On involutions in the Weyl group and $B$-orbit closures in the orthogonal case

We study coadjoint $B$-orbits on ${\mathfrak{n}}^{\ast }$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit ${\mathrm{\Omega }}_w$. We prove that, given basis involutions $\sigma$, $\tau$ in $W$, if the orbit ${\mathrm{\Omega }}_{\sigma }$ is contained in the closure of the orbit ${\mathrm{\Omega }}_{\tau }$ then $\sigma$ is less than or equal to $\tau$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of ${\mathrm{\Omega }}_w$ and present a conjectural description of the closure of ${\mathrm{\Omega }}_w$.