Semigroups in Symmetric Lie Groups

Let $G$ be a connected noncompact semi-simple real Lie group, $\tau $ an involutive automorphism of $G,$ and $L$ a subgroup of $G$ such that $G_{0}^{\tau }\subset L\subset G^{\tau }.$ In this article we give conditions on $x\in G$ such that the semigroup generated by the coset $Lx$ has nonempty interior in $G$. As a consequence we prove that for several $\tau $ the fixed point group $G^{\tau }$ is a maximal semigroup.