Base subsets of symplectic Grassmannians

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