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.
A natural one-parameter family of Kähler quantizations of the cotangent bundle T * K of a compact Lie group K, taking into account the half-form correction, was studied in [C. Florentino, P. Matias, J. Mourão, J.P. Nunes, Geometric quantization, complex structures and the coherent state transform, J. Funct. Anal. 221 (2005) 303-322]. 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 [S. Axelrod, S. Della Pietra, E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991) 787-902]. The BKS pairing map is a composition of (unitary) coherent state transforms of K, introduced in [B.C. Hall, The Segal-Bargmann coherent state transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103-151]. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the Kähler 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. 2005 Elsevier Inc. All rights reserved.
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