Maximality of compression semigroups (Q1346303)

Let \(G\) be a connected semisimple Lie group and \(\tau\) an involution on \(G\). Further let \(L\) be an open subgroup of the group \(G^ \tau\) of \(\tau\)-fixed points and \(P \subset G\) a parabolic subgroup. The semigroup \(S(L,P) = \{g \in G : gLP \subset LP\}\) is called the compression semigroup for \(L\)-orbits of the base point in the flag manifold \(G/P\). The authors prove that compression semigroups for regular symmetric pairs are maximal semigroups. This result explains why in applications one often finds them naturally, for instance when studying maximal domains of extensions of representations.