Langlands functoriality predicts the transfer of automorphic representations along maps of L-groups. In particular, the symmetric power representation Symm^{n-1} of GL(2) should give rise to a lifting from automorphic representations of GL(2) to automorphic representations of GL(n). I will discuss joint work with Jack Thorne, in which we prove the existence of all symmetric power lifts for many cuspidal Hecke eigenforms (for example, those of square-free level).  

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