Characterization of γ-factors: the Asai case

Let $E$ be a separable quadratic extension of a locally compact field $F$ of positive characteristic. Asai γ-factors are defined for smooth irreducible representations πof ${\rm GL}_n(E)$. If σis the Weil-Deligne representation of $\mathcal{W}_E$ corresponding to πunder the local Langlands correspondence, we show that the Asai γ-factor is the same as the Deligne-Langlands γ-factor of the Weil-Deligne representation of $\mathcal{W}_F$ obtained from σunder tensor induction. This is achieved by proving that Asai γ-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 γ-factors under twists by highly ramified characters.

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