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 π of GLn(E). If σ is the Weil–Deligne representation of $\mathcal W_E$ corresponding to π under 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 σ under 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.