Base subsets of polar Grassmannians

Let Δ be a thick building of type X n = C n , D n . Let also G k be the Grassmannian of k-dimensional singular subspaces of the associated polar space Π (of rank n). We write G k for the corresponding shadow space of type X n , k . Every bijective transformation of G k which maps base subsets to base subsets (the shadows of apartments) is a collineation of G k , and it is induced by a collineation of Π if n ≠ 4 or k ≠ 1 .