Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces

Abstract

Abstract Let B be a class of point-line geometries. Given Γ i ∈ B with subspace i for i = 1, 2, does any isomorphism Γ1 − 1 → Γ2 − 2 extend to a unique isomorphism Γ1 → Γ2? It is known to be true if B is the class of almost all projective spaces or the class of almost all nondegenerate polar spaces. We show that this is true for the class of almost all strong parapolar spaces, including dual polar spaces. A special case occurs when Γ1 = Γ2 = Γ has an embedding into a projective space ℙ(V ) that is natural in the sense that Aut(Γ ) ≤ PΓL(V ). Then the question becomes whether ℙ(V ) is also the natural embedding for Γ − . Our result shows that in most cases the stabilizer StabAut(Γ)(Γ − ) is faithful on Γ − and equals Aut(Γ − ) and so the answer is affirmative. We know that there exist some interesting exceptions. These will be covered in a subsequent paper.