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

Abstract

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