Invariant ordering on the simply connected covering of the Shilov boundary of a symmetric domain (Q1002789)

Let \(G\) be a simply connected Hermitian simple Lie group. The Shilov boundary of a symmetric domain belonging to \(G\) has the form \(G/P\), where \(P\) is a maximal parabolic subgroup of \(G\) such that the unipotent radical of \(P\) is abelian. The author proves that the homogeneous space \(G/P\) possesses exactly one pair of opposite continuous invariant orderings and that these orderings are group orderings (which means that there exist orderings on \(G\) with determining semigroup \(S\) such that \(x\leq y\) for \(x, y \in G/P\) if and only if there exists an \(s\in S\) such that \(x = sy \)). Moreover, if \(P_1\) is a parabolic subgroup of \(G\) and there is an invariant ordering on \(G/P_{1}\), then \(P_1\) is conjugate to some subgroup of \(P\).