Nonhomogeneous $G$-spaces of compact Lie groups

A duality theorem for a compact $G$-manifold $M$ of a compact Lie group $G$ is proved. For the case when all the orbits in $M$ are principal, it is proved that the quotient space of the complexification of $M$ by the action of the associated algebraic group $G^c$ is isomorphic to the quotient space of $M$ by the action of $G$.