Semigroups in symmetric Lie groups

Let G be a Lie group and L subset of G a Lie subgroup. We give necessary and sufficient conditions for a family of cosets of L to generate a subsemigroup with nonempty interior in G. We apply these conditions to symmetric pairs (G, L) where L is a subgroup of G such that G(0)(tau) subset of L subset of G' and r is an involutive 0 automorphism of G. As a consequence we prove that for several r the fixed point group G(tau) is a maximal semigroup.18113514