Lattices generated by two orbits of subspaces under finite classical groups

Let be the n-dimensional vector space over a finite field Fq, and let Gn be the symplectic group Spn(Fq) where n=2ν; or the unitary group Un(Fq) where . For any two orbits M1 and M2 of subspaces under Gn, let L1 (resp. L2) be the set of all subspaces which are sums (resp. intersections) of subspaces in M1 (resp. M2) such that M2⊆L1 (resp. M1⊆L2). Suppose L is the intersection of L1 and L2 containing {0} and . By ordering L by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in these lattices and classifies their geometricity.