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

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