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 TβK. The unitary
parallel transport of this connection establishes the equivalence of
(geometric) quantizations of TβK 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.Comment: to appear in Journal of Functional Analysi