Characterization of \gamma-factors: the Asai case

TL;DR

This paper characterizes Asai b3-factors for representations of bca0groups over quadratic extensions in positive characteristic, linking them to Deligne-Langlands factors and proving their stability under certain twists.

Contribution

It establishes that Asai b3-factors coincide with Deligne-Langlands b3-factors via tensor induction and characterizes them through local and global properties.

Findings

Asai b3-factors match Deligne-Langlands b3-factors for bca0representations.

Characterization of Asai b3-factors by local properties and global functional equations.

Proved stability of b3-factors under twists by highly ramified characters.

Abstract

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.