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

摘要

Using linear periods on the mirabolic subgroup of GL(n,F), for F a nonarchimedean local field, we give a list of the maximal Levi subgroups of 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) 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 p for a generalised Steinberg representation St_k(p) of GL(2m, F) to have a local model.