Linear and Shalika local periods for the mirabolic group, and some consequences

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