Adapted complex structures and geometric quantization

Abstract

A compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure $J_{A}$ is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure $J_S$ is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of $J_{A}$ under an appropriate family of diffeomorphisms exists and agrees with $J_S$.