Characterization of γ-factors: the Asai case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai \gamma-factors are defined for smooth irreducible representations \pi of ${\rm GL}_n(E)$. If \sigma is the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to \pi under the local Langlands correspondence, we show that the Asai \gamma-factor is the same as the Deligne-Langlands \gamma-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from \sigma under tensor induction. This is achieved by proving that Asai \gamma-factors are characterized by their local properties together with their role in global functional equations for $L$-functions. As an immediate application, we establish the stability property of \gamma-factors under twists by highly ramified characters.