  Using linear periods on the mirabolic subgroup of $GL(n,F)$, for $F$ a non
archimedean local field, we give a list of the maximal Levi subgroups of
$GL(n,F)$ which can distinguish a discrete series, and a generic
representation. We also obtain the functional equation of the local
exterior-square $L$-function of a generic representation of $GL(n,F)$ when
$n=2m$ is even. Then we discuss the relation between Shalika models and local
models for representations of $GL(2m,F)$. Finally we give a necessary and
sufficient condition on the cuspidal representation $\rho$ for a generalised
Steinberg representation $St_k(\rho)$ of $GL(2m,F)$ to have a local model.
