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

Let q (2 ν + δ + l) be a (2 ν + δ + l) -dimensional vector space over the finite field fq. in this paper we assume that q is a finite field of odd characteristic,and o 2 ν + δ + l,δ (fq) the singular orthogonal groups of degree 2 ν + δ + l over fq. let ℳ be any orbit of subspaces under o 2 ν + 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 q (2 ν + δ + l) is assumed to be 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. © 2012 you gao and xinzhi fu.