Level-raising and symmetric power functoriality, I

As the simplest case of Langlands functoriality, one expects the existence of the symmetric power Sⁿ(π ) , where π is an automorphic representation of GL(2,A) and A denotes the adeles of a number field F . This should be an automorphic representation of GL(N,A) ( $N=n+1)$ . This is known for $n=2,3$ and $4$ . In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume F totally real, and the initial representation π of classical type.