  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 $L^{2}(K)$ 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
$L^{2}(K)$ and a certain weighted $L^{2}$-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
$-\tau$) within $L^{2}(K)$, followed by a polarization--changing geometric
quantization evolution (for complex time $+\tau$). 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 Thiemann's "complexifier" method (which generalizes the
construction of adapted complex structures). We will also investigate some
properties of the generalized CSTs, and discuss how their existence can be
understood in terms of Mackey's generalization of the Stone-von Neumann
theorem.
