An essay on self-dual generalized quadrangles

We collect some known facts about a self-dual generalized quadrangle (GQ) and consider especially the number of absolute points of a duality. The only known finite self-dual GQs are the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T_2(\mathcal O)}$$\end{document} constructed by J. Tits where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal O}$$\end{document} is a translation oval in a finite desarguesian projective plane of even order. We review a construction of these GQs in enough detail to study the sets of absolute points of certain dualities, but for more details about these GQs see the monograph Finite Generalized Quadrangles (Europ. Math. Soc., Zurich 2009). After some generalities about the incidence matrix of a finite (GQ) we return to the study of self-dual examples.