Base subsets of polar Grassmannians

Abstract

Let $\Delta$ be a thick building of type ${\text{X}}_n={\text{C}}_n,{\text{D}}_n$. Let also ${\mathcal{G}}_k$ be the Grassmannian of $k$-dimensional singular subspaces of the associated polar space $\Pi$ (of rank $n$). We write ${\mathfrak{G}}_k$ for the corresponding shadow space of type ${\text{X}}_{n,k}$. Every bijective transformation of ${\mathcal{G}}_k$ sending base subsets to base subsets (the shadows of apartments) is a collineation of ${\mathfrak{G}}_k$, and it is induced by a collineation of $\Pi$ if $n\ne 4$ or $k\ne 1$.