On a symplectic manifold $M$ with a hamiltonian action by a group $K$, it is natural to perform geometric quantization with respect to $K$-invariant polarizations. We first consider the case of the cotangent bundle $T^\ast K$ (based on arXiv:2301.10853), where parallel transport along the geodesics yields a realization of the Peter--Weyl Theorem.

In the second part, we dicuss the universal role of construction for the cotangent bundle plays in the parametrization of invariant Mabuchi geodesic rays for general hamiltonian $K$-manifolds.

This talk is based on joint work in progress with A.Ferreira, J.Hilgert, O.Kaya, J.Mourão and J.P.Nunes.