On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group

A natural one-parameter family of Kahler 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. Mourao, 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 Blatmer-Kostant-Stemberg (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 Chem-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 Seggal-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 Kahler 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. (c) 2005 Elsevier Inc. All rights reserved.