p is sufficiently large, with applications to higher filtrations, Steinberg modules, and the Donkin conjecture.

Contribution

It establishes the existence of p-filtrations for dual Weyl modules under certain conditions and explores several important applications in representation theory.

Findings

Dual Weyl modules have p-filtrations for sufficiently large p.

Applications to p^n-filtrations and modules with Steinberg tensor factors.

Progress on the Donkin conjecture regarding p-filtrations.

Abstract

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$-filtrations.