Level-raising and symmetric power functoriality, I

As the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. 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 $\pi $ of classical type.