S ep 2 01 9 PARTIAL COHERENT STATE TRANSFORMS , G × 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 × 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 × G-invariant function, h, and one G × T -invariant function, f . Acting with these complex time Hamiltonian flows on G × G-invariant Kähler structures gives new G×T -invariant, but not G×G-invariant, Kähler structures on T G. We study the Hilbert spaces Hτ,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 × T -invariant Hamiltonian flows also generate families of mixed polarizations P0,σ, σ ∈ C, 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].