Base subsets of symplectic Grassmannians

Abstract

Let $V$ and $V'$ be $2n$-dimensional vector spaces over fields $F$ and $F'$. Let also $\Omega: V\times V\to F$ and $\Omega': V'\times V'\to F'$ be non-degenerate symplectic forms. Denote by $\Pi$ and $\Pi'$ the associated $(2n-1)$-dimensional projective spaces. The sets of $k$-dimensional totally isotropic subspaces of $\Pi$ and $\Pi'$ will denoted by ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect ${\mathcal G}_{k}$ and ${\mathcal G}'_{k}$ by so-called base subsets. We show that every mapping of ${\mathcal G}_{k}$ to ${\mathcal G}'_{k}$ sending base subsets to base subsets is induced by a symplectic embedding of $\Pi$ to $\Pi'$.