Abstract: We consider a family of adapted complex structures on the cotangent bundle of a Lie group and find the BKS pairing relating the corresponding half-form quantisation. We show that the resulting bundle of quantum Hilbert spaces over the space of polarisations is flat. The vertical polarisation as a limit of complex polarisations yields the coherent state transform (or the Segal-Bargmann-Hall transform). We show that there is another limit of the complex polarisations that corresponds to the Peter-Weyl theorem. This is a joint work with W. Kirwin.
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