Partial coherent state transforms, \(G\times T\)-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups (Q2180896)

Fix a compact Lie group \(G\) with a maximal torus \(T\). The paper under review studies Kähler structures on the cotangent bundle \(T^{\ast }G\) with its canonical symplectic \(2\)-form which are not \(G\times G\)-invariant. For this purpose, \(G\times T\)-invariant Hamiltonian flows analytically continued to complex time are considered. Hence, examples of \(G\times T\) but not \(G\times G\)-invariant Kähler structures on \(T^{\ast }G\) are obtained and the geometric quantization of \(T^{\ast }G\) with respect to the associated Kähler polarizations is described. Recall that a polarization of \(T^{\ast }G\), in the sense of geometric quantization, is an involutive Lagrangian distribution in the complexified tangent bundle \(T(T^{\ast }G)\otimes \mathbb{C}\). If \(\pi : T^{\ast }G\rightarrow G\) is the canonical projection, then the vertical distribution \(\operatorname{Ker} D\pi \) is such a polarization.