On the Geometry of Symplectic Involutions

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 6= 2k and max(k, n−k) ≥ 5 then any bijective transformation of Hk(Ω) preserving the class of base subsets is induced by a semi-symplectic automorphism of V . For the case when n 6= 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 6= 2k and max(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. MSC 2000: 51N30, 51A50