Symplectic subspaces of symplectic Grassmannians

Let V be a non-degenerate symplectic space of dimension 2n over the field F and for a natural number I < n denote by Cl(V) the incidence geometry whose points are the totally isotropic l-dimensional subspaces of V. Two points U, W of Cl (V) will be collinear when W subset of U-L and dim(U n 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 n W. The isomorphism type of this geometry is denoted by C-n,C-l (F). When char(F) not equal 2 we classify subspaces S of C-l (F) where S congruent to C-m,C-k (F) (C) 2006 Elsevier Ltd. All rights reserved.