Lattices associated with totally isotropic subspaces in classical spaces

Let be one of (2ν+δ)-dimensional classical spaces over the finite field Fq, where δ=0,1, or 2. For 0⩽i<ν, let P0 denote a maximal totally isotropic subspace of , and let Q0 denote an (i+ω,ω)-totally isotropic subspace of contained in P0, here ω=0 or 1. Suppose L(Q0,P0,ω;2ν+δ) denotes the set of all the totally isotropic subspaces U such that U∩P0=Q0 including . Partially ordered by ordinary or reverse inclusion, two families of finite lattices are obtained. This paper discusses their atomic property, geometricity, and compute their characteristic polynomials.