Distinguished representations and exceptional poles of the Asai-L-function

TL;DR

This paper establishes a link between distinguished representations of GL(n,K) over p-adic fields and the presence of an exceptional pole at zero in their Asai L-functions, extending previous criteria and providing explicit computations.

Contribution

It proves the equivalence between distinguished representations and exceptional poles of Asai L-functions, extending Kable's criterion, and computes Asai L-functions for specific GL(2,K) representations.

Findings

Equivalence between distinguished representations and exceptional poles at zero.

Explicit formulas for Asai L-functions of principal series of GL(2,K).

Extension of Kable's criterion to broader classes of representations.

Abstract

In this article, we proove that it is equivalent for a generic irreducible representation of GL(n,K), with K a p-adic field, to be distinguished, and for its Rankin-Selberg Asai L-function to have an exceptional pole at zero. This extends a criterion of Kable claiming that a discrete series representation is distinguished if and only if its Asai L-function has a pole at zero. As an application we compute by local methods Asai L-functions of ordinary representations of GL(2,K), in particular we give a formula for Asai L-functions of principal series representations of GL(2,K).