  Following Lusztig and Vogan, we study the Bruhat $G$-order on the set
$\mathcal{D}$ of rank $1$ local systems on $B$-orbits over an Hermitian
symmetric variety $G/L$. The main aim is to give a combinatorial
characterization similar to the one on the Bruhat order given by Gandini and
Maffei. The results depend on the type of the root system $\Phi(G)$ which we
suppose irreducible. In particular, for $\Phi(G)$ simply laced either all the
local systems are trivial or only a specific subset of orbits (the orbits of
maximum rank) admit a non-trivial local system. In this last case the Hasse
diagram of the order admit two connected components: all the orbits with
trivial local systems and the orbits of maximum rank with the (unique)
non-trivial local system. On every connected component the order coincides with
the Bruhat order.
