Lattices generated by orbits of subspaces under finite singular orthogonal groups I

Let be the (2ν+δ+l)-dimensional vector space over the finite field Fq. In the paper we assume that Fq is a finite field of odd characteristic, and O2ν+δ+l,Δ(Fq) the singular orthogonal groups of degree 2ν+δ+l over Fq. Let M be any orbit of subspaces under O2ν+δ+l,Δ(Fq). Denote by L the set of subspaces which are intersections of subspaces in M and the intersection of the empty set of subspaces of is assumed to be . By ordering L 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 L, and the characteristic polynomial of L.