Galois self-dual cuspidal types and Asai local factors

Let F=Fo be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and σ be its non-trivial automorphism. We show that any σ-self-dual cuspidal representation of GLn.F/ contains a σ-self-dual Bushnell-Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a GLn.Fo/-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.