Let𝔽q(2ν+δ+l)be a(2ν+δ+l)-dimensional vector space over the finite field𝔽q. In this paper we assume that𝔽qis a finite field of odd characteristic, andO2ν+δ+l, Δ(𝔽q)the singular orthogonal groups of degree2ν+δ+lover𝔽q. Letℳbe any orbit of subspaces underO2ν+δ+l, Δ(𝔽q). Denote byℒthe set of subspaces which are intersections of subspaces inℳ, where we make the convention that the intersection of an empty set of subspaces of𝔽q(2ν+δ+l)is assumed to be𝔽q(2ν+δ+l). By orderingℒby ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these latticesℒare geometric lattices.
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