Conjectures about distinction and Asai $L$-functions of generic representations of general linear groups over local fields

Let $K\mathrm{/}F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky’s classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method developed by Cogdell and Piatetski-Shapiro, that the equality of the Rankin-Selberg type Asai $L$-function of generic representations of $GL(n,K)$ and of the Asai $L$-function of the Langlands parameter, is equivalent to the truth of a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representations. As the conjecture is true for principal series representations, this gives the expression of the Asai L-function of such representations.