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 $\pi$ 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.

An important mistake in the previous version has been corrected here