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 C P n . If G ⊂ SU ( n + 1 ) is the isotropy subgroup of a chosen point x 0 ∈ C P n , and ρ : G ⟶ GL ( V ) is a unitary representation, we obtain SU ( n + 1 ) -equivariant holomorphic Hermitian vector bundles on C P n . Next, given any v ∈ End ( V ρ ) ⊗ ( T z 0 0 , 1 C P n ) ∗ 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 C P n arise this way.