On the geometry of symplectic involutions (Q860095)

In a \(2n\)-dimensional vector space \(V\) over a field \(F\) a non-degenerate symplectic form \(\Omega\) is used to define orthogonality. The set of \(2k\)-dimensional subspaces \(U\) with non-denerate \(\Omega | _U\) is called \(H_k(\Omega)\). Any element of \(H_k(\Omega)\) can be generated by \(k\) mutually orthogonal elements of \(H_1(\Omega)\). Then the author is able to prove the following results: If \(n \neq 2k\) and \(k\) or \(n-k \geq 5\) any bijective transformation of \(H_k(\Omega)\) preserving the class of base subsets is induced by a semi-symplectic automorphism of \(V\). For \(n = 2k \geq 14\) the author delivers a weaker result.