The generating rank of the unitary and symplectic Grassmannians

We prove that the Grassmannian of totally isotropic k-spaces of the polar space associated to the unitary group SU2n(F) (n∈N) has generating rank when F≠F4. We also reprove the main result of Blok (2007) [3], namely that the Grassmannian of totally isotropic k-spaces associated to the symplectic group Sp2n(F) has generating rank , when Char(F)≠2.