Beiträge zur Algebra und GeometrieContributions to Algebra and Geometry Vol. 47, No. 2, pp. 435-446 (2006)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home

EMIS Home

## On the Geometry of Symplectic Involutions

### Mark Pankov

Department of Mathematics and Information Technology, University of Warmia and Mazury, {\. Z}olnierska 14A, 10-561 Olsztyn, Poland; e-mail: pankov@matman.uwm.edu.pl

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-symplectic 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.

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
© 2007–2009 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition