  A natural one-parameter family of K\"ahler quantizations of the cotangent
bundle $T^*K$ of a compact Lie group $K$, taking into account the half-form
correction, was studied in \cite{FMMN}. In the present paper, it is shown that
the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and
coincides with the parallel transport of the quantum connection introduced in
our previous work, from the point of view of \cite{AdPW}. The BKS pairing map
is a composition of (unitary) coherent state transforms of $K$, introduced in
\cite{Ha1}. Continuity of the Hermitian structure on the quantum bundle, in the
limit when one of the K\"ahler polarizations degenerates to the vertical real
polarization, leads to the unitarity of the corresponding BKS pairing map. This
is in agreement with the unitarity up to scaling (with respect to a rescaled
inner product) of this pairing map, established by Hall.
