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

TL;DR

This paper explores complex-time Hamiltonian flows on cotangent bundles of compact Lie groups, leading to new invariant Kähler structures and partial coherent state transforms in geometric quantization.

Contribution

It introduces a method to generate non-invariant Kähler structures and associated partial coherent state transforms via complex-time Hamiltonian flows.

Findings

Constructed new $G\times T$-invariant Kähler structures on $T^*G$.

Developed families of mixed polarizations leading to partial coherent state transforms.

Established the connection between these structures and KSH maps in geometric quantization.

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].