Isometric full embeddings of DW(2n−1,q) into DH(2n−1,q2)

We show that there is up to isomorphism a unique isometric full embedding of the dual polar space DW(2n−1,q) into the dual polar space DH(2n−1,q2). We use the theory of valuations of near polygons to study the structure of this isometric embedding. We show that for every point x of DH(2n−1,q2) at distance δ from DW(2n−1,q) the set of points of DW(2n−1,q) at distance δ from x is a so-called SDPS-set which carries the structure of a dual polar space DW(2δ−1,q2). We show that if n is even, then the set of points at distance at most from DW(2n−1,q) is a geometric hyperplane of DH(2n−1,q2) and we study some properties of these new hyperplanes.