  Let $G$ be a reductive linear algebraic group over an algebraically closed
field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such
that the unipotent radical is abelian and a Levi subgroup $L\subseteq P$. We
parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric
variety $G/L$ supposing the root system $\Phi$ is irreducible. For $\Phi$
simply laced we prove a combinatorial characterization of the Bruhat order over
these orbits. We also prove a formula to compute the dimension of the orbits
from combinatorial characteristics of their representatives.
