On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2

Let $V$ be a $2n$-dimensional vector space over a field $\F$ and $\xi$ a non-degenerate alternating form defined on $V$. Let $\Delta$ be the building of type $C_n$ formed by the totally $\xi$-isotropic subspaces of $V$ and, for $1\leq k \leq n$, let $\G_k$ and $\Delta_k$ be the $k$-grassmannians of $\PG(V)$ and $\Delta$, embedded in $W_k=\wedge^kV$ and in a subspace $V_k\subseteq W_k$ respectively, where $\mathrm{dim}(V_k)={2n\choose k} - {2n\choose {k-2}}$. This paper is a continuation of \cite{CPeven2}. In \cite{CPeven2}, focusing on the case of $k = n$, we considered two forms $\alpha$ and $\beta$ related to the notion of \lq being at non maximal distance\rq\, in $\G_n$ and $\Delta_n$ and, under the hypothesis that $\mathrm{char}(\F) \neq 2$, we studied the subspaces of $W_n$ where $\alpha$ and $\beta$ coincide or are opposite. In this paper we assume that $\mathrm{char}(\F)=2$. We determine which of the quadrics associated to $\alpha$ or $\beta$ are preserved by the group $G= \mathrm{Sp}(2n,\F)$ in its action on $W_n$ and we study the subspace $\cal D$ of $W_n$ formed by vectors $v$ such that $\alpha(v,x) = \beta(v,x)$ for every $x\in W_n$. Finally, we show how properties of $\cal D$ can be exploited to investigate the poset of $G$-invariant subspaces of $V_k$ for $k = n-2i$ and $1\leq i \leq [n/2]$.