Classical Subspaces of Symplectic Grassmannians B. N. Cooperstein 1 Introduction and Basic Concepts We assume the reader is familiar with the concepts of a partial linear rank two inci- dence geometry Γ=(P , L) (also called a point-line geometry) and the Lie incidence geometries. For the former we refer to articles in [B] and for the latter see the paper [Co]. The collinearity graph of Γ is the graph (P , Δ) where Δ consists of all pairs of points belonging to a common line. For a point x ∈P we will denote by Δ(x) the collection of all points collinear with x. For points x, y ∈P and a positive integer t a path of length t from x to y is a sequence x 0 = x, x 1 ,...,x t = y such that {x i ,x i+1 }∈ Δ for each i =0, 1,...,t − 1. The distance from x to y , denoted by d(x, y ) is defined to be the length of a shortest path from x to y if some path exists and otherwise is +∞. By a subspace of Γ we mean a subset S such that if l ∈L and l ∩ S contains at least two points, then l ⊂ S. (P , L) is said to be Gamma space if, for every x ∈P , {x}∪ Δ(x) is a subspace. A subspace S is singular provided each pair of points in S is collinear, that is, S is a clique in the collinearity graph of Γ. For a Lie incidence geometry with respect to a “good node” every singular subspace, together with the lines it contains, is isomorphic to a projective space, see [Co]. Clearly the intersection of subspaces is a subspace and consequently it is natural to define the subspace generated by a subset X of P , 〈X 〉 Γ , to be the intersection of all subspaces of Γ which contain X. Note that if (P , L) is a Gamma space and X is a clique then 〈X 〉 Γ will be a singular subspace. Bull. Belg. Math. Soc. 12 (2005), 719–725