Admissible unitary completions of locally ℚ_{𝕡}-rational representations of 𝔾𝕃₂(𝔽)

Let F F be a finite extension of Q p \mathbb {Q}_p , p > 2 p>2 . We construct admissible unitary completions of certain representations of G L 2 ( F ) \mathrm {GL}_2(F) on L L -vector spaces, where L L is a finite extension of F F . When F = Q p F=\mathbb {Q}_p using the results of Berger, Breuil and Colmez we obtain some results about lifting 2 2 -dimensional mod p p representations of the absolute Galois group of Q p \mathbb {Q}_p to crystabelline representations with given Hodge-Tate weights.