Galois self-dual cuspidal types and Asai local factors

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