L2(M) decomposes into a direct sum of finite-dimensional eigenspaces of D, and G acts on each eigenspace, so each eigenspace decomposes into irreducible unitary representations. (I am now looking at complex functions, as these will correspond to functions into euclidean space.) Let { fja } be a basis for the vector space of the j-th irreducible representation.
Approximate g by Σi dyi dyi in the Ck topology
Let zi = Σja cija fja approximate yi in the Ck+1 topology, and let Xiabr = Σ Nj-1/2 cija fjb where the sum is over all j such that the j-th representation is equivalent to r.
Proposition 1. X is equivariant
2. zi = Σar Nr1/2 Xiaar
3. F(X) = P(Σi dzi dzi)
It follows from (2) that X is an embedding if z is, which it will be if it approximates y and y is. (3) implies that F(X) approximates g in the Ck topology.