Propagation of Multiplicity-Freeness Property for Holomorphic Vector Bundles
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite- and infinite-dimensional cases in a systematic and synthetic manner. The key geometric condition in our theorem is an orbit-preserving anti-holomorphic diffeomorphism on the base space, which brings us to the concept of visible actions on complex manifolds.