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 ${\rm GL}_n(F)$ 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 ${\rm GL}_n(F_{\mathsf{o}})$-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.
Highlights
Let F/Fo be a quadratic extension of locally compact non-archimedean fields of odd residual characteristic p and let σ denote the non-trivial element of Gal(F/Fo)
Our proof of Proposition 1.7 is based on a result of Ok [41] proved for any irreducible complex representation of G, and which we prove for any cuspidal representation of G with coefficients in R in Appendix B
When π is in addition unramified, we prove that it is equal to the local Asai epsilon factor LASs(s, π, ψo) obtained via the Langlands–Shahidi method
Summary
Let F/Fo be a quadratic extension of locally compact non-archimedean fields of odd residual characteristic p and let σ denote the non-trivial element of Gal(F/Fo). Consider a cuspidal (irreducible, smooth, complex) representation π of G, and suppose that its Asai L-function LAs(s, π) is non-trivial. A σ-self-dual cuspidal representation π of G with coefficients in R is distinguished if and only if any generic σ-self-dual type (J, λ) contained in π is distinguished, that is, the space HomJ∩Gσ (λ, 1) is non-zero. Let π be a distinguished cuspidal representation of G, and (J, λ) be a generic σ-self-dual type contained in π. The starting point of both papers is the σ-self-dual type theorem for cuspidal R-representations, namely Theorem 1.5, which is proved in Section 4 below. For complex representations, in which case all cuspidal representations are supercuspidal, this implies the two results stated in Paragraph 7.1 (i.e. Theorem 7.1 and Proposition 7.2) which we use in the proof of Theorem 1.1. We use Proposition 5.8, which is proved in [47], for any σ-self-dual supercuspidal R-representation of G
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