On isometric full embeddings of symplectic dual polar spaces into Hermitian dual polar spaces (Q1014512)

Let \({\mathbb K}'\), \({\mathbb K}\) be (finite or infinite) fields such that \({\mathbb K}'\) is a quadratic Galois extension of \({\mathbb K}\) and let \(\theta\) denote the unique nontrivial element in \(\text{Gal}({\mathbb K}'/{\mathbb K})\). The dual polar space \(DW(2n-1,{\mathbb K})\), \(n\geq\,2\), has up to equivalence a unique isometric full embedding into the Hermitian dual polar space \(DH(2n-1,{\mathbb K}',\theta)\) (the proof of this fact in [\textit{B. De Bruyn}, Finite Fields Appl. 14, No.~1, 188--200 (2008; Zbl 1139.51009)] for the finite case can be extended to the infinite case). The Grassmann-embedding of \(DH(2n-1,{\mathbb K}',\theta)\) [compare \textit{B. De Bruyn}, Linear Multilinear Algebra 56, No.~6, 665--677 (2008; Zbl 1155.51001)] induces a projective embedding of \(DW(2n-1,{\mathbb K})\) which is, as the author shows, isomorphic to the Grassmann-embedding of \(DW(2n-1,{\mathbb K})\). Furthermore, the author proves: If \(n\) is even, then the set of points of \(DH(2n-1,{\mathbb K}',\theta)\) at distance at most \(n/2-1\) from \(DW(2n-1,{\mathbb K})\) is a hyperplane of \(DH(2n-1,{\mathbb K}',\theta)\) which arises from the Grassmann-embedding of \(DH(2n-1,{\mathbb K}',\theta)\).