Let $K$ be a compact Lie group. As is well known, $L^2(K)$ can be interpreted as the “position-space” geometric quantization of the cotangent bundle $T^*K$. In this talk, I will describe a “momentum-space” quantization of $T^*K$. I will also explain how this momentum-space quantization is linked to the position-space representation via parallel transport with respect to the Axelrod-della Pietra-Witten-Hitchin connection in a certain Hilbert bundle. In particular, it is a result of Florentino-Matias-Mourao-Nunes that parallel transport along a particular geodesic from position space to an intermediate fiber is the generalized Segal-Bargmann transform. I will explain how their result can be extended to any other interior fiber (thus obtaining generalized Segal-Bargmann transforms for more general complex structures), and moreover that when extended to momentum space, parallel transport from position space yields the Peter-Weyl decomposition. This is joint work with S. Wu.