On \(Q\)-structures of quasisymmetric domains
A Siegel domain \({\mathcal S}_I\) is defined by a data \((U,V,A, {\mathcal C},I)\), where \(U\), \((V,I)\) are real and complex vector spaces of finite dimension, \(I\) denoting a complex structure on \(V\); \({\mathcal C}\) is an open convex cone in \(U\) and \(A\) is an alternating bilinear map \(V \times V \to U\) such that \(A(v, Iv')\) is symmetric and ``\({\mathcal C}\)-positive'', i.e., \(A(v,Iv) \in \overline {\mathcal C} - \{0\}\) for \(v \in V - \{0\}\). Then one has \({\mathfrak S}_I = \{(u,w) \in U_\mathbb{C} \times V + |\text{Im} u - (i/2) A (\overline w,w) \in {\mathcal C}\}\), where \(V_+ = \{v \in V_\mathbb{C} |Iv = iv\}\). The domain \({\mathcal S}_I\) is called ``quasisymmetric'' if (i) \({\mathcal C}\) is self-dual and homogeneous and if (ii) for any \(g_1 \in \Aut (U, {\mathcal C})^\circ\) there exists \(g_2 \in \text{GL} (V)\) such that \(A(g_2v, g_2v') = g_1 A(v,v')\) for all \(v,v' \in V\). The affine automorphism group of \({\mathcal S}_I\), \(\text{Aff} {\mathcal S}_I\), has a natural structure of real algebraic group. The main objective of the paper is to give a complete determination of the \(\mathbb{Q}\)-structures of \(U \times V\) such that \(\text{Aff} {\mathcal S}_I\) along with its linear part is defined over \(\mathbb{Q}\). Such a \(\mathbb{Q}\)- structure exists only if \(I\) is a ``rational point'' in the parameter space \({\mathfrak S} = {\mathfrak S} (V,A, {\mathcal C})\), which is always a symmetric domain of classical type. (The determination of the rational points in such a domain \({\mathfrak S}\) was given in a previous paper: the author, ibid. 43, No. 3, 401-424 (1991; Zbl 0813.32029)). The main result of the paper shows that the \(\mathbb{Q}\)-structure of \(\text{Aff} {\mathcal S}_I\) is essentially determined by that of a semisimple (associative) algebra \({\mathcal A}_1\) with positive involution, in which Lie \(\Aut (U, {\mathcal C})\) has a natural representation. In the Appendix, it is shown that, when \({\mathcal S}_I\) is symmetric, there exists a unique \(\mathbb{Q}\)- structure of \(\Aut {\mathcal S}_I\) extending the given \(\mathbb{Q}\)-structure of \(\text{Aff} {\mathcal S}_I\). [For some corrections and remarks to this article see the following paper].
scientific article; zbMATH DE number 822824
scientific article