Orbit spaces of Hilbert manifolds

Let G be a compact group acting on a Polish group X   by means of automorphisms. It is proved that the orbit space X/GX/G is an ℓ2ℓ2-manifold (resp., homeomorphic to ℓ2ℓ2) provided X is a G-ANR (resp., G  -AR) and the fixed point set XGXG is not locally compact. It is also proved that if a compact group G acts affinely on a separable closed convex subset K   of a Fréchet space with a non-locally compact fixed point set KGKG, then the orbit space K/GK/G is homeomorphic to ℓ2ℓ2. In particular, (1) if C(Y,X)C(Y,X) denotes the space of all maps from a compact metric G-space Y to a non-locally compact Polish ANR (resp., AR) group X, endowed with the compact-open topology and the induced action of G  , then the orbit space C(Y,X)/GC(Y,X)/G is an ℓ2ℓ2-manifold (resp., homeomorphic to ℓ2ℓ2), and (2) if X is an infinite-dimensional separable Fréchet G  -space and cc(X)cc(X) denotes the hyperspace of all non-empty compact convex subsets of X, endowed with the Hausdorff metric topology and the induced action of G  , then the orbit space cc(X)/Gcc(X)/G is homeomorphic to ℓ2ℓ2, whenever the fixed point set cc(X)Gcc(X)G is not locally compact.