Galois self-dual cuspidal types and Asai local factors

Abstract

Let $\mathrm{F}/{\mathrm{F}}_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $\sigma$ be its non-trivial automorphism. We show that any $\sigma$-self-dual cuspidal representation of ${\mathrm{GL}}_n(\mathrm{F})$ contains a $\sigma$-self-dual Bushnell–Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a ${\mathrm{GL}}_n({\mathrm{F}}_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands–Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.