Restricting Rational Modules to Frobenius Kernels

TL;DR

This paper explores when indecomposable modules for a connected reductive group of type A remain indecomposable or have simple socles when restricted to Frobenius kernels, using Schur functors to develop new methods.

Contribution

It introduces novel techniques using Schur functors to analyze indecomposability and socle simplicity of modules over Frobenius kernels for type A groups.

Findings

Identifies conditions for indecomposability over Frobenius kernels

Provides new results on Weyl, induced, and tilting modules

Develops methods applicable to modules in type A groups

Abstract

Let $G$ be a connected reductive group over an algebraically closed field of characteristic $p>0$. Given an indecomposable G-module $M$, one can ask when it remains indecomposable upon restriction to the Frobenius kernel $G_r$, and when its $G_r$-socle is simple (the latter being a strictly stronger condition than the former). In this paper, we investigate these questions for $G$ having an irreducible root system of type A. Using Schur functors and inverse Schur functors as our primary tools, we develop new methods of attacking these problems, and in the process obtain new results about classes of Weyl modules, induced modules, and tilting modules that remain indecomposable over $G_r$.