Complex time evolution in geometric quantization and generalized coherent state transforms

For the cotangent bundle T⁎K of a compact Lie group K, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L2(K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L2(K) and a certain weighted L2-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ) within L2(K), followed by a polarization-changing geometric-quantization evolution (for complex time +τ). In this case, our construction yields the usual generalized Segal-Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemannʼs “complexifier” method (which generalizes the construction of adapted complex structures).