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 $\mathbb{F}$ and $G$ the symplectic group $\mathrm{Sp}(2n,\mathbb{F})$ stabilizing a non-degenerate alternating form $\alpha(.,.)$ of $V$. Let ${\mathcal G}_k$ be the $k$-grassmannian of $\mathrm{PG}(V)$ and $\Delta_k$ the $k$-grassmannian of the $C_n$-building $\Delta$ associated to $G$. Put $W_k:= \wedge^kV$ and $\iota_k:{\mathcal G}_k\rightarrow W_k$ the natural embedding of ${\mathcal G}_k$, sending a $k$-subspace $\langle x_1,...,x_k\rangle$ of $V$ to the 1-subspace $\langle x_1\wedge...\wedge x_k\rangle$ of $W_k$. Let $\varepsilon_k:\Delta_k\rightarrow V_k$ 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 $\alpha$-isotropic $k$-spaces of $V$. We recall that $\mathrm{dim}(V_k) = {{2n}\choose k}-{{2n}\choose{k-2}}$. For $i = 0, 1,...,\lfloor k/2\rfloor$ let $V^{(k)}_{k-2i}$ be the subspace of $W_k$ spanned by the $\iota_k$-images of the $k$-subspaces $X$ of $V$ such that the codimension of $X\cap X^\perp$ in $X$ is at least $2i$. The group $G$ stabilizes each of the subspaces $V^{(k)}_{k-2i}$. Hence it also acts on each of the sections $V^{(k)}_{k-2i}/V^{(k)}_{k-2i+2}$. In \cite{BCP10}, exploiting the fact that the embeddings $\varepsilon_{k-2i}$ are universal when $\mathrm{char}(\mathbb{F}) \neq 2$, Blok and the authors of this paper have proved that if $\mathrm{char}(\mathbb{F})\neq 2$ then $V^{(k)}_{k-2i}/V^{(k)}_{k-2i+2}$ and $V_{k-2i}$ are isomorphic as $G$-modules, for every $i = 1,...,\lfloor k/2\rfloor$. In the present paper we shall prove that the same holds true when $\mathrm{char}(\mathbb{F}) = 2$