Let p be a prime number, F a totally real number field unramified at places above p and D a quaternion algebra of center F split at places above p and at no more than one infinite place. Let v be a fixed place of F above p and r : Gal(F /F) → GL2(Fp) an irreducible modular continuous Galois representation which, at the place v, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of GL2(Fv) over Fp associated to r in the corresponding Hecke-eigenspaces of the mod p cohomology have Gelfand-Kirillov dimension [Fv : Qp], as well as several related results. Contents