A Global View of Equivariant Vector Bundles and Dirac Operators on Some Compact Homogeneous Spaces

In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries (“quantizations”) that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and arguments. Our approach is quite algebraic, using primarily the modules of cross-sections of vector bundles. We carry the development through the construction of Hodge–Dirac operators. The inducing construction is ubiquitous.