Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

The aim here is to describe all isomorphism classes of SU(n+1)-equivariant Hermitian holomorphic vector bundles on the complex projective space CPn. If G⊂SU(n+1) is the isotropy subgroup of a chosen point x0∈CPn, and ρ:G⟶GL(V) is a unitary representation, we obtain SU(n+1)-equivariant holomorphic Hermitian vector bundles on CPn. Next, given any v∈End(Vρ)⊗(Tz00,1CPn)∗ satisfying certain conditions, a new structure of an SU(n+1)-equivariant holomorphic Hermitian vector bundle on this underlying C∞ holomorphic Hermitian bundle is obtained. It is shown that all SU(n+1)-equivariant holomorphic Hermitian vector bundles on CPn arise this way.