  In this paper we use techniques of geometric quantization to give a geometric
interpretation of the Peter--Weyl theorem. We present a novel approach to
half-form corrected geometric quantization in a specific type of non-K\"ahler
polarizations and study one important class of examples, namely cotangent
bundles of compact semi-simple groups $K$. Our main results state that this
canonically defined polarization occurs in the geodesic boundary of the space
of $K\times K$-invariant K\"ahler polarizations equipped with Mabuchi's metric,
and that its half-form corrected quantization is isomorphic to the K\"ahler
case. An important role is played by invariance of the limit polarization under
a torus action.
  Unitary parallel transport on the bundle of quantum states along a specific
Mabuchi geodesic, given by the coherent state transform of Hall, relates the
non-commutative Fourier transform for $K$ with the Borel--Weil description of
irreducible representations of $K$.
