Functoriality for the Exterior Square of Gl4 and the Symmetric Fourth of Gl2

Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic representations of GLN , which satisfies certain canonical properties. To explain, let F be a number field, and let A be its ring of adeles. Let π = ⊗ v πv be a cuspidal (automorphic) representation of GLn(A). In what follows, a cuspidal representation always means a unitary one. Now by the local Langlands correspondence, ∧πv is well defined as an irreducible admissible representation of GLN (Fv) for all v (the work of Harris-Taylor [H-T] and Henniart [He2] on p-adic places and of Langlands [La4] on archimedean places). Let ∧π = ⊗ v ∧πv. It is an irreducible admissible representation of GLN (A). Then Langlands’ functoriality in this case is equivalent to the fact that ∧π is automorphic. Note that ∧(GL2(C)) ' GL1(C) and in fact for a cuspidal representation π of GL2(A), ∧π = ωπ, the central character of π. Furthermore, ∧(GL3(C)) ' GL3(C). In this case, given a cuspidal representation π of GL3(A), ∧π = π̃ ⊗ ωπ, where π̃ is the contragredient of π. In this paper, we look at the case n = 4. Let π = ⊗ v πv be a cuspidal representation of GL4(A). What we prove is weaker than the automorphy of ∧π. We prove (Theorem 5.3.1)