Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials (Q1690567)

Let ASG\((2\nu +l,\nu ;F_{q})\) be the \((2\nu +l)\)-dimensional affine-singular symplectic space over the finite field \(F_{q}\) and ASp\(_{2\nu +l,\nu }(F_{q}) \) be the affine-singular symplectic group of degree \(2\nu +l\) over \(F_{q}\). Let \(O\) be any orbit of flats under ASp\(_{2\nu +l,\nu }(F_{q})\). Denote by \( L^{J}\) the set of all flats which are joins of flats in \(O\) such that \( O\subseteq L^{J}\) and assume the join of the empty set of flats in ASp\( _{2\nu +l,\nu }(F_{q})\) is \(\emptyset \). Ordering \(L^{J}\) by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice \(L^{J}\), when the lattices form geometric lattices, lastly gives the characteristic polynomial of \(L^{J}\).