On the exterior square $\epsilon$-factors of $G{L}_n$

Abstract

Let $\pi$ be a generic representation of ${\mathrm{GL}}_r(F)$, where $F$ is a $p$-adic field. We show using global methods that the $\epsilon$-factors associated to the exterior square of $\pi$ via the Jacquet-Shalika integrals and Langlands-Shahidi methods coincide. Along the way we also give a new proof of the local functional equation in the $p$-adic case following the techniques in [Beuzart-Plessis, Relative trace formulas, Simons Symp.(2021), pages 1-50].

19 pages