Basic differential forms for actions of Lie groups

A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\mathrm{\Sigma }$ which meets each orbit orthogonally. It is shown that the algebra of $G$-invariant differential forms on $M$ which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on $\mathrm{\Sigma }$ which are invariant with respect to the generalized Weyl group of this orbit, under some condition.