Kirillov models and integral structures in p-adic smooth representations of GL2(F)

Let F be a local field of residual characteristic p, and ρ a smooth irreducible representation of GL2(F), realized over the algebraic closure of Qp. Studying its Kirillov model, we exhibit a necessary and sufficient criterion for the existence of an integral structure in ρ. We apply our criterion to tamely ramified principal series, and get a new proof of a theorem of M.-F. Vigneras.