### 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) |
---|---|

Pages (from-to) | 1231-1291 |

Number of pages | 61 |

Journal | Journal of the European Mathematical Society |

Volume | 19 |

Issue number | 4 |

DOIs | |

State | Published - 2017 |

### Fingerprint

### 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

**Adequate subgroups and indecomposable modules.** / Guralnick, Robert; Herzig, Florian; Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 19, no. 4, pp. 1231-1291. https://doi.org/10.4171/JEMS/692

}

TY - JOUR

T1 - Adequate subgroups and indecomposable modules

AU - Guralnick, Robert

AU - Herzig, Florian

AU - Tiep, Pham Huu

PY - 2017

Y1 - 2017

N2 - 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 SL2(pa) 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.

AB - 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 SL2(pa) 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.

KW - Adequate representations

KW - Artin-Wedderburn theorem

KW - Automorphic representations

KW - Complete reducibility

KW - Galois representations

KW - Indecomposable module

KW - Irreducible representations

UR - http://www.scopus.com/inward/record.url?scp=85016446537&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85016446537&partnerID=8YFLogxK

U2 - 10.4171/JEMS/692

DO - 10.4171/JEMS/692

M3 - Article

AN - SCOPUS:85016446537

VL - 19

SP - 1231

EP - 1291

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 4

ER -