Lattices generated by orbits of flats under finite affine-symplectic groups

Let ASG(2ν,Fq) be the 2ν-dimensional affine-symplectic space over the finite field Fq and let ASp2ν(Fq) be the affine-symplectic group of degree 2ν over Fq. For any two orbits M′ and M″ of flats under ASp2ν(Fq), let L′ (resp. L″) be the set of all flats which are joins (resp. intersections) of flats in M′ (resp. M″) such that M″⊆L′ (resp. M′⊆L″) and assume the join (resp. intersection) of the empty set of flats in ASG(2ν,Fq) is ∅ (resp. ). Let L=L′∩L″. By ordering L′, L″, L by ordinary or reverse inclusion, six lattices are obtained. This article discusses when they form geometric lattices.