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

TL;DR

This paper classifies generic distinguished representations of GL(n,K) over p-adic fields and confirms the expected equality between Rankin-Selberg Asai L-functions and those of Langlands parameters.

Contribution

It provides a classification of generic distinguished representations of GL(n,K) in terms of quasi-square-integrable representations and verifies a key L-function equality.

Findings

Classification of generic distinguished representations achieved.

Confirmed the equality of Rankin-Selberg Asai L-functions and Langlands parameter L-functions.

Established a link between distinguished representations and Asai L-functions.

Abstract

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.