Base subsets of symplectic Grassmannians

Let V and V ′ be 2n-dimensional vector spaces over fields F and F ′. Let also : V × V → F and ′: V ′ × V ′ → F ′ be non-degenerate symplectic forms. Denote by and ′ the associated (2n − 1)-dimensional projective spaces. The sets of kdimensional totally isotropic subspaces of and ′ will be denoted by Gk and G ′ k , respectively. Apartments of the associated buildings intersect Gk and G ′ k by so-called base subsets. We show that every mapping of Gk to G ′ k sending base subsets to base subsets is induced by a symplectic embedding of to ′.