On Liftings of Projective Indecomposable $G_{(1)}$-Modules

TL;DR

This paper investigates the conditions under which projective indecomposable modules for the Frobenius kernel of a simple algebraic group can be lifted to larger groups, extending known results and exploring subgroup cases.

Contribution

It advances the understanding of lifting PIMs by proving new results using extension methods and applies these to specific subgroup cases containing maximal tori.

Findings

PIMs lift to G when p ≥ 2h-2.

PIMs also lift to G_{(1)}H for certain subgroups H.

Extension techniques effectively analyze the lifting problem.

Abstract

Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p$, with Frobenius kernel $G_{(1)}$. It is known that when $p\ge 2h-2$, where $h$ is the Coxeter number of $G$, the projective indecomposable $G_{(1)}$-modules (PIMs) lift to $G$, and this has been conjectured to hold in all characteristics. In this paper, we explore the lifting problem via extensions of algebraic groups, following the work of Parshall and Scott who in turn build upon ideas due to Donkin. We prove various results which augment this approach, and as an application demonstrate that the PIMs lift to $G_{(1)}H$, for particular closed subgroups $H \le G$ which contain a maximal torus of $G$.