Embedded polar spaces revisited

Abstract

Pseudo-quadratic forms have been introduced by Tits in his Buildings of spherical type and finite $BN$-pairs (1974), in view of the classification of polar spaces. A slightly different notion is proposed by Tits and Weiss. In this paper we propose a generalization. With its help we will be able to clarify a few points in the classification of embedded polar spaces. We recall that, according to Tits’ book, given a division ring $K$ and an admissible pair $\left(\sigma ,\epsilon \right)$ in it, the codomain of a $\left(\sigma ,\epsilon \right)$-quadratic form is the group $\overline{K}:=K∕K_{\sigma ,\epsilon }$, where $K_{\sigma ,\epsilon }:={\left\{t-t^{\sigma }\epsilon \right\}}_{t\in K}$. Our generalization amounts to replace $\overline{K}$ with a quotient $\overline{K}∕\overline{R}$ for a subgroup $\overline{R}$ of $\overline{K}$ such that ${\lambda }^{\sigma }\overline{R}\lambda =\overline{R}$ for any $\lambda \in K$. We call generalized pseudo-quadratic forms (also generalized $\left(\sigma ,\epsilon \right)$-quadratic forms) the forms defined in this more general way, keeping the words pseudo-quadratic form and $\left(\sigma ,\epsilon \right)$-quadratic form for those defined as in Tits’ book. Generalized pseudo-quadratic forms behave just like pseudo-quadratic forms. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if $q:V\to \overline{K}∕\overline{R}$ is a non-trivial generalized pseudo-quadratic form and $f:V\times V\to K$ is its sesquilinearization, the points and the lines of $PG\left(V\right)$ where $q$ vanishes form a subspace $S_q$ of the polar space $S_f$ associated to $f$. In this paper, after a discussion of quotients and covers of generalized pseudo-quadratic forms, we shall prove the following, which sharpens a celebretated theorem of Buekenhout and Lefèvre. Let $e:S\to PG\left(V\right)$ be a projective embedding of a non-degenerate polar space $S$ of rank at least $2$; then $e\left(S\right)$ is either the polar space $S_q$ associated to a generalized pseudo-quadratic form $q$ or the polar space $S_f$ associated to an alternating form $f$. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding $e$ as above is dominant if and only if either $e\left(S\right)=S_q$ for a pseudo-quadratic form $q$ or $char\left(K\right)\ne 2$ and $e\left(S\right)=S_f$ for an alternating form $f$.