Basic Differential Forms for Actions of Lie Groups, Ii

The assumption in the main result of [2] is removed Let G be a Lie group which acts isometrically on a Riemannian manifold M . A section of the Riemannian G-manifold M is a closed submanifold Σ which meets each orbit orthogonally. In this situation the trace on Σ of the G-action is a discrete group action by the generalized Weyl group W (Σ) = NG(Σ)/ZG(Σ), where NG(Σ) := {g ∈ G : g.Σ = Σ} and ZG(Σ) := {g ∈ G : g.s = s for all s ∈ Σ}. A differential form φ ∈ Ω(M) is called G-invariant if g∗φ = φ for all g ∈ G and horizontal if φ kills each vector tangent to a G-orbit. We denote by Ωphor(M) G the space of all horizontal G-invariant p-forms on M which are also called basic forms. In the paper [2] it was shown that for a proper isometric action of a Lie group G on a smooth Riemannian manifold M admitting a section Σ the restriction of differential forms induces an isomorphism Ωphor(M) G ∼= −→ Ω(Σ) (Σ) between the space of horizontal G-invariant differential forms on M and the space of all differential forms on Σ which are invariant under the action of the generalized Weyl group W (Σ) of the section Σ, under the following assumption: For each x ∈ Σ the slice representation Gx → O(Tx(G.x)) has a generalized Weyl group which is a reflection group. In this paper we will show that this result holds in general, without any assumption. Notation is as in [2], which is used throughout. For more information on G-manifolds with sections see the seminal paper [3]. 1991 Mathematics Subject Classification. Orbits, sections, basic differential forms.