Compression semigroups on open orbits on real flag manifolds
Let \((G,H)\) be an irreducible semisimple symmetric pair, \(P \subset G\) a parabolic subgroup. Suppose that the \(L\)-orbit of the base point in the flag manifold \(G/P\) is open. The semigroup \(S(L,P) = \{g \in G : gLP \subset LP\}\) is called the compression semigroup. The authors show that if \(P\) is minimal and \(S(L,P) = G\), then \((G,H)\) is Riemannian and give a geometric characterization of those cases where \(S(L,P)\) has non-empty interior different from \(G\). If \(G/H\) is a symmetric space of regular type, it is shown under certain additional assumptions that \(S(L,Q)\), where \(Q\) is some parabolic subgroup, is an Ol'shanskij semigroup.
scientific article; zbMATH DE number 742701