Adequate subgroups and indecomposable modules

TL;DR

This paper extends the concept of adequate subgroups to all absolutely irreducible modules of dimension p in characteristic p, proving adequacy for certain groups and classifying indecomposable modules of low dimension.

Contribution

It generalizes the notion of adequacy to modules where p divides the dimension and classifies indecomposable modules of small dimension in characteristic p.

Findings

Adequacy holds for almost all irreducible representations of SL_2(p^a).

Classifies indecomposable modules of dimension less than 2p-2.

Answers Serre's question on complete reducibility in low-dimensional classical groups.

Abstract

The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL_2(p^a) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p-2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.