Base subsets of polar Grassmannians

Abstract

Let $\Delta$ be a thick building of type $\textsf{X}_{n}=\textsf{C}_{n},\textsf{D}_{n}$. 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 $\textsf{X}_{n,k}$. 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 $k\ne 1$.