On geometry of symplectic involutions

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Let V be a 2n-dimensional vector space over a field F and Ω be a non-degenerate symplectic form on V. Denote by Hₖ(Ω) the set of all 2k-dimensional subspaces U⊂ V such that the restriction Ω|ᵤ is non-degenerate. Our main result (Theorem 1) says that if n≠ 2k and (k,n-k)≥ 5 then any bijective transformation of Hₖ(Ω) preserving the class of base subsets is induced by a semi-simplectic automorphism of V. For the case when n≠ 2k this fails, but we have a weak version of this result (Theorem 2). If the characteristic of F is not equal to 2 then there is a one-to-one correspondence between elements of Hₖ(Ω) and symplectic (2k,2n-2k)-involutions and Theorem 1 can be formulated as follows: for the case when n≠ 2k and (k,n-k)≥ 5 any commutativity preserving bijective transformation of the set of symplectic (2k,2n-2k)-involutions can be extended to an automorphism of the symplectic group.