# A relation between automorphic forms on GL(2) and GL(3)

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## Abstract

Let ρn denote the standard n-dimensional representation of GL(n,C) and ρn2 its symmetric square. For each automorphic cuspidal representation π of GL(2,A) we introduce an Euler product L(s,π,ρ22) of degree 3 which we prove is entire. We also prove that there exists an automorphic representation II of GL(3)—“the lift of π”—with the property that L(s,II,ρ3) = L(s,π,ρ22). Our results confirm conjectures described in a more general context by R. P. Langlands [(1970) Lecture Notes in Mathematics, no. 170 (Springer-Verlag, Berlin-Heidelberg-New York)].

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