Intertwining of simple characters in GL(n)

Abstract: Let $F$ be a non-Archimedean local field and let $G$ be the general linear group $G = \text{\rm GL}_n(F)$. Let $\theta_1$, $\theta_2$ be simple characters in $G$. We show that $\theta_1$ intertwines with $\theta_2$ if and only if $\theta_1$ is endo-equivalent to $\theta_2$. We also show that any simple character in $G$ is a $G$-type.