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 (J. Funct. Anal. 122 (1994) 103-151) 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 (Comm. Math. Phys. 131 (1990) 347-380) and Axelrod et al. (J. Differential Geom. 33 (1991) 787-902).