Lattices generated by orbits of subspaces under finite singular unitary group and its characteristic polynomials

Let be the (n+l)-dimensional vector space over the finite field , and the singular Unitary groups of degree n+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in , where the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , when the lattices form a geometric lattice, and the characteristic polynomial of .