Symplectic subspaces of symplectic Grassmannians

Let V be a non-degenerate symplectic space of dimension 2 n over the field F and for a natural number l < n denote by C l ( V ) the incidence geometry whose points are the totally isotropic l -dimensional subspaces of V . Two points U , W of C l ( V ) will be collinear when W ⊂ U ⊥ and dim ( U ∩ W ) = l − 1 and then the line on U and W will consist of all the l -dimensional subspaces of U + W which contain U ∩ W . The isomorphism type of this geometry is denoted by C n , l ( F ) . When char ( F ) ≠ 2 we classify subspaces S of C l ( F ) where S ≅ C m , k ( F ) .