Nonreversibility of subsemigroups of semi-simple Lie groups (Q1186846)

A subsemigroup \(S\) of a Lie group \(G\) is called reversible if \(aS\cap bS\neq\emptyset\neq Sa\cap Sb\) for any two \(a,b\in S\). Equivalently one has the condition that \(SS^{-1}\) and \(S^{-1}S\) are subgroups of \(G\). The author shows that for a semisimple Lie group \(G\) with finite center a semigroup \(S\) with non-empty interior can only have this property if \(S=G\). The proof depends on the observation that the reversibility can be phrased in terms of transitivity of semigroup actions and the author's theorem that under the described hypothesis a semigroup can only act transitively on flag manifolds for minimal parabolics if it is equal to \(G\).