Partial coherent state transforms, $G\times T$-invariant K\"ahler 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 $G\times T$-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 $G\times G$-invariant function, $h$, and one $G\times T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $G\times G$-invariant K"ahler structures gives new $G\times T$-invariant, but not $G\times G$-invariant, K"ahler structures on $T^{\ast }G$. We study the Hilbert spaces ${\mathcal{H}}_{\tau ,\sigma }$ corresponding to the quantization of $T^{\ast }G$ with respect to these non-invariant K"ahler structures. On the other hand, by taking the vertical Schr"odinger polarization as a starting point, the above $G\times T$-invariant Hamiltonian flows also generate families of mixed polarizations $$\mathcal{P}$_{0,\sigma}, \sigma \in {\mathbb C}, {\rm Im}(\sigma) >0$. 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 K"ahler structure on the leaves of a foliation of $T^{\ast }G$. The geometric quantization of $T^{\ast }G$ with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [KMN1,KMN2].