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 GL_n(E). If σ is the Weil-Deligne representation of 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 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.