Multiplicity one for the mod p cohomology of Shimura curves: the tame case

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Let F be a totally real field, p an unramified place of F dividing p and r: Gal(F/F)₂(Fₚ) a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis associates to r an admissible smooth representation of GL₂(Fₚ) on the mod p cohomology of Shimura curves attached to indefinite division algebras which split at p. When r|Gal(Fₚ/Fₚ) is tamely ramified and generic (and under some technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level p. In particular, it depends only on r|Gal(Fₚ/Fₚ) and verifies a multiplicity one property.