Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

Let 𝔽 q ( 2 ν + δ + l ) be a ( 2 ν + δ + l ) -dimensional vector space over the finite field 𝔽 q . In this paper we assume that 𝔽 q is a finite field of odd characteristic, and O 2 ν + δ + l , Δ ( 𝔽 q ) the singular orthogonal groups of degree 2 ν + δ + l over 𝔽 q . Let ℳ be any orbit of subspaces under O 2 ν + δ + l , Δ ( 𝔽 q ) . 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 q ( 2 ν + δ + l ) is assumed to be 𝔽 q ( 2 ν + δ + l ) . By ordering ℒ by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices ℒ are geometric lattices.