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

Let K/F be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky’s classification asserts that generic representations are parabolically induced from quasi-squareintegrable representations. We show, following a method developed by Cogdell and PiatetskiShapiro, that expected equality of the Rankin-Selberg type Asai L-function of a generic representation of GL(n, K), and the Asai L-function of its 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.