Maximal semigroups in semi-simple Lie groups (Q2750940)

Let \(G\) be a real simple Lie group with finite center. The author characterizes the set of maximal subsemigroups of \(G\) in terms of their action on real flag manifolds for \(G\). To this end he introduces the concept of \({\mathcal B}\)-convex set in a flag manifold \(M\): \(A\) subset \(C\) of \(M\) is \({\mathcal B}\)-convex if it is the intersection of Bruhat cells. The main result is then that a subsemigroup \(S\) of \(G\) is maximal if and only if there exists a minimal flag manifold \(G/P\) (i.e. \(P\) is a maximal parabolic) and a \({\mathcal B}\)-convex set \(C\) with non-empty interior int\(C\) such that \(S=\{g\in G\mid gK \subseteq K\}\), where \(K\) is the closure of int\(C\) in \(G/P\). The proof of this result depends heavily on the author's previous work on compression semigroups and control sets on flag manifolds. The paper also shows how previously identified maximal semigroups (e.g. in the context of total positivity) can be found using the new theorem.