Published online by Cambridge University Press: 15 September 2014
We study a Hermitian form $h$ over a quaternion division algebra $Q$ over a field ($h$ is supposed to be alternating if the characteristic of the field is two). For generic $h$ and $Q$, for any integer $i\in [1,\;n/2]$, where $n:=\dim _{Q}h$, we show that the variety of $i$-dimensional (over $Q$) totally isotropic right subspaces of $h$ is $2$-incompressible. The proof is based on a computation of the Chow ring for the classifying space of a certain parabolic subgroup in a split simple adjoint affine algebraic group of type $C_{n}$. As an application, we determine the smallest value of the $J$-invariant of a non-degenerate quadratic form divisible by a $2$-fold Pfister form; we also determine the biggest values of the canonical dimensions of the orthogonal Grassmannians associated to such quadratic forms.