On a series of modules for the symplectic group in characteristic 2

Let V be a 2n-dimensional vector space defined over an arbitrary field IF and G the symplectic group Sp(2n, F) stabilizing a non-degenerate alternating form alpha(.,.) of V. Let G(k) be the k-grassmannian of PG(V) and Delta(k) the k-grassmannian of the C-n-building Delta associated to G. Put W-k := Lambda V-k and let iota(k) : G(k) -> W-k be the natural embedding of G(k) sending a k-subspace < x(1), ... , x(k)> of V to the I-subspace (x(1) Lambda ... Lambda x(k)) of W-k. Let epsilon(k) -> V-k be the embedding of Delta(k) induced by iota(k), where V-k is the subspace of W-k spanned by the iota(k)-images of the totally a-isotropic k-spaces of V. Recall that dim(V-k) = ((2n)(k)) - ((2n)(k-2)) For i = 0,1, ... , left perpendiculark/2right perpendicular let V-k-2i((k)) be the subspace of W-k spanned by the iota(k)-images of the k-subspaces X of V such that the codimension of X X boolean AND X-perpendicular to in X is at least 2i. The group G stabilizes each of the subspaces V-k-2i((k)). Hence it also acts on each of the sections (Vk-2iVk-2i+2(k))-V-(k)/. In [5], exploiting the fact that the embeddings ek_2i are universal when char(F) not equal 2, Blok and the authors of this paper have proved that if char(F) not equal 2 then V-k-2i((k))/V-k-2i+2((k)) and Vk-2i are isomorphic as C-modules, for every i = 1, ... , left perpendiculark/2right perpendicular. In the present paper we shall prove that the same holds true when char(F) = 2.