Conjectures about distinction and Asai $L$-functions of generic representations of general linear groups over local fields
Résumé
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-square-integrable representations. Assuming a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representations, we show, following a method developed by Cogdell and Piatetski-Shapiro, that the Rankin-Selberg type Asai L-function of a generic representation of $GL(n,K)$, is equal to the canonical Asai $L$-function of the Langlands parameter. As the conjecture is true for principal series representations, this gives the expression of the Asai L-function of such representations.
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