Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 47, No. 2, pp. 435446 (2006) 

On the Geometry of Symplectic InvolutionsMark PankovDepartment of Mathematics and Information Technology, University of Warmia and Mazury, {\. Z}olnierska 14A, 10561 Olsztyn, Poland; email: pankov@matman.uwm.edu.plAbstract: Let $V$ be a $2n$dimensional vector space over a field $F$ and $\Omega$ be a nondegenerate 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 nondegenerate. Our main result (Theorem 1) says that if $n\ne 2k$ and $\max(k,nk)\ge 5$ then any bijective transformation of ${\mathfrak H}_{k}(\Omega)$ preserving the class of base subsets is induced by a semisymplectic 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 onetoone correspondence between elements of ${\mathfrak H}_{k}(\Omega)$ and symplectic $(2k,2n2k)$involutions and Theorem 1 can be formulated as follows: for the case when $n\ne 2k$ and $\max(k,nk)\ge 5$ any commutativity preserving bijective transformation of the set of symplectic $(2k,2n2k)$involutions can be extended to an automorphism of the symplectic group. Keywords: hyperbolic symplectic geometry, symplectic group, Grassmannian Classification (MSC2000): 51N30, 51A50 Full text of the article:
Electronic version published on: 19 Jan 2007. This page was last modified: 5 Nov 2009.
© 2007 Heldermann Verlag
