On the dimensions of the root groups of full subquadrangles of Moufang quadrangles arising from algebraic groups (Q1281155)

Moufang quadrangle is said to be rational, if it arises from an algebraic group over an algebraically closed field by taking the ``rational points'' with respect to a subfield. Note that every Moufang quadrangle in characteristic \(\neq 2\) is a classical or rational (and the dimensions \((s,t)\) of a Moufang quadrangle can be defined as the dimension of any central or axial elation). In the finite case, a Moufang quadrangle has order \((q^s,q^t)\) for some prime power \(q\). Let \(\Gamma\) be a rational Moufang quadrangle, and suppose that \(\Gamma'\) be a rational Moufang subquadrangle of \(\Gamma\). Subquadrangle \(\Gamma'\) is called full, if every point of \(\Gamma\) on any line of \(\Gamma'\) belongs to \(\Gamma'\) (the dual notion of full subquadrangle will be called an ideal subquadrangle). Let \(x\) be a point of \(\Gamma\) not contained in \(\Gamma'\). Let \(\mathcal O\) be the set of points of \(\Gamma'\) collinear with \(x\). It is easy to see that every line of \(\Gamma'\) is incident with precisely one point of \(\mathcal O\). Hence \(\mathcal O\) is an ovoid of \(\Gamma'\), which we call a subtended ovoid (subtended by \(\Gamma\) and \(x\)). Main Result. Let \(\Gamma\) be a rational Moufang quadrangle with dimensions \((s,t)\) over the field \(\mathbf K\); let \(\Gamma'\) be a rational Moufang full subquadrangle with dimensions \((s,t')\) over \(\mathbf K\); let \(\mathcal O\) be an ovoid in \(\Gamma'\) subtended by \(\Gamma\) and \(x\); let \(p\) be any point of \(\mathcal O\) and let \(U_p\) be the subgroup of the whole group about \(p\) stabilizing \(\mathcal O\). Then (i) \(U_p\) acts regularly on \({\mathcal O}-\{p\}\). In particular, the group \(U\) generated by all \(U_a\), \(a\in \mathcal O\), acts 2-transitively on \(\mathcal O\); (ii) \(U_p\) is nilpotent of class at most 2. Moreover, both \([U_p,U_p]\) and \(U_p/[U_p,U_p]\) can be given the structure of a vector space over \(\mathbf K\), and the sum of the dimensions of these vector spaces is equal to \(s+t'\); (iii) \(s+t'\leq t\) and equality holds if and only if every line of \(\Gamma\) contains at least one point of \(\Gamma'\); (iv) the ovoid \(\mathcal O\) contains all elements of \(\{a,b\}^\bot\) which belongs to \(\Gamma'\), for all \(a,b\in \mathcal O\), \(a\neq b\). Applied to the finite case, a subquadrangle of order \((q^s,q^{t'})\) of some Moufang quadrangle of order \((q^s,q^t)\) satisfies \(q^sq^{t'}\leq q^t\), which is exactly the Thas inequality.