## Abstract

Let *F* be a *p*-adic field with uniformizer *π*. Given a smooth mod *p*-representation of *G* = GL_{2}(*F*)/*π*, we consider its subspace of invariants under the action of the pro-*p*-Iwahori subgroup and get this way a functor taking values in the category of right modules over the pro-*p*-Hecke algebra with characteristic *p*. We show that if *F* = ℚ* _{p}*, this functor restricted to the representations generated by their pro-

*p*-invariants is an equivalence of categories, and give examples of other

*p*-adic fields for which it is not the case.

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