### Abstract

The notion of adequate subgroups was introduced by Jack Thorne [60]. 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 [61], 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.

Original language | English (US) |
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Pages (from-to) | 1231-1291 |

Number of pages | 61 |

Journal | Journal of the European Mathematical Society |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2017 |

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### Keywords

- Adequate representations
- Artin-Wedderburn theorem
- Automorphic representations
- Complete reducibility
- Galois representations
- Indecomposable module
- Irreducible representations

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*19*(4), 1231-1291. https://doi.org/10.4171/JEMS/692