On a series of modules for the symplectic group in characteristic 2. (Q2869125)

Let \(V\) be a \(2n\)-dimensional vector space defined over an arbitrary field \(F\) and \(G\) the symplectic group \(\mathrm{Sp}(2n,F)\) stabilizing a non-degenerate alternating form \(\alpha(.,.)\) of \(V\). Let \(G_k\) be the \(k\)-th 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 let \(l_k\colon G_k\to W_k\) be the natural embedding of \(G_k\), sending a \(k\)-subspace \(\langle x_1,\ldots,x_k\rangle\) of \(V\) to the \(1\)-subspace \(\langle x_1\wedge\cdots\wedge x_k\rangle\) of \(W_k\). Let \(\varepsilon_k\colon\Delta_k\to V_k\) be the embedding of \(\Delta_k\) induced by \(l_k\), where \(V_k\) is the subspace of \(W_k\) spanned by the \(l_k\)-images of the totally \(\alpha\)-isotropic \(k\)-space of \(V\). In [Bull. Belg. Math. Soc. - Simon Stevin 18, No. 1, 1-29 (2011; Zbl 1264.20039)], exploiting the fact that the embedding \(\varepsilon_{k-2i}\) is universal when \(\mathrm{char}(F)\neq 2\), \textit{R. J. Blok} and the authors of this paper have proved that if \(\mathrm{char}(F)\neq 2\) then \(V_{k-2i}^{(k)}/V_{k-2i+2}^{(k)}\) and \(V_{k-2i}\) are isomorphic as \(G\)-modules, for every \(i=1,\ldots,[\frac{k}{2}]\). In the present paper they prove that the same holds true when \(\mathrm{char}(F)=2\).NEWLINENEWLINEFor the entire collection see [Zbl 1253.00023].