Local-to-global extensions for GL n in non-zero characteristic: a characterization of γ F ( s , π, Sym 2 , ψ) and γ F ( s , π, ∧ 2 , ψ)

Abstract

Let $F$ be a locally compact field of positive characteristic. In his thesis, the second author used the Langlands-Shahidi method to define $\gamma$-factors for the symmetric square and exterior square representations of the dual group of ${\rm GL}_n(F)$. We prove here that such $\gamma$-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 ${\rm GL}_n(F)$ 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 $\ell$-adic Galois representations, due to O.~Gabber and N.~Katz, and transport it via the global Langlands correspondence proved by L.~Lafforgue.