  Let K be a connected Lie group of compact type and let T*(K) be its cotangent
bundle. This paper considers geometric quantization of T*(K), first using the
vertical polarization and then using a natural Kahler polarization obtained by
identifying T*(K) with the complexified group K_C. The first main result is
that the Hilbert space obtained by using the Kahler polarization is naturally
identifiable with the generalized Segal-Bargmann space introduced by the author
from a different point of view, namely that of heat kernels. The second main
result is that the pairing map of geometric quantization coincides with the
generalized Segal-Bargmann transform introduced by the author. This means that
in this case the pairing map is a constant multiple of a unitary map. For both
results it is essential that the half-form correction be included when using
the Kahler polarization. Together with results of the author with B. Driver,
these results may be seen as an instance of "quantization commuting with
reduction."
