Projective Subgrassmannians of Polar Grassmannians

Abstract

In this short note, completing a sequence of studies, we consider the $k$-Grassmannians of a number of polar geometries of finite rank $n$. We classify those subspaces that are isomorphic to the $j$-Grassmannian of a projective $m$-space. In almost all cases, these are parabolic, that is, they are the residues of a flag of the polar geometry. Exceptions only occur when the subspace is isomorphic to the Grassmannian of $2$-spaces in a projective $m$-space and we describe these in some detail. This Witt-type result implies that automorphisms of the Grassmannian are almost always induced by automorphisms of the underlying polar space.