On geometry of symplectic involutions

Abstract

Let $V$ be a $2n$-dimensional vector space over a field $F$ and $\Omega$ be a non-degenerate symplectic form on $V$. Denote by ${\mathfrak{H}}_k(\Omega )$ the set of all $2k$-dimensional subspaces $U\subset V$ such that the restriction $\Omega {|}_U$ is non-degenerate. Our main result (Theorem 1) says that if $n\ne 2k$ and $\max (k,n-k)\ge 5$ then any bijective transformation of ${\mathfrak{H}}_k(\Omega )$ preserving the class of base subsets is induced by a semi-simplectic automorphism of $V$. For the case when $n\ne 2k$ 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 ${\mathfrak{H}}_k(\Omega )$ and symplectic $(2k,2n-2k)$-involutions and Theorem 1 can be formulated as follows: for the case when $n\ne 2k$ and $\max (k,n-k)\ge 5$ any commutativity preserving bijective transformation of the set of symplectic $(2k,2n-2k)$-involutions can be extended to an automorphism of the symplectic group.