Appendix A: Adequacy of representations of finite groups of Lie type

Abstract

Thorne introduced the notion of adequate representations as a weakening of the big representations used by Wiles and Taylor and others. In this appendix to Dieulefait's paper, Automorphy of Symm5(GL(2)) and base change, we show that certain representations of SL(2,q) are adequate. This is used by Dieulefait to prove results about Hecke eigenforms of level 1 and newforms. We also prove some general results about adequacy for representations of finite groups of Lie type in the natural characteristic.