Partial coherent state transforms, $G \times T$-invariant K\"ahler structures and geometric quantization of cotangent bundles of compact Lie groups

Abstract

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 $T^*G$. We study the Hilbert spaces ${\mathcal H}_{\tau,\sigma}$ corresponding to the quantization of $T^*G$ 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 $T^*G$. 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].