On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces

Let n ≥ 2, let K,K′ be fields such that K′ is a quadratic Galoisextension of K and let θ denote the unique nontrivial element in Gal(K′/K). Suppose the symplectic dual polar space DW (2n− 1,K) is fully and isometrically embedded into the Hermitian dual polar space DH(2n − 1,K′, θ). We prove that the projective embedding of DW (2n − 1,K) induced by the Grassmann-embedding of DH(2n − 1,K′, θ) is isomorphic to the Grassmann-embedding of DW (2n−1,K). We also prove that if n is even, then the set of points of DH(2n − 1,K′, θ) at distance at most n2 −1 from DW (2n−1,K) is a hyperplane of DH(2n − 1,K′, θ) which arises from the Grassmann-embedding of DH(2n− 1,K′, θ).