Geometric quantization, complex structures and the coherent state transform

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group $K$ is related with a Hermitian connection associated to a natural one-parameter family of complex structures on $T^*K$. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of $T^*K$ for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin and Axelrod, Della Pietra and Witten.