Rook placements in $A_n$ and combinatorics of $B$-orbit closures

Let $G$ be a complex reductive group, $B$ be a Borel subgroup of G, $\nt$ be the Lie algebra of the unipotent radical of $B$, and $\nt^*$ be its dual space. Let $\Phi$ be the root system of $G$, and $\Phi^+$ be the set of positive roots with respect to $B$. A subset of $\Phi^+$ is called a rook placement if it consists of roots with pairwise non-positive inner products. To each rook placement $D$ one can associate the coadjoint orbit $\Omega_D$ of $B$ in $\nt^*$. By definition, $\Omega_D$ is the orbit of $f_D$, where $f_D$ is the sum of root covectors corresponging to the roots from $D$. We find the dimension of $\Omega_D$ and construct a polarization of $\nt$ at $f_D$. We also study the partial order on the set of rook placements induced by the incidences among the orbits associated with rook placements.