Local-to-global extensions for GL_n in non-zero characteristic: A characterization of γ_f(s, π, Sym², Ψ) and γ_f(s, π, ∇², Ψ)

Let F be a locally compact field of positive characteristic. In his thesis, the second author used the Langlands-Shahidi method to define γ-factors for the symmetric square and exterior square representations of the dual group of GL_n(F). We prove here that such γ-factors are characterized by local properties - including a multiplicativity property with respect to parabolic induction - and their role in global functional equations for L-functions. For this, we prove that a given cuspidal representation of GL _n(F) is the component at some place of a global automorphic representation, the ramification of which is controlled at other places. In fact, we use an equivalent result for l-adic Galois representations, due to O. Gabber and N. Katz, and transport it via the global Langlands correspondence proved by L. Lafforgue.