Partial coherent state transforms, G x T-invariant Kahler structures and geometric quantization of cotangent bundles of compact Lie groups

In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain G x T-invariant functions on the cotangent bundle of a compact connected Lie group G with maximal torus T. Namely, we will take the Hamiltonian flows of one G x G-invariant function, h, and one G x T-invariant function, f. Acting with these complex time Hamiltonian flows on G x G-invariant Kahler structures gives new Gx T-invariant, but not G x G-invariant, Kahler structures on T* G. We study the Hilbert spaces corresponding to the quantization of T* G with respect to these non-invariant Kahler structures. On the other hand, by taking the vertical Schrodinger polarization as a starting point, the above G x T-invariant Hamiltonian flows also generate families of mixed polarizations P-0(,sigma), sigma is an element of C, Im sigma > 0. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kahler structure on the leaves of a foliation of T* G. The geometric quantization of T* G with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [11,12]. (C) 2020 Elsevier Inc. All rights reserved.