Galois self-dual cuspidal types and Asai local factors

Abstract

Let $F/F_{\mathsf{o}}$ 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 ${\mathrm{GL}}_n(F)$ 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 ${\mathrm{GL}}_n(F_{\mathsf{o}})$-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.

arXiv comment61 pages, final version to appear in JEMS

References in corpus (1)

• On automorphic L-functions in positive characteristic

Cited by in corpus (5)

• Test vectors for finite periods and base change
• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by a unitary involution
• Characterisation of the poles of the $\ell$-modular Asai $L$-factor
• Distinction and Base Change
• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by an orthogonal involution