Base subsets of symplectic Grassmannians

Abstract

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

12 pages