Partial coherent state transforms, $G \times T$-invariant Kähler 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 Kähler structures gives new $G\times T$-invariant, but not $G\times G$-invariant, Kähler structures on $T^*G$. We study the Hilbert spaces ${\mathcal H}_{τ,σ}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kähler structures. On the other hand, by taking the vertical Schrödinger polarization as a starting point, the above $G\times T$-invariant Hamiltonian flows also generate families of mixed polarizations $\mathcal{P}_{0,σ}, σ\in {\mathbb C}, {\rm Im}(σ) >0$. 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 Kähler 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 [KMN1,KMN2].