Learning Roadmap for Borel - Weil - Bott Theorem

The two, broadly defined things you need to know are Lie theory and (complex) differential geometry. The specific things from each topic are

Lie groups/algebras

Highest weight theory of compact Lie groups/complex semisimple Lie groups. One needs to build up to the theorem that irreducible complex representations of a Lie algebra are parametrized by dominant positive weights.

Most books (including Brain Hall's) build up to this. I personally like Compact Lie Groups by Sepanski if one already understand the basics of manifolds, as its very concise, has good exercises, and concludes with a proof of the Borel-Weil theorem.

Differential geometry

One needs to know what a holomorphic vector bundle over a complex manifold is and their Dolbeault cohomology. Thus the following general differential geometry background is needed, all of which the last can be found in Lee's Smooth Manifolds

• Vector bundles over a manifold (I think Lee mostly talks about real vector bundles but complex ones are just replacing $\mathbb R$ with $\mathbb C$).

• Differential forms.

• de Rham theory.
• Connections on vector bundles (maybe not necessary but I think it's useful).

After having a solid differential geometry background, in my opinion the best place to learn the necessary complex geometry is part 1 of the freely available notes by Moroianu, titled Lectures on Kahler Geometry.

For the Borel-Weil-Bott theorem proper, I would see Sepanski's text and also the paper Representations in Dolbeault Cohomology by Zierau.