Transitive actions of semigroups in semi-simple Lie groups
Given a connected semisimple Lie group \(G\) with finite center and a subsemigroup \(S\) with interior points, the authors point out a natural way of associating to \(S\) a parabolic subgroup \(P\) of \(G\) and an associated flag manifold \(B=G/P\). The notion of a contractive sequence in \(G\) with respect to \(B\) is introduced, and it is shown that for a closed subgroup \(L\) of \(G\), the action of \(S\) on \(G/L\) is transitive if and only if the action of \(L\) on \(B\) is minimal (no proper invariant closed subsets) and \(L\) contains a contractive sequence. For the case that \(G\not= L\) is simple, the theorem specializes to the result that if \(G/L\) is compact, then \(S\) is transitive on \(G/L\) if and only if \(S=G\). The restriction that \(G/L\) be compact corrects an earlier claim of the authors; they provide a counterexample to show that the theorem can fail if \(G/L\) is not compact.
scientific article; zbMATH DE number 1251164
scientific article