Cuspidal representations of $GL(n,F)$ distinguished by a maximal Levi subgroup, with $F$ a non-archimedean local field

Abstract

Let $ρ$ is a cuspidal representation of $GL(n,F)$, with $F$ a non archimedean local field, and $H$ a maximal Levi subgroup of $GL(n,F)$. We show that if $ρ$ is $H$-distinguished, then $n$ is even, and $H\simeq GL(n/2,F)\times GL(n/2,F)$.