# 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.

## Key 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].

---
Source: https://tomesphere.com/paper/1907.05232