Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups II

Let be the (2ν+1+l)-dimensional vector space over the finite field Fq. In the paper we assume that Fq is a finite field of characteristic 2, and Ps2ν+1+l,2ν+1(Fq) the singular pseudo-symplectic groups of degree 2ν+1+l over Fq. Let M be any orbit of subspaces under Ps2ν+1+l,2ν+1(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 questions when the lattices L form a geometric lattice.