Lattices generated by orbits of totally isotropic flats under finite affine-classical groups

Let ACG(n,Fq) be one of n-dimensional affine-classical spaces over a finite field Fq, and let M be an orbit of totally isotropic flats under the corresponding affine-classical group. Denote by L the set of all the intersections of totally isotropic flats in M. If we partially order L by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This paper discusses their geometricity and computes their characteristic polynomials.