Distinction of some induced representations

TL;DR

This paper investigates when certain induced representations of general linear groups over quadratic extensions of p-adic fields are distinguished with respect to the base field, advancing the classification of such representations.

Contribution

It establishes that the normalized parabolic induction of the tensor product of a quasi-square-integrable representation and its Galois conjugate is distinguished, contributing to the classification of distinguished representations.

Findings

Induced representation is distinguished with respect to the base field.

Provides a criterion for distinction in terms of Galois conjugates.

Advances understanding of generic representations over p-adic fields.

Abstract

Let $K/F$ be a quadratic extension of $p$-adic fields, $\sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $\pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $\pi^{\vee}$ the smooth contragredient of $\pi$, and by $\pi^{\sigma}$ the representation $\pi\circ \sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $\pi^\vee \otimes \pi^\sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.