Non-big subgroups for l large

Lifting theorems form an important collection of tools in showing that Galois representations are associated to automorphic forms. (Key examples in dimension n>2 are the lifting theorems of Clozel, Harris and Taylor and of Geraghty.) All present lifting theorems for n>2 dimensional representations have a certain rather technical hypothesis---the residual image must be `big'. The aim of this paper is to demystify this condition somewhat. For a fixed integer n, and a prime l larger than a constant depending on n, we show that n dimensional mod l representations which fail to be big must be of one of three kinds: they either fail to be absolutely irreducible, are induced from representations of larger fields, or can be written as a tensor product including a factor which is the reduction of an Artin representation in characteristic zero. Hopefully this characterization will make the bigness condition more comprehensible, at least for large l.