  Let H be a connected reductive group defined over a non-archimedean local
field F of characteristic p>0. Using Poincar\'e series, we globalize
supercuspidal representations of H(F) in such a way that we have control over
ramification at all other places, and such that the notion of distinction with
respect to a unipotent subgroup (indeed more general subgroups) is preserved.
In combination with the work of Vincent Lafforgue on the global Langlands
correspondence, we present some applications, such as the stability of
Langlands-Shahidi \gamma-factors and the local Langlands correspondence for
classical groups.
