Let G be a complex reductive group, B be a Borel subgroup in G, n be the Lie algebra of the unipotent radical of B , and n * be its dual space.Let be the root system of G, and let + be the set of positive roots with respect to B .A subset of + 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 D of B in n * .By definition, D is the orbit of f D , where f D is the sum of root covectors corresponding to the roots from D .We find the dimension of D and construct a polarization of n at f D .We also study the partial order on the set of rook placements induced by the incidences among the closures of orbits associated with rook placements.
Ignatyev et al. (Wed,) studied this question.