Distinction and Asai $L$-functions for generic representations of general linear groups over p-adic fields

Nadir Matringe (1)

1 UEA, School of Mathematics.

Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form. In the present work, we classify genric distinguished representations of the group $GL(n,K)$ in terms of inducing quasi-square-integrable representations. This has as a consequence the truth of the expected equality between the Rankin-Selberg type Asai $L$-function of a generic representation, and the Asai $L$-function of its Langlands parameter.

Mots clés: Distinguished representations; Asai $L$-functions

Domaines: Théorie des représentations [math.RT]

Nadir Matringe. Distinction and Asai $L$-functions for generic representations of general linear groups over p-adic fields. 2009. ⟨hal-00362334v5⟩