What are the representations of $U(n)$ that induce (see [link text](https://mathoverflow.net/questions/5772/principal-bundles-representations-and-vector-bundles)) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known fact that $\mathbb{CP}^n = SU(n+1)/U(n)$). 


Also, what are the representations that induce the line bundles?



Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space
The group U(n-1) has an abelian factor U(1) and a semisimple factor SU(n-1). 


I'll answer first the second part of the question: Since line bundles are induced from one dimensional character representations, and the semismple component has no non-trivial character representations, thus the line bundles are induced from character representations of the U(1) component. These representations are indexed by a single integer. 


The answer to the first part of the question is that the bundle of holomorphic one forms is induced from the representation of U(n-1) which has a unit central charge in the U(1) part and equal to the contragredient fundamental n-1 dimensional representation of the SU(n-1) component. One of the ways to see that is say from the explicit expression of the U(n) action on CP(n-1) in the affine coordinates realized as a rational transformation. When restricted to U(n-1), one can see that the action of the U(1) is through its unit central charge representation and the action of SU(n-1) a multiplication of the affine coordinate vector by the reciprocal of the group element in the fundamental representation.


For higher holomorphic form bundles, the bundle is induced from the (irreducible) antisymmetric powers of the contragredient representations of SU(n-1) accomanied by the representation with the corresponding multiple central charge of U(1).


