On the Bruhat $\mathcal{G}$-order between local systems on the B-orbits of a Hermitian symmetric variety

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 $G\mathrm{/}L$. 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 $\mathrm{\Phi }(G)$ which we suppose irreducible. In particular, for $\mathrm{\Phi }(G)$ 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.