On the exterior square $\varepsilon$-factors of $GL_n$

Abstract

Let $π$ be a generic representation of ${\mathrm{GL}}_r(F)$, where $F$ is a $p$-adic field. We show using global methods that the $\varepsilon$-factors associated to the exterior square of $π$ 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].