Let $K$ be a compact Lie group. I will review the construction of Mabuchi geodesic families of $K\times K$–invariant Kähler structures on $T^\ast K$, via Hamiltonian flows in imaginary time generated by a strictly convex invariant function on $\operatorname{Lie}K$, and the corresponding geometric quantization. At infinite geodesic time, one obtains a rich mixed polarization of $T^\ast K$, the Kirwin-Wu polarization, which is then continuously connected to the vertical polarization of $T^\ast K$. The geometric quantization of $T^\ast K$ along this family of polarizations is described by a generalized coherent state transform that, as geodesic time goes to infinity, describes the convergence of holomorphic sections to distributional sections supported on Bohr-Sommerfeld cycles. These are in correspondence with coadjoint orbits $O_{\lambda+\rho}$. One then obtains a concrete (quantum) geometric interpretation of the Peter-Weyl theorem, where terms in the non-abelian Fourier series are directly related to geometric cycles in $T^\ast K$. The role of a singular torus action in this construction will also be emphasized. This is joint work with T. Baier, J. Hilgert, O. Kaya and J. Mourão.