紧李群的部分相干态变换、$G \times T$不变Kähler结构和余切丛的几何量子化
Partial coherent state transforms, $G \times T$-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups
摘要：
本文研究了具有极大环面$T$的紧致连通李群$G$的余切丛上的某些$G\times T$ -不变函数的Hamilton流到复时的解析延拓.也就是说，我们将取一个$G\times G$不变函数$h$和一个$G\times T$不变函数$f$的哈密顿流.在$G\times G$不变的Kähler结构上作用这些复时间Hamilton流，得到了新的$G\times T$不变的，但不是$G\times G$不变的，$T^*G$上的Kähler结构.我们研究了Hilbert空间${\mathcal H}_{τ,σ}$对应的量子化的$T^*G$关于这些非不变的Kähler结构。另一方面，以垂直薛定谔极化为出发点，上述$G\times T$不变哈密顿流也产生混合极化族$\mathcal{P}_{0,σ}, σ\in {\mathbb C}, {\rm Im}(σ) >0$ .这些混合极化的每一个都是由一个可积的实分布和一个复分布的直和全局给出的，复分布定义了$T^*G$叶理叶上的凯勒结构。相对于这些混合偏振的$T^*G$的几何量子化产生酉部分相干态变换，对应于如在[KMN 1，KMN 2]中定义的KSH映射。
英文摘要：
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].
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