The B-orbits on a Hermitian symmetric variety: the characteristic 2 case and the Bruhat G-order

Let G be a connected reductive linear algebraic group over an algebraically closed field. Fix a torus T and a Borel subgroup B and suppose there is a parabolic subgroup P such that G/L is a Hermitian symmetric variety. The Borel subgroup B acts on G/L with a finite number of orbits. The set of these orbits can be ordered through the Bruhat order. We give parametrizations for the B-orbits in cases where the base field has characteristic 2. We also give a characterization for the Bruhat order among these orbits and a formula to compute the dimension. Consider now the same situation over the complex field and for every B-orbit the set of B-equivariant rank 1 local systems up to isomorphisms. Following Lusztig and Vogan we can order the pairs orbit-local system through the Bruhat G-order. We will show a combinatorial characterization of the Bruhat G-order. This can be used to improve the computation of the Kazhdan-Lusztig-Vogan polynomial.