Polar spaces and embeddings of classical groups

Abstract

Given polar spaces $(V,β)$ and $(V,Q)$ where $V$ is a vector space over a field $K$, $β$ a reflexive sesquilinear form and $Q$ a quadratic form, we have associated classical isometry groups. Given a subfield $F$ of $K$ and an $F$-linear function $L:K\to F$ we can define new spaces $(V,Lβ)$ and $(V,LQ)$ which are polar spaces over $F$. The construction so described gives an embedding of the isometry groups of $(V,β)$ and $(V,Q)$ into the isometry groups of $(V,Lβ)$ and $(V,LQ)$. In the finite field case under certain added restrictions these subgroups are maximal and form the so called {\it field extension subgroups} of Aschbacher's class $\curlyc{3}$ \cite{aschbacher}. We give precise descriptions of the polar spaces so defined and their associated isometry group embeddings. In the finite field case our results give extra detail to the account of maximal field extension subgroups given by Kleidman and Liebeck \cite[p112]{kl}.