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

Let F 2ν δ l q be a 2ν δ l -dimensional vector space over the finite field Fq. In this paperwe 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, where we make the convention that the intersection of an empty set of subspaces of F 2ν δ l q is assumed to be F 2ν δ l q . By ordering L by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices L are geometric lattices.