Distinction of some induced representations

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 $\Delta$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $\Delta^{\vee}$ the smooth contragredient of $\Delta$, and by $\Delta^{\sigma}$ the representation $\Delta\circ \sigma$, we show that representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $\Delta^\vee \otimes \Delta^\sigma$, 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.